<?xml version="1.0" standalone="yes"?>
<Paper uid="P99-1039">
  <Title>Alternating Quantifier Scope in CCG*</Title>
  <Section position="4" start_page="303" end_page="305" type="metho">
    <SectionTitle>
3 Scope Alternation and Skolem Entities
</SectionTitle>
    <Paragraph position="0"> If indefinites can be assumed to have a referential translation as an arbitrary object, rather than a meaning related to a traditional existential generalized quantifier, then other supposed quantifiers, such as some/a few/two saxophonists may also be better analyzed as referential categories.</Paragraph>
    <Paragraph position="1"> We will begin by assuming that some is not a quantifier, but rather a determiner of a (singular) arbitrary object. It therefore has the following pair of subject and complement categories:  (10) a. some := (T/(T\NP))/N:~p.7~7.q(arb'p) b. some := (T\(T/NP))/N: ~,pS~q.q(arb'p)  In this pair of categories, the constant arb' is the function identified earlier from properties p to entities of type e with that property, such that those entities are functionally related to any universally quantified NPs that have scope over them at the level of logical form. If arblp is not in the extent of any universal quantifier, then it yields a unique arbitrary constant individual.</Paragraph>
    <Paragraph position="2"> We will assume that every has at least the gen- null eralized quantifier determiner given at (3), repeated here: (11) a. every := (T/(T\NP))/N :</Paragraph>
    <Paragraph position="4"> These assumptions, as in Park's related account, provide everything we need to account for all and only the readings that are actually available for the Geach sentence (5b), repeated here: (12) Every boy admires, and every girl detests, some saxophonist.</Paragraph>
    <Paragraph position="5"> The &amp;quot;narrow-scope saxophonist&amp;quot; reading of this sentence results from the (backward) referential category (10b) applying to the translation of Every boy admires and every girl detests of type S/NP (whose derivation is taken as read), as in (13). Crucially, if we evaluate the latter logical form with respect to a database after this reduction, as indicated by the dotted underline, for each boy and girl that we examine and test for the property of admiring/detesting an arbitrary saxophonist, we will find (or in the sense of Lewis (1979) &amp;quot;accommodate&amp;quot; or add to our database) a potentially different individual, dependent via the Skolem functors sk(~ and sk~r2 upon that boy or girl. Each conjunct thereby gives the appearance of including a variable bound by an existential within the scope of the universal.</Paragraph>
    <Paragraph position="6"> The &amp;quot;wide-scope saxophonist&amp;quot; reading arises from the same categories as follows. If Skolemization can act after reduction of the object, when the arbitrary object is within the scope of the universal, then it can also act before, when it is not in scope, to yield a Skolem constant, as in (14). Since the resultant logical form is in all important respects model-theoretically equivalent to the one that would arise from a wide scope existential quantification, we can entirely eliminate the quantifier reading (4) for some, and regard it as bearing only the arbitrary object reading (10). 5  gories do not yield a reading in which the boys admire the same wide scope saxophonist but the girls detest possibly different ones* Nor do they yield one in which the girls also all detest the same saxophonist, but not necessarily the one the boys admire* Both facts are necessary consequences of the monotonic nature of CCG as a theory of grammar, without any further assumptions of parallelism conditions* null In the case of the following scope-inverting relative of the Geach example, the outcome is subtly different* (15) Some woman likes and some man detests every saxophonist* The scope-inverting reading arises from the evaluation of the arbitrary woman and man after combination with every saxophonist, within the scope of the universal:</Paragraph>
    <Paragraph position="8"> The reading where some woman and some man appear to have wider scope than every saxophonist arises from evaluation of (the interpretation of) the residue of right node raising, some woman likes and some man detests, before combination with the gen- null eralized quantifier every saxophonist. This results in ' and sk~nan liking two Skolem constants, say skwoma n every saxophonist, again without the involvement of a true existential quantifier: (17) Vx.saxophonist' x --+  and' (likes'x skrwo,nan)(detests' x sk~nan ) These readings are obviously correct. However, row scope versions of the existential donkey in (6). since Skolemization of the arbitrary man and woman has so far been assumed to be free to occur any time, it seems to be predicted that one arbitrary object might become a Skolem constant in advance of reduction with the object, while the other might do so after. This would give rise to further readings in which only one of some man or some woman takes wide scope--for example: 6</Paragraph>
    <Paragraph position="10"> and' ( likes' x SUwoma n ) (detestS' x( Sk~nanx ) ) Steedman (1991) shows on the basis of possible accompanying intonation contours that the coordinate fragments like Some woman likes and some man detests that result from right node raising are identical with information structural units of utterances--usually, the &amp;quot;theme.&amp;quot; In the present framework, readings like (18) can therefore be eliminated without parallelism constraints, by the further assumption that Skolemization/binding of arbitrary objects can only be done over complete information structural units--that is, entire themes, rhemes, or utterances. When any such unit is resolved in this way, all arbitrary objects concerned are obligatorily bound. 7 While this account of indefinites might appear to mix derivation and evaluation in a dangerous way, this is in fact what we would expect from a mono~I'he non-availability of such readings has also been used to argue for parallelism constraints. Quite apart from the theoretically problematic nature of such constraints, they must be rather carefully formulated if they are not to exclude perfectly legal conjunction of narrow scope existentials with explicitly referential NPs, as in the following: (i) Some woman likes, and Fred detests, every saxophonist. 71 am grateful to Gann Bierner for pointing me towards this solution.</Paragraph>
    <Paragraph position="11">  tonic semantics that supports the use of incremental semantic interpretation to guide parsing, as humans appear to (see below).</Paragraph>
    <Paragraph position="12"> Further support for a non-quantificational analysis of indefinites can be obtained from the observation that certain nominals that have been talked of as quantifiers entirely fail to exhibit scope alternations of the kind just discussed. One important class is the &amp;quot;non-specific&amp;quot; or &amp;quot;non-group-denoting counting&amp;quot; quantifiers, including the upward-monotone, downward-monotone, and non-monotone quantitiers (Barwise and Cooper 1981) such as at least three, few, exactly five and at most two in examples like the following, which are of a kind discussed by Liu (1990), Stabler (1997), and Beghelli and Stowell (1997): (19) a. Some linguist can program in at most two programming languages.</Paragraph>
    <Paragraph position="13"> b. Most linguists speak at least three /few/exactly five languages.</Paragraph>
    <Paragraph position="14"> In contrast to true quantifiers like most and every, these quantified NP objects appear not to be able to invert or take wide scope over their subjects. That is, unlike some linguist can program in every programming language which has a scope-inverting reading meaning that every programming language is known by some linguist, (19a) has no reading meaning that there are at most two programming languages that are known to any linguist, and (19b) cannot mean that there are at least three/few/exactly five languages, languages that most linguists speak.</Paragraph>
    <Paragraph position="15"> Beghelli and Stowell (1997) account for this behavior in terms of different &amp;quot;landing sites&amp;quot; (or in GB terms &amp;quot;functional projections&amp;quot;) at the level of LF for the different types of quantifier. However, another alternative is to believe that in syntactic terms these noun-phrases have the same category as any other but in semantic terms they are (plural) arbitrary objects rather than quantifiers, like some, a few, six and the like. This in turn means that they cannot engender dependency in the arbitrary object arising from some linguist in (19a). As a result the sentence has a single meaning, to the effect that there is an arbitrary linguist who can program in at most two programming languages.</Paragraph>
  </Section>
  <Section position="5" start_page="305" end_page="306" type="metho">
    <SectionTitle>
4 Computing Available Readings
</SectionTitle>
    <Paragraph position="0"> We may assume (at least for English) that even the non-standard constituents created by function composition in CCG cannot increase the number of quantifiable arguments for an operator beyond the limit of three or so imposed by the lexicon. It follows that the observation of Park (1995, 1996) that only quantified arguments of a single (possibly composed) function can freely alternate scope places an upper bound on the number of readings.</Paragraph>
    <Paragraph position="1"> The logical form of an n-quantifier sentence is a term with an operator of valency 1, 2 or 3, whose argument(s) must either be quantified expressions or terms with an operator of valency 1, 2 or 3, and so on. The number of readings for an n quantifier sentence is therefore bounded by the number of nodes in a single spanning tree with a branching factor b of up to three and n leaves. This number is given by a polynomial whose dominating term is b tdeggb'that is, it is linear in n, albeit with a rather large constant (since nodes correspond up to 3! = 6 readings). For the relatively small n that we in practice need to cope with, this is still a lot of readings in the worst case.</Paragraph>
    <Paragraph position="2"> However, the actual number of readings for real sentences will be very much lower, since it depends on how many true quantifiers are involved, and in exactly what configuration they occur. For example, the following three-quantifier sentence is predicted to have not 3 ! = 6 but only 4 distinct readings, since the non-quantifiers exactly three girls and some book cannot alternate scope with each other independently of the truly quantificational dependency-inducing Every boy.</Paragraph>
    <Paragraph position="3"> (20) Every boy gave exactly three girls some book~ This is an important saving for the parser, as redundant analyses can be eliminated on the basis of identity of logical forms, a standard method of eliminating such &amp;quot;spurious ambiguities.&amp;quot; Similarly, as well as the restrictions that we have seen introduced by coordination, the SVO grammar of English means (for reasons discussed in Steedman 1996) that embedded subjects in English are correctly predicted neither to extract nor take scope over their matrix subject in examples like the following: null (21) a. *a boy who(m) I know that admires John Coltrane b. Somebody knows that every boy admires some saxophonist.</Paragraph>
    <Paragraph position="4"> As Cooper 1983 points out, the latter has no readings where every boy takes scope over somebody.</Paragraph>
    <Paragraph position="5"> This three-quantifier sentence therefore has not 3 ! = 6, not 2! * 2! = 4, but only 2! * 1 = 2 readings.</Paragraph>
    <Paragraph position="6"> Bayer (1996) and Kayne (1998) have noted related  restrictions on scope alternation that would otherwise be allowed for arguments that are marooned in mid verb-group in German. Since such embeddings are crucial to obtaining proliferating readings, it is likely that in practice the number of available readings is usually quite small.</Paragraph>
    <Paragraph position="7"> It is interesting to speculate finally on the relation of the above account of the available scope readings with proposals to minimize search during processing by building &amp;quot;underspecified&amp;quot; logical forms by Reyle (1992), and others cited in Willis and Manandhar (1999). There is a sense in which arbitrary individuals are themselves under-specified quantitiers, which are disambiguated by Skolemization. However, under the present proposal, they are disambiguated during the derivation itself.</Paragraph>
    <Paragraph position="8"> The alternative of building a single under-specified logical form can under some circumstances dramatically reduce the search space and increase efficiency of parsing--for example with distributive expressions in sentences like Six girls ate .five pizzas, which are probably intrinsically unspecified. However, few studies of this kind have looked at the problems posed by the restrictions on available readings exhibited by sentences like (5b).</Paragraph>
    <Paragraph position="9"> The extent to which inference can be done with the under-specified representations themselves for the quantifier alternations in question (as opposed to distributives) is likely to be very limited. If they are to be disambiguated efficiently, then the disambiguated representations must embody or include those restrictions. However, the restriction that Geach noted seems intrinsically disjunctive, and hence appears to threaten efficiency in both parsing with, and disambiguation of, under-specified representations. null The fact that relatively few readings are available and that they are so tightly related to surface structure and derivation means that the technique of incremental semantic or probabilistic disambiguation of fully specified partial logical forms mentioned earlier may be a more efficient technique for computing the contextually relevant readings. For example, in processing (22) (adapted from Hobbs and Shieber 1987), which Park 1995 claims to have only four readings, rather than the five predicted by their account, such a system can build both readings for the S/NP every representative of three companies saw and decide which is more likely, before building both compatible readings of the whole sentence and similarly resolving with respect to statistical or contextual support: (22) Every representative of three companies saw some sample.</Paragraph>
  </Section>
class="xml-element"></Paper>