Alternating Quantifier Scope in CCG* 
Mark Steedman 
Division of Informatics, 
University of Edinburgh, 
2 Buccleuch Place, 
Edinburgh EH8 9LW, UK 
steedman@cogsc i. ed. ac. uk 
Abstract 
The paper shows that movement or equivalent 
computational structure-changing operations of any 
kind at the level of logical form can be dispensed 
with entirely in capturing quantifer scope ambi- 
guity. It offers a new semantics whereby the ef- 
fects of quantifier scope alternation can be obtained 
by an entirely monotonic derivation, without type- 
changing rules. The paper follows Fodor (1982), 
Fodor and Sag (1982), and Park (1995, 1996) in 
viewing many apparent scope ambiguities as arising 
from referential categories rather than true general- 
ized quantitiers. 
1 Introduction 
It is standard to assume that the ambiguity of sen- 
tences like (1) is to be accounted for by assigning 
two logical forms which differ in the scopes as- 
signed to these quantifiers, as in (2a,b): 1 
(1) Every boy admires some saxophonist. 
(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. 
This process has on occasion been explained 
in terms of "quantifier movement" 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 Ja- 
cobson, 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. 
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 "Curried". A convention of "left associativi- 
ty" is assumed, so that admires'yx is equivalent to (admires'y)x. 
equivalent computational operations of "quantify- 
ing in" or "storage" 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 trans- 
parency between syntax and semantics from which 
the monotonic theories begin. Given the assump- 
tions of syntactic/semantic transparency and mono- 
tonicity that are usual in the Frege-Montague tra- 
dition, 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 deriva- 
tion via type-changing operations; or enriching the 
lexicon and the semantic ontology. 
It is standard in the Frege-Montague tradition to 
begin by translating expressions like "every boy" 
and "some saxophonist" into "generalized quanti- 
tiers" in effect exchanging the roles of arguments 
like NPs and functors like verbs by a process of 
"type-raising" the former. In terms of the notation 
and assumptions of Combinatory Categorial Gram- 
mar (CCG, Steedman 1996) the standard way to in- 
corporate generalized quantifiers into the semantics 
of CG deterbainers is to transfer type-raising to the 
lexicon, assig~g the following categories to deter- 
miners like every and some, making them functions 
from nouns to "type-raised" noun-phrases, where 
the latter are simply the syntactic types correspond- 
ing 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 
301 
(T is a variable over categories unique to each in- 
dividual 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 stan- 
dard in pure Categorial Grammar, the further in- 
clusion of type-raised arguments engenders deriva- 
tions in which objects command subjects, as well as 
more traditional ones in which the reverse is true. 
Given the categories in (3) and (4), these alterna- 
tive derivations will deliver the two distinct logi- 
cal forms shown in (2), entirely monotonically and 
without involving structure-changing operations. 
However, linking derivation and scope as simply 
and directly as this makes the obviously false pre- 
diction that in sentences where there is no ambi- 
guity of CCG derivation there should be no scope 
ambiguity. In particular, object topicalization and 
object right node raising are derivationally unam- 
biguous in the relevant respects, and force the dis- 
placed 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. 
b. Every boy admires, and every girl detests, 
some saxophonist. 
Both sentences have a narrow scope reading in 
which every individual has some attitude towards 
some saxophonist, but not necessarily the same sax- 
ophonist. This observation appears to imply that 
even the relatively free notion of derivation provided 
by CCG is still too restricted to explain all ambigu- 
ities arising from multiple quantifiers. 
Nevertheless, the idea that semantic quantifier 
scope is limited by syntactic derivational scope has 
some very attractive features. For example, it imme- 
diately explains why scope alternation is both un- 
bounded 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 re- 
stricted 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 saxo- 
phonists, it does not have a reading where the sax- 
ophonist 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. 
There does not even seem to be a reading involving 
separate wide-scope saxophonists respectively tak- 
ing scope over boys and girls--for example where 
the boys all admire Coltrane and the girls all detest 
Lester Young. 
These observations are very hard to reconcile 
with semantic theories that invoke powerful mech- 
anisms like abstraction or "Quantifying In" and its 
relatives, or "Quantifier Movement." 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 saxo- 
phonist should have to take the same scope in both 
conjuncts. The same applies if quantifiers are gener- 
ated 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 gram- 
mar 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 heuris- 
tics based on (partial) orderings on the operations 
concerned, and raising questions about complete- 
ness and practical parsability. All of these ques- 
tions have been addressed by Hendriks and others, 
but the result has been to dramatically raise the ratio 
of mathematical proofs to sentences analyzed. 
It seems worth exploring an alternative response 
to these observations concerning interactions of sur- 
2Such observations have been countered by the invocation 
of a "parallelism condition" on coordinate sentences, a rule of 
a very expressively powerful "transderivational" kind that one 
would otherwise wish to avoid. 
3For example, in order to obtain the narrow scope object 
reading for sentence (5b), Hendriks (1993), subjects the cate- 
gory of the transitive verb to "argument lifting" to make it a 
function over a type-raised object type, and the coordination 
rule must be correspondingly semantically generalized. 
302 
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. 
For example, in order to capture the narrow-scope 
object reading for Geach's right node raised sen- 
tence (5b), in whose CCG derivation the object must 
command everything else, the present paper fol- 
lows Park in assuming that the narrow scope read- 
ing arises from a non-quantificational interpretation 
of some scecophonist, one which gives rise to a read- 
ing 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. 
The claim that many noun-phrases which have 
been assumed to have a single generalized quan- 
tifier 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 dis- 
tinctions in scope of traditional quantifiers. There 
is widespread recognition that many such distinc- 
tions arise instead from a rich ontology of different 
types of (collective, distributive, intensional, group- 
denoting, arbitrary, etc.) individual to which nom- 
inal expressions refer. (See for example Webber 
1978, Barwise and Perry 1980, Fodor and Sag 1982, 
Fodor 1982, Fine 1985, and papers in the recent col- 
lection 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 "arbi- 
trary objects" (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 re- 
semble the Skolem terms that are generated by in- 
ference rules like Existential Elimination in proof 
theories of first-order predicate calculus. 
The rest of the paper will argue that arbitrary ob- 
jects 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 al- 
ternation is not available), but also certain cases of 
anomalous scopal binding which are unexplained 
under any of the alternatives discussed so far. 
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/. 
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 of- 
fered 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/de- 
serves a subsidy 
One popular reaction to this observation has been 
to try to generalize the notion of scope, as in Dy- 
namic 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/. 
I shall assume for the sake of argument that "a 
donkey" 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 quan- 
tifier 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 pro- 
noun is in the scope of the quantifiers that bind any 
variables in the Skolem term, it will include a vari- 
able that is outside the scope of its binder, and fail 
to refer. 
This analysis is similar to but distinct from 
the analyses of Cooper (1979) and Heim (1990), 
41 assume that arb p "knows" what scopes it is in by the same 
mechanism whereby a bound variable pronoun "knows" about 
its binder. Whatever this mechanism is, it does not have the 
power of movement, abstraction, or storage. An arbitrary ob- 
ject is deterministically bound to all scoping universals. 
303 
who assume that a donkey translates as a quanti- 
fied expression, and that the entire subject every 
farmer who owns a donkey establishes a contextu- 
ally 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 stan- 
dard donkey sentence and know that farmers may 
own more than one donkey, we will probably in- 
fer 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 sen- 
tence (attributed to Jeff Pelletier), and the knowl- 
edge that some people have more than one dime in 
their pocket. 
(9) Everyone who had a dime in their pocket put 
it in the parking meter. 
The reason is that we know that the reason for 
putting a dime into a parking meter, unlike the rea- 
son 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 Compu- 
tational 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 un- 
published work by Kratzer. Skolem functors are 
closely related to, but distinct from, "Choice Func- 
tions" (see Reinhart 1997, Winter 1997, Sauerland 
1998, and Schlenker 1998 for discussion. Webber's 
1978 analysis is essentially a choice functional anal- 
ysis, as is Fine's.) 
3 Scope Alternation and Skolem Entities 
If indefinites can be assumed to have a referen- 
tial translation as an arbitrary object, rather than a 
meaning related to a traditional existential gener- 
alized quantifier, then other supposed quantifiers, 
such as some/a few/two saxophonists may also be 
better analyzed as referential categories. 
We will begin by assuming that some is not a 
quantifier, but rather a determiner of a (singular) ar- 
bitrary 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 en- 
tities 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. 
We will assume that every has at least the gen- 
eralized quantifier determiner given at (3), repeated 
here: 
(11) a. every := (T/(T\NP))/N : 
LpSkq.Vx.px -+ qx 
b. every := (T\(T/NP))/N: 
 p. .Vx.px qx 
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. 
The "narrow-scope saxophonist" reading of this 
sentence results from the (backward) referential cat- 
egory (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 dot- 
ted underline, for each boy and girl that we exam- 
ine and test for the property of admiring/detesting 
an arbitrary saxophonist, we will find (or in the 
sense of Lewis (1979) "accommodate" or add to our 
database) a potentially different individual, depen- 
dent 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 ex- 
istential within the scope of the universal. 
The "wide-scope saxophonist" reading arises 
from the same categories as follows. If Skolem- 
ization can act after reduction of the object, when 
the arbitrary object is within the scope of the uni- 
versal, 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 
5Similar considerations give rise to apparent wide and nar- 
304 
(\]3) 
(14) 
Every boy admires and every girl detests some saxophonist 
S/NP S\(S/NP) 
• Lr.and'(Vy.boy'y --+ admires'xy)(Vz.girl'z --+ detests'xz) • kq.q(arb'sd) 
S: and' (Vy.boy'y -+ admires' ( arb' sax~)y) (Vz.girl' z -+ detests' ( arb' sd )z~ 
S " and' (Vy.boy'y --+ admires' (sk~ax, y)y) (Vz.girl' z --+ detests' (sk~,tr 2 z) z) 
Every boy admires and every girl detests 
• Lx.and' (Vy.boy'y --+ admires xy) (Vz.girl'z --~ detests'xz) 
some saxophonist 
S\(S/NP) 
: 2~t.q( arb' sax I) 
• ; • 
< 
S : and' (Vy.boy'y --+ admires' sk~,vcy ) (Vz•girl'z --+ detests' sk~axZ ) 
Consistent with Geach's observation, these cate- 
gories do not yield a reading in which the boys ad- 
mire 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 sax- 
ophonist, but not necessarily the one the boys ad- 
mire• Both facts are necessary consequences of the 
monotonic nature of CCG as a theory of grammar, 
without any further assumptions of parallelism con- 
ditions• 
In the case of the following scope-inverting rel- 
ative of the Geach example, the outcome is subtly 
different• 
(15) Some woman likes and some man detests ev- 
ery saxophonist• 
The scope-inverting reading arises from the evalua- 
tion of the arbitrary woman and man after combina- 
tion with every saxophonist, within the scope of the 
universal: 
(16) Vx•saxophonist' x --+ 
/ / / / / and (likes x(skwomanX) )(detests x(skmanX) ) 
The reading where some woman and some man ap- 
pear 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- 
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 read- 
ings in which only one of some man or some woman 
takes wide scope--for example: 6 
(18) Vx.saxophonist' x --+ 
and' ( likes' x SUwoma n ) (detestS' x( Sk~nanx ) ) 
Steedman (1991) shows on the basis of pos- 
sible accompanying intonation contours that the 
coordinate fragments like Some woman likes and 
some man detests that result from right node rais- 
ing are identical with information structural units 
of utterances--usually, the "theme." In the present 
framework, readings like (18) can therefore be elim- 
inated 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 the- 
oretically 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. 
305 
tonic semantics that supports the use of incremental 
semantic interpretation to guide parsing, as humans 
appear to (see below). 
Further support for a non-quantificational analy- 
sis of indefinites can be obtained from the observa- 
tion that certain nominals that have been talked of 
as quantifiers entirely fail to exhibit scope alterna- 
tions of the kind just discussed. One important class 
is the "non-specific" or "non-group-denoting count- 
ing" quantifiers, including the upward-monotone, 
downward-monotone, and non-monotone quanti- 
tiers (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 Stow- 
ell (1997): 
(19) a. Some linguist can program in at most two 
programming languages. 
b. Most linguists speak at least three 
/few/exactly five languages. 
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 program- 
ming language which has a scope-inverting read- 
ing meaning that every programming language is 
known by some linguist, (19a) has no reading mean- 
ing that there are at most two programming lan- 
guages 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. 
Beghelli and Stowell (1997) account for this be- 
havior in terms of different "landing sites" (or in GB 
terms "functional projections") 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 ob- 
jects rather than quantifiers, like some, a few, six and 
the like. This in turn means that they cannot engen- 
der 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 program- 
ming languages. 
4 Computing Available Readings 
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 (possi- 
bly composed) function can freely alternate scope 
places an upper bound on the number of readings. 
The logical form of an n-quantifier sentence is a 
term with an operator of valency 1, 2 or 3, whose ar- 
gument(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 sen- 
tence 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 t°gb'- 
that is, it is linear in n, albeit with a rather large 
constant (since nodes correspond up to 3! = 6 read- 
ings). For the relatively small n that we in practice 
need to cope with, this is still a lot of readings in the 
worst case. 
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 inde- 
pendently of the truly quantificational dependency- 
inducing Every boy. 
(20) Every boy gave exactly three girls some book~ 
This is an important saving for the parser, as redun- 
dant analyses can be eliminated on the basis of iden- 
tity of logical forms, a standard method of eliminat- 
ing such "spurious ambiguities." 
Similarly, as well as the restrictions that we have 
seen introduced by coordination, the SVO grammar 
of English means (for reasons discussed in Steed- 
man 1996) that embedded subjects in English are 
correctly predicted neither to extract nor take scope 
over their matrix subject in examples like the fol- 
lowing: 
(21) a. *a boy who(m) I know that admires John 
Coltrane 
b. Somebody knows that every boy admires 
some saxophonist. 
As Cooper 1983 points out, the latter has no read- 
ings where every boy takes scope over somebody. 
This three-quantifier sentence therefore has not 3 ! = 
6, not 2! • 2! = 4, but only 2! • 1 = 2 readings. 
Bayer (1996) and Kayne (1998) have noted related 
306 
restrictions on scope alternation that would other- 
wise 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 read- 
ings is usually quite small. 
It is interesting to speculate finally on the relation 
of the above account of the available scope readings 
with proposals to minimize search during process- 
ing by building "underspecified" logical forms by 
Reyle (1992), and others cited in Willis and Man- 
andhar (1999). There is a sense in which arbitrary 
individuals are themselves under-specified quanti- 
tiers, which are disambiguated by Skolemization. 
However, under the present proposal, they are dis- 
ambiguated during the derivation itself. 
The alternative of building a single under- 
specified logical form can under some circum- 
stances 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 un- 
specified. However, few studies of this kind have 
looked at the problems posed by the restrictions on 
available readings exhibited by sentences like (5b). 
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 disam- 
biguated 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 repre- 
sentations. 
The fact that relatively few readings are available 
and that they are so tightly related to surface struc- 
ture and derivation means that the technique of in- 
cremental semantic or probabilistic disambiguation 
of fully specified partial logical forms mentioned 
earlier may be a more efficient technique for com- 
puting the contextually relevant readings. For ex- 
ample, 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 build- 
ing 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. 
5 Conclusion 
The above observations imply that only those so- 
called quantifiers in English which can engender 
dependency-inducing scope inversion have interpre- 
tations corresponding to genuine quantifiers. The 
others are not quantificationai at all, but are various 
types of arbitrary individuals translated as Skolem 
terms. These give the appearance of taking nar- 
row scope when they are bound to truly quantified 
variables, and of taking wide scope when they are 
unbound, and therefore "take scope everywhere." 
Available readings can be computed monotonically 
from syntactic derivation alone. The notion of syn- 
tactic derivation embodied in CCG is the most pow- 
erful limitation on the number of available read- 
ings, and allows all logical-form level constraints 
on scope orderings to be dispensed with, a result 
related to, but more powerful than, that of Pereira 
(1990). 

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