Quantifier Scope, Lexical Semantics, and Surface Structure Constituency

University of Pennsylvania ScholarlyCommons

IRCS Technical Reports Series Institute for Research in Cognitive Science

December 1996

Jong Cheol Park University of Pennsylvania

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Park, Jong Cheol, "Quantifier Scope, Lexical Semantics, and Surface Structure Constituency" (1996). IRCS Technical Reports Series. 106. https://repository.upenn.edu/ircs_reports/106

University of Pennsylvania Institute for Research in Cognitive Science Technical Report No. IRCS-96-28.

This paper is posted at ScholarlyCommons. https://repository.upenn.edu/ircs_reports/106 For more information, please contact [email protected]. Quantifier Scope, Lexical Semantics, and Surface Structure Constituency

Abstract We present a novel conjecture concerning the scope ambiguities that arise in sentences including multiple nonreferential quantifiers. eW claim that many existing theories of the phenomenon fail to correctly limit the set of readings that such sentences engender by failing to distinguish between referential and non-referential quantifiers. Once the distinction is correctly drawn, we show that surface syntax can be made, via an extended notion of surface constituency, to identify the set of available differently-scoped readings for such sentences. We examine various English constructions to show that the scopings predicted by the conjecture are the only ones that are available to human language understanders. We show how to incorporate this conjecture into a theory of quantifier scope, yb couching it in a unification-based Combinatory Categorial Grammar framework and implementing it in SICStus Prolog. Finally, we compare the proposal with related approaches to quantifier scope ambiguity.

Comments University of Pennsylvania Institute for Research in Cognitive Science Technical Report No. IRCS-96-28.

This technical report is available at ScholarlyCommons: https://repository.upenn.edu/ircs_reports/106 , Institute for Research in Cognitive Science

Quantiﬁer Scope, Lexical Seman- tics, and Surface Structure Constituency

Jong Cheol Park

University of Pennsylvania 3401 Walnut Street, Suite 400A Philadelphia, PA 19104-6228

December 1996

Site of the NSF Science and Technology Center for Research in Cognitive Science

IRCS Report 96--28

Quantier Scop e Lexical Semantics

and Surface Structure Constituency

Jong C Park

UniversityofPennsylvania

We present a novel conjectureconcerning the scope ambiguities that arise in sentences

including multiple nonreferential quantiers We claim that many existing theories of

the phenomenon fail to correctly limit the set of readings that such sentences engender

by failing to distinguish between referential and nonreferential quantiers Once the dis

tinction is correctly drawn we show that surfacesyntaxcan be made via an extended

notion of surfaceconstituency to identify the set of available dierentlyscopedreadings

for such sentences We examine various English constructions to show that the scop

ings predicted by the conjectureare the only ones that are available to human language

understanders We show how to incorporate this conjecture into a theory of quantier

scope by couching it in a unicationbased Combinatory Categorial Grammar framework

and implementing it in SICStus Prolog Final ly we compare the proposal with related

approaches to quantier scope ambiguity

Intro duction

The semantics of sentences containing quantiers can b e dicult to predict Particularly

when a sentence contains multiple quantiers the scop e p ossibilities for each quantier

mayinteract in unexp ected ways with each other and with other syntactic prop erties of

the sentence Many theories of quantier scop e have b een prop osed in the literature most

of them variants either of quantier raising as prop osed byMay or of quantifying

in as prop osed byMontague Both prop osals op erate under the assumption that

the semantics of quantiers can b e characterized by abstraction according to whichNP

semantics can b e pulled out of the original NP p osition and take the rest of the sentential

semantics or some part thereof under its scop e According to these prop osals whether

two NPs mayormay not alternate their relative scop e order can only b e determined after

the two NPs are individually abstracted out Despite numerous mo dications of these

original prop osals they still app ear to fall short of explanatory and descriptive adequacy

for reasons that are discussed in Section b elow

In this pap er we presentanovel conjecture that predicts when two nonreferential

quantiers are or are not ambiguous with resp ect to their relative scop e This approach

ties scop e ambiguity in a language to co ordination in the language Which substrings

serve as scop e islands can b e predicted from which substrings can b e co ordinated We

claim that the conjecture makes predictions that are b oth explanatory and descriptively

adequate To substantiate this claim this pap er fo cuses on three kinds of English con

structions that allowmultiple NPs in a single grammatical sentence complex NPs con

Department of Computer and Information Science UniversityofPennsylvania Philadelphia PA

Email parklinccisupennedu

A preliminary sketch app ears in Park

Jong Park A Lexical Theory of Quantier Scop e

taining PPs complex NPs containing Whrelatives and transitiveattitude verbs We

also give a theory of quantier scop e that is couched in Combinatory Categorial Gram

mar CCG formalism and implemented in SICStus Prolog

The pap er is structured as follows Section motivates and lays out the conjecture

for scop e ambiguity Section argues whywe need to distinguish referential NP inter

pretations from quanticational NP interpretations in semantics following Fo dor and

Sag Section presents a comp etence theory of quantier scop e couched in a

unicationbased CCG framework While CCG is chosen for this task since its notion of

constituency meshes well with that assumed in the conjecture it should also b e p ossi

ble to sp ell out the theory in other grammar formalisms Section lays out theoretical

predictions on scop e readings Section compares the presentapproach with traditional

approaches to quantier scop e Complete prolog co de for the example sentences consid

ered in this pap er and some sample runs are given in an app endix

Surface Constituency Conjecture

Consider the following sentences

a Every representative of a companysaw most samples

b Some studentwillinvestigate two dialects of every language

Hobbs and Shieb er made a claim based on quantier binding at LF that out of the

six combinatorial ways of ordering the three quantiers ie every a and most sentence

a has one missing scop e reading in which every representative outscop es most

sampleswhic h in turn outscop es acompany This scop e reading is certainly unavailable

from sentence a Notice that in this claim Hobbs Shieb er implicitly assumed

that among the available ve readings is the one in which acompany outscop es most

sampleswhich in turn outscop es every representative Let us call this Hobbs Shieb ers

reading The reading would b e true of a situation in which there is a company such

that most samples were individually seen bytheentire representatives of that particular

companyWe agree that Hobbs Shieb ers reading is available from sentence a

May claimed that sentence b has a reading in which every language outscop es

some student which in turn outscop es two dialects Let us call this Mays reading This

reading would b e true of a situation in which for each language there is a p ossibly

dierent studentsuch that he or she will investigate two dialects of that language

Again we agree that Mays reading is available from sentence b Notice that these

two readings share an interesting pattern where the two NPs NP prep NP andNP

ignoring the word order give rise to a scop e order in whichNP outscop es NP whichin

turn outscop es NP This pattern suggests that standard English constituent structure

or even the extended notion of surface constituency discussed b elow do es not limit the

range of available readings

Nevertheless weshow in Section that the kind of scop e relation implicated in

Hobbs Shieb ers account of their reading is unavailable for quanticational NPs eg

at least two companies or few companies in place of acompany This is due to the kind of

functional dep endency inherentinquanticational scop e relations to b e discussed later

There is an inherent realworld connection b etween languages and dialects This connection app ears

to interfere with the said scop e relation in suchaway that mightoverride an otherwise unavailable

scop e relation This p otential interference would go awayifwe replace two dialects with two aspects

Bonnie Webb er and TonyKroch pc The change makes the fact clearer that the said scop e

reading is available indep endentofsucharealworld connection

Jong Park A Lexical Theory of Quantier Scop e

The reason Hobbs Shieb ers reading is available for sentence a is we b elieve that

acompany can b e interpreted referentially Heim We know following Fo dor and

Sag that while referential NPs app ear to take matrix scop e they do not really

participate in the kind of scop e relations that quanticational NPs do Most crucially

referential NPs are interpreted relatively indep endently of the rest of the NPs in the same

sentence and the rest of the NPs are interpreted as if referential NPs are more or less

prop er nouns It is thus theoretically essential to distinguish referential NP interpretations

from quanticational NP interpretations in semantics

Given this semantic distinction and setting referential readings aside sentence

a has exactly four quanticational readings whereas sentence b has ve quan

ticational readings as shown b elow The symbol refers to the outscoping relation

Every rep of a companysaw most samples Some studentwillinvtwo dialects of every language

every rep acomp most samp two dial every lang some student

acomp every rep most samp every lang two dial some student

most samp every rep acomp some student two dial every lang

most samp acomp every rep some student every lang two dial

every lang some student two dial

Table

Quanticationall y Available Readings

We claim that the following conjecture precisely captures this dierence in the num

berofavailable readings and esp ecially the fact that only Mays sentence allows a reading

in which the quantiers intercalate in the sense discussed earlier for the said pattern

We rst make the following denition

cconstituent A string s of words of a sentence S in a language L is a co ordi

nating constituentorcconstituentunderS if and only if L has a grammatical

sentence S which is exactly like S except that s is co ordinated with another string

The qualication under S will b e omitted whenever the context makes it obvious

For example b oth loves and wil l marry are cconstituents as Every man loves and wil l

marry some woman is a grammatical English sentence We will use the term qquantiers

resp ectively rquantiers to refer to quanticational quantiers resp ectively referential

quantiers We also dene cpatterns as follows

While plural NPs show this functional dep endency clearly there is no comparable wayof

determining if nonreferential singular NPs suchas one company result in the same kind of scop e

order as in Hobbs Shieb ers reading Occams razor rules however that such NPs do not

See the forthcoming discussion as to the ob ject quantier most outscoping sub ject quantier

Notice that this version of cconstituency is exactly the CCG notion of surface constituency

Steedman

Jong Park A Lexical Theory of Quantier Scop e

cpattern Supp ose that sentence S contains qquantiers Q and Q There is

a constituency pattern or cpattern for qquantiers Q and Q in S i there is

achoice of NP NP Aand B suchthatS has the form

NP S NP

where Q resp Q is the head quantier of NP resp NP and A and B are

b oth cconstituents

conjecture Supp ose that sentence S contains qquantiers Q and Q Then it

is imp ossible for Q and Q to alternate in scop e ie their scop e relative to each

other is xed unless a there is a cpattern in S for Q and Q or b there is a

choice of qquantiers Q and Q in S where Q resp Q maybe Q resp Q

such that there is a p ossibly dierent cpattern in S for the pairs of qquantiers

Q and Q Q and Q andQ and Q In the case of a the two qquantiers

may alternate their relative scop e and anyqquantiers that maybepresentinA

are outscop ed by b oth Q and Q In the case of b the relative scope between

Q and Q is determined indirectly by the relation b etween Q and Q

Note that this conjecture never states that a scop e ordering is always p ossible it can

only rule readings out We b elieve that scop e orderings not ruled out by the conjecture

usually are available but there is at least one counterexample The conjecture do es not

forbid ambiguity for No printers print no documents but the sentence happ ens to b e

unambiguous so other factors p erhaps p eculiar to noseemtobeatwork Notice also

that according to recent claims quantiers like few or most do not outscop e sub ject

quantiers when they are in the ob ject p osition Beghelli Szab olcsi The

conjecture do es not rule out this p ossibility either While weleave further details to

future work it should b e p ointed out that the new upper b ounds in scop e p ossibilities

set by the conjecture are meantfor al l quantiers that are nonreferentially used

To see how the conjecture works consider sentence a again whose cpatterns

are shown in T able The cpattern p indicates the p ossibilityfor every rep and a

Left NP A NP Right

p every rep of a comp saw most samp

Table

Four CPatterns Every representative of a companysaw most samples

company to alternate their relative scop e p indicates the p ossibilityfor every rep

and most samp to alternate their relative scop e No other cpatterns are p ossible Thus

the sentence is predicted to have up to four readings Notice that Hobbs Shieb ers

reading is not among them p is the only cpattern that might directly relate acomp

to most sampbut acomp saw most samp is not a cconstituent under the sentence as

the structure in a is ungrammatical This do es not mean however that the scop e

We need a further condition such that the fragment A has twoneighb or NPs as its direct semantic

arguments This condition will b e discussed with resp ect to the sentences in and

Jong Park A Lexical Theory of Quantier Scop e

between acomp and most samp is necessarily xed since every rep works as Q for the

clause b in the conjecture where Q coincides with Q The cpattern p do es not

apply for the scop e relation b etween every rep and most sampsinceof a comp saw is not

a cconstituent as the structure in b is ungrammatical Square brackets indicate

the intended co ordination

a Every representative of a companysaw most samples and an institute

insp ected a few samples

b Every representative of a companysaw and of an institute insp ected

most samples

Consider now sentence b whose cpatterns are shown in Table The cpattern

Left NP A NP Right

m some stu will inv twodialofevery lang

m some stu will inv twodial of every lang

m some stu will invtwo dial of every lang

Table

Three CPatterns Some studentwillinvestigate two dialects of every language

m indicates the p ossibility for some stu and two dial to alternate their relative scop e

Likewise m tells us that two dial and every lang can alternate their relativescopeThe

cpattern m further indicates the p ossibility for some stu and every lang to alternate

their relative scop e in which two dial is outscop ed by b oth of the qquantiers Together

they tell us that the sentence can haveuptove readings correctly including Mays

reading The cpattern m go es through due to the structure implied in the following

grammatical sentence

Some studentwillinvestigate two dialects of but may collect most cases of co or

dination in every language

We can thus tentatively conclude that the conjecture explains the sub jectob ject asym

metry at semantics in English with resp ect to the twosentences in Let us examine a

few more examples to see how and what the conjecture predicts b efore explaining why

a Mary thinks that exactly three men danced with more than four women

b At least two girls think that John danced with more than four women

c At least two girls think that exactly three men danced with Susan

It is obvious that sentence a is semantically ambiguous We b elieve that sentence

b is likewise semantically ambiguous cf Lasnik and Uriagereka page

As for sentence c there are conicting semantic judgments bynative sp eakers

The conjecture predicts that sentence a can b e ambiguous since exactly three

men and more than four women may alternate their relativescopeas danced with and

Thewellknown thattrace phenomenon shown b elow might suggest that emb edded sub ject

quantier do es not outscop e matrix sub ject quantier assuming that Whtraces and QRtraces are

governed by the same constraint However it app ears that native sp eakers do not base semantic

judgments on the presenceabsence of the complementizer cf Steedman

a Who do you think that t danced with Susan

b Who do you think t danced with Susan

Jong Park A Lexical Theory of Quantier Scop e

the emb edded clause are cconstituents The conjecture also predicts that sentence

b can b e ambiguous since think that John danced with is a cconstituent as evidenced

below

At least two girls think that John danced with but doubt that Bob even talked

to more than four women

The conjecture as constrained further in fo otnote predicts that sentence c is

unambiguous This is b ecause while the following structure in is marginally ac

ceptable the semantics of the fragment think that takes two arguments one NPtyp e but

another Styp e For the condition to go through they need to b e two NPtyp es

At least two girls think that exactly three men but most b oys doubt that more

than two men danced with Susan

Again the conjecture thus predicts that there is a p otential semantic asymmetry b e

tween emb edded ob ject quantier and emb edded sub ject quantier in a thatclause

complement of an extensional verb suchasthink Notice that Montagovian quantifying

in correctly generates the de re reading for the following sentence apparently pro ducing

a scop e order in which a unicorn outscop es the matrix sub ject quantier

Every valiant knight b elieves that a unicorn is approaching from the mountain

This app ears to contradict the prediction by the conjecture However it is clear that de re

interpretation of a unicorn inside an opaque context is strongly related to its referential

interpretation as the name suggests Since there is a distributional dierence b etween

referential and quanticational NP interpretations to b e argued in the next section this

reading is not relevant to the present consideration regarding nonreferential quantiers

Finally consider the following pair of sentences

a Two professors who interviewed every student wrote a letter

b Two professors whom every student admired wrote a letter

Recall that there is a wellknown island condition on emb edded NPs in a relative clause

Ross so that the following syntactic extraction is considered ungrammatical

I havemetevery student whom two professors whom t admired wrote a letter

i i

Again movementbased theories of quantier scop e suchasvariants of quantier rais

ing accounts make use of this condition in predicting the range of available scop e read

ings This kind of observation is considered theoryneutral so that other theories suchas

variants of quantifyingin also consider it necessary to make use of a related stipulation

such as Complex Noun Phrase Constraint CNPC that blo cks emb edded quantiers

from outscoping head quantiers Ro dman Hendriks

The sentence pattern Mary thinks that P and Q for emb edded clauses P and Q is syntactically

ambiguous b etween Mary thinks that P and Q and Mary thinks that P and Q

The sentence a is due to Janet Fo dor pc

Jong Park A Lexical Theory of Quantier Scop e

One can show however that unlikeemb edded subject NPs emb edded object NPs

can outscop e head quantiers though marginallyasshown in sentence a b elow

And it do es not app ear that these NPs must b e syntactic ob jects as relativeclause nal

NPs also showthischaracteristics as in b Notice that referential NPs do not show

this dierence at all to b e discussed in Section

a FBI agent Starling contacted morethanthreerelatives who knew every

victim of the infamous Dr Lector

b Most businessmen who grew up in almost every big city talk fast but

most businessmen who grew up in Chicago talk rather slowly

The conjecture predicts that these sentences are ambiguous since who knew and who

grew up in are all cconstituents and b oth of them taketwo NPtyp e arguments Notice

that a contrary prediction is correctly made for sentence b since the pattern two

professors whom every student is not a cconstituent as evidenced b elow

Two professors whom every student and most deans whom every girl admired

wrote a letter

There are many other English constructions that need to b e tested but the ab ove

constructions already provide go o d examples to identify the striking phenomenon

Let us now consider the implication of the conjecture The conjecture predicts when

an NP quantier suchasNP is allowed to outscop e another temp orally preceding NP

quantier suchasNP in a grammatical sentence The reason that this works can b e

attributed to the fragments a and b b eing cconstituents that b is a cconstituent

assures the relative semantic autonomy or selfsuciency of the fragment itself and

that a is a cconstituent implies that NP and NP work as twosemantic arguments of

the fragment much like a transitiveverb having two semantic arguments In order to

showwhy the conjecture explains English sub jectob ject asymmetry in scop e readings

consider the following simplied surface structures

NP NP

z z

Quantier Head TV Quantier Head a

z z

S O

NP NP NP NP

z z z z

b Quantier Head P Quantier Head TV Quantier Head P Quantier Head

z z

S O

English is a congurational language in which the standard word order of a gram

matical sentence is SVO as shown in a ab ove Transitiveverbs normally exp ect

two arguments S and O on their two sides When the NPs are mo died further as in b

the transitiveverb still exp ects to receivetwo arguments or NP and NP but these

We appreciate Mark Steedman for this sentence structure

In the CCG formulation to b e shown shortly the syntactic category of the fragments is N nN NP

ie one of the arguments is of noun typ e N This is the result of the category of the relative pronoun

who which is assigned the category N nN S nNP Alternativelywe can adjust the categories for

quantiers and nouns to accommo date the category N nNP S nNP for relative pronouns in

order to implement the conjecture more literally at the exp ense of clarity of implementation

The reader is referred to Park for further constructions including control and ditransitive

verbs many more examples of extraction and co ordinate structures

Wehave seen also that we need to force the implication ab ove since otherwise sentences like

c will b e incorrectly determined to b e ambiguous

Jong Park A Lexical Theory of Quantier Scop e

two arguments are rst mo died by NP and NP resp ectively b efore they are made

available for the transitiveverb The fact that English allows the fragment TV NP P

but not the fragment PNP TV to b e a cconstituent implies not only that NP is

still the same argumentthat TV can accept but also that NP is not This makes

sense since we exp ect a p ostmo dier suchasPNP to b e something like a transducer

function that takes a normal NP to yield another normal NP In particular the pres

ence of such a p ostmo dier should aect neither the grammaticality nor the semantic

integrity of the rest of the sentence It is thus natural to exp ect that the transitiveverb

will not b e able to accept such a complex ob ject directly as one of its arguments In

other words English sub jectob ject asymmetry in scop e readings is the direct result of

its standard word order where the mo died head part of a complex object NP but not

that of a complex subject NP is temp orally adjacent to the transitiveverb We need a

crosslinguistic study to see how this kind of observation works in languages other than

English but it is b eyond the scop e of the present pap er

Quanticational Readings and Functional Dep endency

This section shows why referential readings should b e distinguished from quanticational

reading x and why functional dep endency b ears signicance with resp ect to quan

ticational readings x

Referential NP Interpretations

This section presents a claim that one must distinguish referential and quanticational

NP interpretations in semantics We review some evidence for this claim in whichthe

two kinds of interpretations show distributional dierences

A student in the syntax class cheated on the nal exam

When the sp eaker of the sentence has a particular p erson in mind for the studentin

question say John the sub ject NP is taken to b e used referentially In this reading the

sentence would b e false if John didnt cheat on the nal exam even if there was another

student say Bob who did the deed A p ossible resp onse to this sentence would b e No

a student in the syntax class could not nd the instructions on the nal exam On the

other hand when the sp eaker used sentence to simply assert the fact that there was

one p ossibly more such student the sentence would b e truthful as long as there iswas

one such individual even if the individual is not the one whom the sp eaker had in mind

In this reading the sub ject NP is taken to b e used quanticationally It is granted

however that the two readings of sentence do not dep end much on surface structure

to make a convincing case for a distributional dierence b etween them For this consider

the following sentences

a John ov erheard the rumor that every student of mine had b een called

b efore the dean

b John overheard the rumor that a student of mine had b een called b efore

the dean

If P is excluded from the fragments that they exp ect further arguments is lost in the semantics

The data and as well as the related observations are from Fo dor and Sag

This reading improves with some student in place of a student

Jong Park A Lexical Theory of Quantier Scop e

The emb edded sub ject p osition of a complex NP is known to b e a syntactic island Ross

as mentioned b efore which explains whysentence is ungrammatical

John met every student whom each teacher overheard the rumor that t had

b een called b efore the dean

This syntactic phenomenon has also b een utilized in semantics to constrain the movement

of quantiers in Government and Binding theories which can thus explain whysentence

a do es not have a reading in which every student outscop es the rumor a p ossibly

dierent but uniquely identiable rumor for eachstudent However it is obvious that

this constraint do es not apply to referential NPs as sentence b does havean

interpretation in which there is a certain student suchthatJohnoverheard the rumor

that he or she had b een called b efore the dean In this reading the denotation of the NP

a student of mine is not dep endent up on the kind of rumor that John overheard As such

referential NP interpretations do not seem to b e so much constrained as quanticational

NP interpretations are in taking matrix scop e

a Eachteacher overheard the rumor that every student of mine had b een

called b efore the dean

b Each teacher overheard the rumor that a student of mine had b een called

b efore the dean

Sentence a has only two readings one with the same rumor for all the teachers

and the other with a p ossibly dierentversion of rumor for each teacher Incidentally

this is exactly what the conjecture would predict Notice that every student of mine can

not outscop e any of the two NPs We know that a student of mine in b can take

matrix scop e if it is referentially interpreted The question is if it is p ossible for the NP

to b e outscop ed byanyofthetwo NPs p ossibly placed b etween the two This as the

reader can verify is imp ossible The only readings that are available are ones in which a

student app ears to outscop e b oth each teacher and the rumor In other words referential

NP interpretations can only take matrix scop e not intermediate scop e Given the

evidence presented so far Fo dor and Sag conclude that a theory of indenites

in our case quantiers can b e made parsimonious if referential and quanticational NP

interpretations are distinguished in semantics

Based on this semantic distinction we will fo cus exclusively on quanticational NP

interpretations in identifying the connection b etween syntax and semantics as mani

fested byquantier scop e As for referential NP interpretations including other typ es of

NPs there are renewed interests in dynamic NP interpretations following the lead of a

discourse representation theory by Kamp or the le change semantics by Heim

There have also b een recent attempts to combine the two asp ects for instance in

theories of scop e byPo esio and Reyle While the quanticational asp ect

of these theories do es not app ear to present a comprehensive and explanatory answer to

There are cases esp ecially in intensional contexts where referential NPs do not necessarily take

matrix scop e as exemplied in the sentences b elow Dan Hardt pc

I dreamed that I was a teacher and in mydreamIoverheard the rumor that a studentofmine

had b een called b efore the dean

See also the discussion with resp ect to sentence where de re interpretations may not necessarily

b e equated with matrix scop e However the p ointhereisthatthetwotyp es of NP interpretations

show a noticeable dierence regarding surface syntax

Jong Park A Lexical Theory of Quantier Scop e

the kind of data the current pap er is concerned with there is no doubt that a unied

theory for b oth referential and quanticational NP interpretations is desirable

There are some apparentcounterexamples Wehaveshown earlier why Hobbs

Shieb ers reading can b e explained by a referential acompany This reading will b e

discussed in more detail in Section Now consider sentence a The prominent

reading called conjunctiveorcumulative is true of a situation in which there are three

hunters and ve tigers such that the said event happ ened b etween the two parties

a Three hunters shot at ve tigers

b Three Frenchmen visited ve Russians

Most imp ortantly the reading of this kind can not b e addressed by a linear order b etween

the two NP denotations This is why Hintikka dened the notion of branching

quantiers in his gametheoretic semantics subsequently endorsed and extended by Bar

wise and Westerstahl among others Sentence b is argued to have

a similar reading Partee Webb er It is interesting to note however that

conjunctiveorcumulative readings of this kind do not obtain when there is a strong

lexical preference of quantiers towards taking functional scop e eg a or when

there is no p ossibility for a referential NP interpretation eg b Higginb otham

Krifka Hence we b eliev e that it is reasonable to assume that cumulative

readings are not in the range of quanticational scop e readings since the involved NPs

either one of them or b oth must b e interpreted referentially

a EachFrenchman visited ve Russians

b Few Frenchmen visited ve Russians

There is another sentence shown b elow in a that May claimed has

a related branching reading citing the account of Hintikka May notes that

for the reading to obtain b oth of the the head quantiers must b e outscop ed bythe

corresp onding mo difying quantiers Notice that this kind of reading do es not obtain from

sentence b where b oth of the head quantiers have a nonreferential interpretation

We claim therefore that the reading in question if it exists is also an instance where

the NPs are used referentially though the denotations of the complex NPs have a little

more structure than those of the simple NPs

a Some article byevery author is referred to in some essaybyevery critic

b Every article bysomeauthorisreferredtoinevery essaybysomecritic

While the data considered here are not sucienttoprovethevalidity of the conjec

ture fullywe b elieve that the conjecture is shown to b ehave reasonably on some of the

most discussed apparentcounterexamples

Functional Dep endency

This section shows that quanticational readings always exhibit a kind of functional

dep endency b etween the scop e related NP denotations We claim that this prop erty can

b e utilized to sharp en p eoples intuition to determine the availability of a particular

reading by maximizing the way scop erelated NP denotations are laid out Note that

the kind of scop erelated functional dep endency that we are interested in here is truly

semantic and distinct from the kind of pragmatic dep endency that makes sentence

un ambiguous

Jong Park A Lexical Theory of Quantier Scop e

Every professional mother gives birth to at most two babies

The claim is that in quanticational readings the semantic ob jects denoted byan

outscop ed quantied NP dep end functionally up on the semantic ob jects denoted bythe

outscoping quantied NPFor instance consider sentence a b and c show

its two p ossible logical forms in rstorder logic

a Every man loves some woman

b mmanm w w omanw lovesm w

c wwomanw mmanm lovesm w

Toevaluate the logical form b truthconditionallywe should makethechoice of

an individual for w functionally dep endentuponthechoice of each individual for m

since otherwise there would b e no semantic truthconditional dierence b etween

b and c This is usually captured byskolemizing the variable w in b

We argue that this kind of scop erelated functional dep endency shows up b etween any

two NPs connected by a scop e relation regardless of whether the reading has a group

interpretation or a distributivein terpretation

What is signicant with this functional dep endency is that it amplies the connection

between individuals related by scop e ordering to such a degree that it b ecomes evident

that some connections and therefore the related scop e ordering are not warranted by

the sentence at hand Consider the following sentence a variant of a

Two representatives of three companies saw four samples

The following shows six logical forms in a generalized quantier format Barwise and

Co op er Hobbs and Shieb er

a three companies two representatives four samples

threeccompctworreprofrcfourssampssawrs

b two representatives three companies four samples

tworreprthreeccompcofrcfourssampssawrs

c four samples three companies two representatives

fourssampsthreeccompctworreprofrcsawrs

d four samples two representatives three companies

fourssampstworreprthreeccompcofrcsawrs

e three companies four samples two representatives

threeccompcfourssampstworreprofrcsawrs

f two representatives four samples three companies

fourssampsthreeccompcsawrs tworreprofrc

The four readings a through d are selfevidently available For instance

the logical form a is true of a situation in which there are three companies suchthat

each such company has two representatives suchthateachsuch representativesaw four

samples Likewise the logical form d is true of a situation in which there are four

Bare numerals are more likely to receive referential interpretations On the other hand they can

also b e assumed to have implicit premo diers suchasexactly at least etc which strengthen

quanticationalinterpretations For the following discussion we will assume the premo dier exactly

without losing generality

Each logical form is preceded by the corresp onding scop e ordering

Jong Park A Lexical Theory of Quantier Scop e

samples such that each sample was seen bytwo representatives suchthateach such

representative is one of three companies

Notice however that the reading corresp onding to the logical form f would

b e immediately excluded by Hobbs and Shieb er or anyone else due to the fact

that it is not p ossible to construct a sensible mo del related to the sentence Notice as

Hobbs Shieb er p ointed out that among the six logical forms only this one contains a

free variable c underlined Hobbs and Shieb er s consequent suggestion to utilize

an unbound variable constraint or uvc as a semantic lter for available logical forms

would thus b e acceptable provided that all the other ve readings were available An

approach to incorp orating this kind of a logical condition in a logicbased system has

also b een pursued in much subsequentwork including Keller Carp enter

Pereira We should also p oint out that this kind of condition maybe

needed in one form or another in order to explain natural language pronouns as b ound

variables This is a separate issue however

We claim that in addition to the reading f the reading corresp onding to

e is also unavailable due to the kind of functional dep endency it requires of its mo del

This reading shares the same scop e order with Hobbs Shieb ers reading in whichthe

latter can b e explained with a referential interpretation of acompanyTo see whyitis

imp ossible for a quanticational threecompanies to lead to the reading e let us

rst assume that all the relevant quantied NPs have a distributive sense as group senses

will only simplify the matter The following situation would supp ort the reading

There were three companies such that there were four samples for eachsuch company

such that each of those samples was seen bytwo representatives of that company Cru

cially samples seen by representatives of dierent companies were not necessarily the

same

We claim that this is not what the sentence says The reader is urged to use hisher

own intuition to verify this Figure shows a pictorial layout of a mo del supp orting this

reading

According to the present theory the reason that the reading is excluded is that the

surface structure is NP of NP verb NP It is not due to the lexical semantics of the

nouns and the verb involved Notice also that the uvc do es not exclude this unavailable

reading

A Lexical Theory of Quantier Scop e

This section presents a theory of quantier scop e that captures the conjecture Section

intro duces a version of unicationbased Combinatory Categorial Grammar framework

in which the theory is couched Section prop oses a dual quantier representation for

quantier semantics

Combinatory Categorial Grammar

Categorial Grammars or CGs are a class of grammar formalisms originally prop osed

y BarHillel The reader is referred by Ajdukiewics and further develop ed b

to Wo o d for a general intro duction to CGs CGs enco de syntactic information

in a categorial lexicon where each lexical entry sp ecies how the corresp onding lexeme

is interpreted syntactically In the following sample lexical entries the op erator

Park shows the formal denition of its syntax and semantics

Jong Park A Lexical Theory of Quantier Scop e

companies

samples

representatives

functional dependency

see-relation

of-relation

Figure

A Mo del Supp orting the Reading threecompanies four samples two representatives

connects lexemes and categories

a john np b slept snnp

a enco des the fact that john is syntactically a noun phrase or np b enco des the

fact that sl ept is a syntactic constituent that when combined with another constituent

of category np on its left results in a constituent of category s The directional symb ols

or slashes n and have the following intended interpretations in rules of function

application The symb ols and abbreviate the corresp onding rules

a X Y Y b Y X nY

X X

When the constituent X nY has another constituent Y on its left the rule b can

b e applied to cancel out the argument category Y with the constituent Y leaving the

result category X for the combined constituent as shown b elow

John slept

np snnp

The derivation np snnp sis achieved by resp ectively replacing the values np and

snnp with the patterns Y and X nY in the rule where the pattern Y is unied with

the value np and the pattern X with the value s

We will use the expressions aconstituent of category x and aconstituent x interchangeably

Notice that we are using the Prolog convention to distinguish variables from constants

Jong Park A Lexical Theory of Quantier Scop e

There are a xed number of elementary categories suchass np and n Categories

are dened recursively as the smallest set that contains elementary categories or cate

gories separated by a directional symb ol Categories asso ciate to the left by default The

following shows another derivation

every man loves some woman

npn n snnpnp npn n

np np

snnp

Combinatory CGs or CCGs extend the purely applicative CGs describ ed ab oveto

include a limited set of combinatory rules corresp onding to combinators suchastyp e

raising T function comp osition B function substitution S etc for the combination of

two adjacent linguistically realized or phonologically nonempty categories Steedman

Rules of typ e raising and function comp osition are shown b elow

a Typ e Raising forward T b Typ e Raising backward T

X X

T T

TT nX T nTX

c Function Comp osition B d Function Comp osition B

X Y YZ Y nZXnY

B B

XZ X nZ

With the combinatory rules based on combinators T and B can have the following

derivation among others

every man loves some woman

npn n snnpnp npn n

np np

ssnnp

snp

In this derivation the category of every man is typ e raised from np to ssnnp using

the forward typ e raising rule in a where the placeholders X and T are replaced

with np and s resp ectively The new category ssnnp is consistent with the syntactic

characteristics of English sub ject NPs which normally exp ect a VP constituent snnp on

their right to result in a sentence constituent s In the derivation the fragment every

man loves is analyzed to b e of category snp or one that exp ects a constituent np on its

right to result in a constituent sBothofthetwo fragments snp and snnp are p erfect

CCGconstituents

Jong Park A Lexical Theory of Quantier Scop e

There is a lexical alternativetothesyntactic typ e raising in For instance

prop er nouns can b e assigned raised categories suchasssnnpandsnsnp etc in

the lexicon Likewise quantiers can b e assigned similar raised categories exp ecting a

noun category on their right suchasssnnpn and snsnpn etc The derivation

shows an example with a raised sub ject NP quantier and the derivation with

a raised ob ject NP quantier

every man loves some woman

ssnnpn n snnpnp npn n

ssnnp np

snp

every man loves some woman

npn n snnpnp snnpnsnnpnpn n

np snnpnsnnpnp

snnp

The fact that there is an alternative derivation such as or in addition to the

more standard derivation is crucial for sentences containing co ordination or parasitic

gap as p ointed out by Steedman among others For instance the co ordination

in sentence a forces the fragment every man loves to b e combined rst and the

co ordination in b forces loves a dog to b e combined rst

a Every man loves but most women hate a dog

b Every man loves a dog but hates a cat

Both of the derivations and contain not only typ eraised categories but also

unraised category npn As far as this particular example go es the unraised category

can b e avoided as shown in the following derivations

every man loves some woman

ssnnpn n snnpnp snsnpn n

ssnnp snsnp

snp

every man loves some woman

ssnnpn n snnpnp snnpnsnnpnpn n

ssnnp snnpnsnnpnp

snnp

The immediate question is if it is always p ossible to nd an alternative derivation with

out unraised categories The following section prop oses a dual quantier representation

in which b oth raised and unraised categories are asso ciated with a prop er quantier se

mantics We argue that without unraised categories the resulting theory is not only more

complicated to design but also unable to account for the full range of scop e readings

Jong Park A Lexical Theory of Quantier Scop e

Connecting Syntax and Semantics

A prop er characterization of the range of grammatical scopings would dep end crucially

on howwecho ose to dene the syntax for the semantic representation The goal here is

to make the connection b etween syntax and semantics as transparent as p ossible and

we will try to use a minimal semantic representation For this purp ose we prop ose the

following dual representation for quantier semantics

a QuantierModeVarRestrictionBody

b QuantierRestriction

a enco des the widescop e quantier semantics with explicit scop e information and

b the degenerate quantier semantics with no corresp onding scop e information We

relate the representation a to typ eraised NP categories suchasssnnporsnsnp

These categories always contain s category which can b e asso ciated with a full sentential

semantics for the required scop e b o dy The quantier in b is called degenerate in the

sense that the op erator corresp onding to the quan tier lacks the general abilitytotake

scop e over something else The representation b is used for unraised np category which

do es not allow the sp ecication of full sentential semantics for scop e information

shows an example widescop e quantier representation

a More than three men sneezed

b thr ee M manM sneezedM

Examples of degenerate quantier representation will b e shown along with the relevant

lexical enco ding

There are twoways of asso ciating semantic information with syntactic information

under the present framework as shown b elow for the transitiveverb loves

a loves snnpnp nx ylovesx y

b loves s lovesX Y nnp X np Y

The metho d a relates each whole lexical category to an appropriate semantic form

usually a higherorder expression separated by the colon op erator This represen tation

The symb ol in b is for a further syntactic distinction b etween widescop e and degenerate

op erators It should not b e confused with the usual annotation on ungrammatical sentences

Incidentally the representation a further generalizes the generalized quantier format such as

shown earlier in that the optional premo dier is put into one of the argument p ositions ie M ode

of an op erator that corresp onds to a natural language quantier This allows the op erator

completely determined even when the numeral has a missing premo dier and thus is considered

p otentially ambiguous In the representation this ambiguity is carried over in a variable whichmay

b e instantiated bychoice later on with a contextdep endent information In the present description

of the theorywewillcho ose to translate a missing premo dierinto the symbol

While there is a clear characteristic distinction b etween degenerate quantier semantics and

referential quantier semantics to b e noted shortlytheymight turn out to b e more closely related

with each other than assumed here Weleave op en the issue of further explicating the relation For

the moment we should say that degenerate quantier semantics is unrelated to referential NP

semantics or sp ecic indenites whose denotations are determined contextually In a sense the

degenerate representation b is a syntactic sugar for a widescop e quantier representation in

a in which the scop e information corresp ondingto Body is missing Just as the widescop e

ternal distinction b etween group vs quantier semantics do es not commit to the semanticsin

distributiveNPinterpretations the degenerate quantier semantics are not committed to sucha

distinction either One can alternatively think of the degenerate quantier semantics as intro ducing

a kind of DRTstyle existential variable whose denotation is determined according to where it

app ears in a logical representation We appreciate Matthew Stone for this suggestion

The symbol n in the semantics is a keyb oard substitute for the lamb da op erator

Jong Park A Lexical Theory of Quantier Scop e

requires an ability to p erform a limited higherorder term unication Categorial rules of

combination can accommo date this metho d with the following revision

a X Y F Y A X F A

b Y AXnY F X F A

The metho d b relates each elementary category to an appropriate semantic form

separated by the colon op erator The semantic form itself do es not involve a higherorder

expression and the representation can b e manipulated by a rstorder term unication

alone Also this metho d allows reduction at compile time via a Prolog programming

technique known as partial execution Pereira and Shieb er Jowsey Steedman

Park

These twoapproaches are logically equivalent as long as the unication for a and

b ab ove are higherorder We will show a theory based on the second approach metho d

b in the interest of implementing it in normal ie not higherorder though not

pure Prolog

With lexical typ e raising eachquantier is assigned a numb er of lexical entries

Numeral quantiers that can optionally have a premo dier need further entries a

and b showtwo lexical entries among many others for a n umeral quantier that is

missing a premo dier

a three s thr eeXNSs S nnp X n X N

b three s thr eeXNSns Snp X n X N

The derivation shows how the premo dier at least is combined with the numeral

three in this framework with an additional entry for three among others bythe

use of theoryinternal elementary categories suchasql and qm This technique can also

handle idiomatic expressions

three s thr eeM X N Ss S nnp X n X N nql M

at least three

ql q m least q m least s thr eeM X N Ss S nnp X n X N nql M

s thr eeXNSs S nnp X n X N

every man

s ev er y XNSs S nnp X n X Nn X manX

s ev er y XmanX Ss S nnp X

The derivation shows how the wide scop e sub ject NP semantics is derived To explain

pro cedurally how the derivation go es through the pattern X N is rst unied with the

pattern X manX in whichthevariable N is unied with manX This value of N

is then carried over to the other instance of N in the pattern ev er y XNS for the

result

But see b elow for the degenerate quantier semantics The reader is referred to the discussion of

the signicance of rstorder unication in Mo ore and Park among others

The present implementation simulates a secondorder term matching via the univ op erator

Jong Park A Lexical Theory of Quantier Scop e

The derivations in and showhow the wide and narrow scop e interpre

tations of some woman are resp ectively obtained from the sentence Every man loves

some woman Each derivation is divided into two separate derivations for typ ographical

reasons

a every man loves

s ev er y X manX Ss S nnp X s lov X Y nnp X np Y

s ev er y X manX lov X Y np Y

every man loves some woman

s ev er y X manX lovX Y np Y s someYwmnY Sns Snp Y

s someYwmnY every X manX lovX Y

loves some woman

s lov X Y nnp X np Y s someYwmnY Snnp X ns S nnp X np Y

s someYwmnY lovX Y nnp Y

every man loves some woman

s ev er y XmanX Ss S nnp Y s someYwmnY lovX Y nnp X

s ev er y X manX someYwmnY lovX Y

In each of the derivations loves works as the constituent a in the conjecture while the

entire sentence corresp onds to the constituent b The derivations app ear to suggest that

readings are derivationdep endent For instance when loves is combined rst with some

woman it leads to a reading in which some woman is outscop ed but when loves is com

bined rst with every man it leads to a reading in which the scop e ordering is reversed

This prediction is in general valid but the availability of the degenerate quantier seman

tics gives a result that mayovercome the apparent derivationdep endency of readings

For instance consider the following sentence cf Geach

Every girl admired but most b oys detested one saxophonist

Without the degenerate interpretation of one saxophonist it is predicted that one sax

ophonist can only b e interpreted to outscop e b oth every girl and most boys since every

girl admired and likewise the second conjunct must b e interpreted b efore it is asso ciated

with the ob ject NPAsGeach argues this is not a valid prediction since p eople

get b oth wide scop e reading and narrow scop e reading of one saxophonist The degen

erate interpretation of one saxophonist takes care of the narrow scop e reading of one

saxophonistasshown b elow Notice that the unraised NP category for one saxophonist

is used in the derivation

every girl admired one saxophonist

s ev er y X g ir lX admX Y np Y np oneY saxY

s ev er y X g ir lX admX oneY saxY

If one saxophonist were interpreted referentially the resulting logical form would b e interpreted in

suchaway that the denotation of one saxophonist is determined indep endent of the individual

denotations of men This shows whywe need to distinguish referential and degenerate quantier

interpretations in semantics

Jong Park A Lexical Theory of Quantier Scop e

Theoretical Interpretations

This section shows how the constructions discussed in Section are accounted for in

the present theory The data are discussed in three subsections complex NPs containing

PPs complex NPs containing Whrelatives and attitude verbs

Complex NPs containing PP

The sub ject NP in the following sentence has twoquantiers

Two representatives of three companies showed up

The category for the prep osition of enco des the fact that it is the head of a PP

of n X N of X Y nn X N np Y

The grammaticality of the following sentence indicates that the noun category for repre

sentatives for instance should b e typ e raised from n to nnnnsothatrepresentatives

and of will b e able to combine by function comp osition

At least two representatives of and more than ve applicants of three companies

came to the part y

The mo difying NP threecompanies can either take the rest of the complex NP as an

argument or work as an argument of the prep osition The following shows the category

for the former

two representatives of three companies

ssnnpn nnnn nnnnp ssnnpnssnnpnp

nnp

ssnnpnp

ssnnp

and b elowshowhow the derivation yields an interpretation in which three

companies outscop es two representatives

two representatives of

s tw oXNSs S nnp X n X Nn X Nn X N nn X repX see

n X repX of X Y np Y

s tw oXrepX of X Y Ss S nnp X np Y

two representativesof three companies

see s thr eeYcompY Ss S nnp X ns S s S nnp X np Y

s thr eeYcompY twoXrepX of X Y Ss S nnp X

Notice that this interpretation is structurally identical to that of a simple NP In other

words a further combination of this interpretation with that of the verb saw in sentence

a b elowwould result in a scop e ordering in which b oth quantiers in the sub ject

NP are outscop ed by the ob ject quantier Similarly a further combination of this inter

We will continue to ignore referential quantier interpretations

Jong Park A Lexical Theory of Quantier Scop e

pretation with that of the verb phrase saw four samples would yield a scop e ordering in

whichbothquantiers in the sub ject NP outscop e the ob ject quantier

The other p ossibility for the category of threecompanies should allow the derivation

of the CCG constituent of threecompanies so that two representatives may outscop e three

companies With the category nnnnp for the prep osition of the immediate solution

is to use the base or unraised category np for threecompaniesWehave argued earlier

that this category is applicable to degenerate quantiers Since other quantiers can

outscop e a degenerate quantier this gives the result we exp ect as shown b elow in

which two representatives outscop es threecompanies While it is true that in this form

threecompanies would not b e able to outscop e any other quantiers in the ob ject NP

this is not a problem since it do es not participate in any further scop e ordering due to

its placement inside the restriction not inside the b o dy

two representatives of three companies

see n X Nn X N nn X repX see np thr eecomp

n X N of X thr eecompnn X N

n X repX of X thr eecomp

s tw oXrepX of X thr eecompSs S nnp X

As an alternative to the latter ordering we can utilize another category for the

prep osition of as shown b elow with the desired derivation

of n X N S nn X N s S ns of X Y np Y

two representatives of three companies

see see see s thr eeYcompY Sns Snp Y

n X N thr eeYcompY ofX Y nn X N

n X repX thr eeYcompY ofX Y

s tw oXrepX thr eeYcompY ofX Y Ss S nnp X

Both and pro duce logically equivalentsemantic forms so the new category

makes available a more standard logical form at the exp ense of redundancy of derived

semantic forms Also the theory that presupp oses the category has the burden of

justifying the category nnnsnsnp for the prep osition on purely syntactic grounds

Weknow that sentence a has four readings and b ve readings

a Two representatives of three companies saw four samples

b Most students studied two asp ects of every language

First the two derivations and or in conjunction with the derivations

of the kinds in and correctly give rise to four dierently scop ed readings for

sentence a Toshow that the readings allowed under the conjecture are the only

ones that are predicted by the theorywemust show that the theory do es not derivethe

following scop e relations

eecompanies a two representatives four samples thr

b threecompanies four samples two representatives

Notice that weshowan reduced restriction for the degenerate semantics of threecompanies The

unreduced representation should b e three X comp X as similarly shown in

Jong Park A Lexical Theory of Quantier Scop e

Toshow that the reading a is not derived by the theory notice rst that as so on as

two representatives outscop es threecompanies the semantics of threecompanies is put

into a restriction whereas the semantics of four samples is put into a scop e b o dySo

it is syntactically imp ossible to derivesuch a scop e relation where four samples comes

between two representatives and threecompanies in that order

As for the reading b when threecompanies outscop es anything three must b e

assigned a widescop e quantier semantics When the semantics for the sub ject complex

NP which includes that of threecompanies is derived nothing can come b etween

threecompanies and two representatives as shown in This makes imp ossible for

four samples outscop e two representatives Notice also that when threecompanies is

assigned a widescop e semantics two representatives can not b e assigned a degenerate

semantics as there is no typ e raised category T that allows the following derivation to

go through

two representatives of three companies

npn nnnn or Tn n

npnp or npsnsnp T

ssnnp

Since there is no other p ossible scop e order the theory correctly derives only and exactly

four readings for sentence a

As for sentence b the question is if the theory can predict and derive the ad

ditional reading in which every language outscop es most students which in turn outscop es

two aspects The following shows that it do es

studied two asp ects of

s studiedX Y nnp X np Y np tw oN n Nn Nn N nn Y asptY see

s studiedX tw oY asptY of Y Znnp X np Z

The derived category syntactically works just like that of a transitiveverb except

that the semantic asso ciation is dierent Notice the use of a degenerate category for the

quantier two As the following complete derivation for the reading in question shows

two aspects is outscop ed byboth some student and every language The details of the

initial lexical entries for them are suppressed here for typ ographical reasons

most students studied two asp ects of every language

ssnnp s studiedX tw oY asptY of Y Znnp X np Z snsnp

s mostXstuX studiedX tw oY asptY of Y Znp Z

s ev er y Y lang Y most X stuX studiedX tw oY asptY of Y Z

The theory do es not generate the reading whose scop e relation is two aspects most

students every language for a similar reason shown earlier for sentence a

Notice that the following related sentence do es not have a reading corresp onding to

the one derived in

At least two asp ects of every language confused most students

Jong Park A Lexical Theory of Quantier Scop e

The successful derivation for such a reading would require the recognition of the fragment

of every language confused as a constituent This is syntactically imp ossible In CCG

terms this is explained by the fact that the category n is not an argumenttyp e of the

category of a transitiveverb confused

of every language confused

nnnnp npn n snnpnp

nnn

The prop osed theory thus explains the identied English sub jectob ject asymmetry

Complex NPs with WhRelatives

Consider the following sentences with sub ject Whrelatives

a Two professors who interviewed every student wrote a letter

b Two professors whose students admiredmostdeans wrote several letters

c Two professors interviewed three students most pictures of whom pleased

exactly two judges

Wehave argued that sentence a mayhave a reading in which every student

outscop es two professorswhich in turn outscop es a letter And as shown earlier the

conjecture predicts this as long as who interviewed is a cconstituent In order to see if the

theory predicts this as well we need to consider rst how the lexical entries corresp onding

to Whrelatives are dened

shows the category for sub ject Whrelative who cf Steedman

who n X N S nn X N s S nnp X

The theory do es consider the fragment who interviewed as a constituent as the following

two derivations show

two professors who interviewed every student

ssnnpn nnnn s inter v X Y nnp X np Y see

f or r ef er ence

n X N inter v X Y nn X N np Y

n X pr of X inter v X Y np Y

s tw oXprofX inter v X Y Ss S nnp X np Y

s ev er y Y stuY twoXprofX inter v X Y Ss S nnp X

two professors who interviewed every student

ssnnpn nnnn s inter v X Y nnp X np Y np ev er y stu

n X N inter v X Y nn X N np Y

n X N inter v X ev er y stunn X N

n X pr of X inter v X ev er y stu

s tw oXprofX inter v X ev er y stuSs S nnp X

In fact this only guarantees a p ossible intercalation of quantiers not matrix scop e for every

language Notice that this is the reading corresp onding to Hobbs Shieb ers reading

Jong Park A Lexical Theory of Quantier Scop e

Compare the derivation with b oth of which utilize a degenerate quantier

semantics As for the need to have an extra category such as for a widescop e

semantics of threecompanies but still equivalent to the reading derived in the

present derivation do es not need such an additional category since every student can

simply b e assigned the category snpnsnnpnp for such a derivation To complete

such a derivation every student must b e combined with interviewed rst

Since the sentence in whichtheemb edded ob ject quantier outscop es the head quan

tier requires the comp osition of fragments such as the conjuncts in we can predict

that sp eakers who do not tolerate those readings would also regard sentence as un

grammatical In CCG terms this level of tolerance could b e measured by the willingness

of typ eraising the noun category from n to nnnn or by the willingness of combining

a common noun with a relative pronoun

Two professors who interviewed and three deans who visited every student

wrote a letter

Consider nowsentence b As with normal readings one can think of several

relations b etween professors and studen ts for the readings that are available from the

sentence In the following formulation of the lexical item whosewe assume that all the

available readings involve a relation in which for eachsuch professor every studentof

hers admired deans This decision is not motivated theoryinternally

whose n Z N ev er y XNof X Z Snn Z N s S nnp X n X N

The fragment whose students admired in sentence is pro cessed as follows

whose students admired

n X stuX snnpnp

n Z N ev er y X stuX of X Z Snn Z N s S nnp X

n Z N ev er y X stuX of X Z admX Y nn Z N np Y

This gives exactly the same result as b efore except that the implicit quantier every is

correctly outscop ed by other quantiers

Consider piedpiping sentence c Following Szab olcsi Morrill

and Steedman we need to assume extra categories for whom so that the fragment

every picture of whom maywork as a normal sub ject Whrelative This is done by raising

the typ e of whomasshown b elow

s S nnp X np Z whom n Z N S nn Z N s S nnp X ns S

It is clear that typ eraising over islandinducing relative pronouns would b e harder than

typ eraising over prep ositions as predicted also by Steedman Semantic island condition

would stipulate the former as completely imp ossible cf Hendriks

Ideallywe need a mapping function that converts oneplace predicate suchas stuX into

twoplace predicates suchasstuX Z Suchatwoplace predicate will replace the conjoined

restrictions such N of X Z There are other instances that show this problem

Jong Park A Lexical Theory of Quantier Scop e

and showhow to derive the semantics for the fragment most pictures of whom

pleased

most pictures of whom

s most XNSs S nnp X n X Nn X Nn X N nn X picX

n X picX of X Z np Z

s mostXpicX of X Z Ss S nnp X np Z

n Z N most X picX of X Z Snn Z N s S nnp X

most pictures of whom pleased

s plsdX Y nnp X np Y

n Z N mostXpicX of X Z plsdX Y nn Z N np Y

The following sentences contain nonsub ject Whrelatives

a Two professors whom every student admired wrote a letter

b Two professors whose students most janitors liked wrote a letter

c Two professors abiography of whom three journalists wrote interviewed

most students

The lexical entry shows the category for a sub ject Whrelative whom Steedman

The category exp ects an argument of category snpwhichisasentence missing

an ob ject NP

whom n X N S nn X N s Snp X

The conjecture predicts that sentence unlikesentence do es not have a reading

or readings in whichtheemb edded quantier outscop es the head quantier Wehave

shown that the characterization predicts this without invoking a constraint suchasthe

Complex Noun Phrase Constraint and the like Consider how the present theory predicts

this as well

First the relative pronoun whom cannot b e combined directly with the emb edded

sub ject NP since the following derivation is imp ossible The derivation is imp ossible even

with unraised emb edded sub ject NP categories

whom every student

n X N S nn X N s Snp X s ev er y YstuY Ss S nnp Y

Ignoring the lefthand part of the relative pronoun whom for the moment the only

case in which the derivation is successful is when whom combines with the entire emb ed

ded clause or every student admired The following shows the derivation

whom every student admired

s ev er y YstuY Ss S nnp Y s admir edY X nnp Y np X

s ev er y YstuY admir edY X np X

n X N ev er y YstuY admir edY X nn X N

Notice that the combination of every student and admired forces the op erator every to

take the narrow scop e with resp ect to the remaining quantiers including the head

quantier as shown b elow

Jong Park A Lexical Theory of Quantier Scop e

two professors whom every student admired

ssnnpn nnnn n X N ev er y YstuY admir edY X nn X N

n X pr of X ev er y Y stuY admir edY X

s tw oXprofX ev er y YstuY admir edY X Ss S nnp X

When the result combines with the rest of the sentence it will give rise to only two read

ings Notice that the result do es not change even if weinvoke the degenerate semantics

for the head quantier as shown b elow

two professors whom every student admired

np tw oX N n X N nnnn n X N ev er y YstuY admir edY X nn X N

n X pr of X ev er y YstuY admiredY X

s tw oX pr of X ev er y Y stuY admir edY X

Notice that the quantier every is inside the degenerate quantier twoThus the theory

never generates logical forms in whichtheemb edded sub ject quantier outscop es the

head quantier

As for sentence b the lexical entry of whose is shown b elow

whose n Z N ev er y XNof X Z Snn Z N s Snp X n X N

The corresp onding derivation for sentence b is similarly done

Finally the following entry shows the category for whom in the ob ject piedpiping

sentence c Further details are omitted

whom n Z N S nn Z N s Snp X ns S s S nnp X np Z

Attitude Verbs

Consider the following sentences again

a Mary thinks that exactly three men danced with more than four women

b At least two girls think that John danced with more than four women

c At least two girls think that exactly three men danced with Susan

We will assume the following simplied categories for think and the complementizer that

The elementary category ss corresp onds to the S no de in Xbar theories

a think s think X S nnp X ss S

b that ss thatS s S

The theory generates two scop e relations but three distinct readings for sentence

a shows a class of p ossible derivations for the reading in which exactly three

men outscop es more than four women

Mary thinks that exactly three men danced with more than four women

ssnnpsnnpss sss ssnnp snnpnp snnpnsnnpnp

s four Z w mnZ danY Znnp Z

s think mar y thatthr ee Y manY four Z w mnZ danY Z

Jong Park A Lexical Theory of Quantier Scop e

shows another class of p ossible derivations for a reading in which morethanfour

women outscop es exactly three men

Mary thinks that exactly three men danced with more than four women

ssnnpsnnpss sss ssnnp snnpnp snsnp

s thr eeYmanY danY Znp Z

s four Z w mnZ three Y manY danY Z

s think mar y thatfour Z w mnZ threeYmanY danY Z

There is another class of derivations for another reading in which morethanfour

women outscop es exactly three menasshown b elow

Mary thinks that exactly three men danced with more than four women

ssnnpsnnpss sss ssnnp snnpnp snsnp

s thr eeYmanY danY Znp Z

s think mar y thatthr ee Y manY danY Znp Z

s four Z w mnZ think mar y thatthr eeYmanY danY Z

The theory predicts two scop e relations and three distinct readings for sentence

b The logical forms that are generated by the theory are shown b elow

a tw oXgirlX thinkX thatfour Z w mnZ danjohn Z

b four Z w mnZ twoXgirlX think X thatdanjohn Z

c tw o X g ir lX four Z w mnZ thinkX that danjohn Z

The theory predict only one scop e relation and one reading for sentence c

This is due to the fact that emb edded sub ject quantier never escap es the argument

p osition of the that op erator The theory generates the following logical form

tw o X g ir lX think X thatthr eeYmanY danY susan

As a further example consider the following sentence

At least two girls think that exactly three men danced with more than four women

The theory predicts three scop e relations and four distinct readings cf App endix

Comparisons with Related Work

This section compares the presentaccountofquantier scop e with two paradigmatic

accounts of quantier scop e

Quantifyingin Accounts

Quantifyingin is a technique originally prop osed byMontague for de re NP inter

pretations Consider for instance the following sentence which is traditionally regarded

as semantically ambiguous due to the intensional op erator asso ciated with the verb seeks

John seeks a unicorn

Jong Park A Lexical Theory of Quantier Scop e

The idea is that in one of the readings of the sentence there do es not havetobea

unicorn actually existing in the real world for the sentence to make sense In order to

represent this reading or de dicto reading Montague prop osed to assign a function from

p ossible worlds to sets of prop erties where prop erties are functions from p ossible worlds

to characteristic functions to the ob ject of the relation seeks cf DowtyWall and Peters

The de re reading on the other hand app ears to presupp ose the existence of

such a unicorn in the real world The way Montague prop osed to make the denotation

for such a unicorn rigid with resp ect to p ossible worlds is to syntactically take apart the

computation of the NP semantics for aunicorn from that of the rest of the sentence and

to put back the two semantics together via the quantifyingin rules S and T This

eectively creates the logical form P xS x where P is the NP semantics whose

op erator quanties into the opaque context S x to bind the variable x that replaces the

NP in question Notice that the op erator is insensitive to the distance b etween itself

and the variable and in particular to the intervening NP semantics Montague further

prop osed to utilize this rule schemata to account for scop e ambiguities due to extensional

transitiveverbs suchasnds Again quantifyingin makes any NP outscop e the rest of

the sentence and the outscoping relation b etween NPs is determined by the arbitrary

order of invoking quantifyingin

Co op er prop osed a mo deltheoretic version of quantifyingin by utilizing se

mantic storage but the p ower of the two prop osals is still equivalent Keller has

later prop osed to structure the store mechanism so that the order of retrieving the sim

ple NP semantics from complex NP semantics do es not create syntactically illformed

logical forms This issue has also b een addressed by Hobbs and Shieb er and Car

penter b oth of whom identied the problem as one of dealing with free variables

None of these revisions address the problem p ointed out in this pap er esp ecially regard

ing Hobbs Shieb ers reading that we related to sentence a Carp enter

prop osed Natural Deduction scoping schemes that capture Montagovian quantifying

in utilizing assumption intro duction Scop e Intro duction SI and assumption discharge

Scop e Elimination SE SI resp ectively SE corresp onds to Co op ers store resp ectively

retrieve mechanism and Carp enters prop osal overgenerates readings in the same way

as Co op ers since no further surface structure information is checked at the time of SE

or Co op ers retrieve All of the systems that utilize some version of quantifyingin in

cluding the prop osal by Hendriks b elow generate b oth Hobbs Shieb ers reading

and Mays reading since the mo difying NP quantier of a complex NP can b e taken out

of the rest of the NP semantics except when it is inside a relative clause which has an

explicit no de such as a relative pronoun that is known to blo cksuch op eration Crucially

prep ositions are not known to b ehave as such

Hendriks prop osed syntactic typ e shifting rules argument raisinglow ering

and value raising as a middle ground b etween Montagovian syncategorematic prop osal

and Co op ers mo deltheoretic prop osal Roughly sp eaking if ob ject argument raising

is p erformed on the semantics of the transitiveverb b efore sub ject argument raising

the ob ject quantier will b e outscop ed by the sub ject quantier and vice versa Since

argument raising can b e done at any p ointofsemantic derivation one can always nd a

way of letting an NP quantier escap e from a given semantics The following shows an

example where Hendriks was able to derive readings from

Fred claims that every scho olb oybelieves that a mathematician wrote Through

the lo oking glass

Jong Park A Lexical Theory of Quantier Scop e

a cl aimf v boy v bel iev ev umathu writeu L

b cl aimf v boy v umathu bel iev ev writeu L

c cl aimf umathu v boy v bel iev ev writeu L

v boy v bel iev ev writeu L d umathu cl aimf

e v boy v cl aimf bel iev ev umathu writeu L

f v boy v cl aimf umathu bel iev ev writeu L

g v boy v umathu cl aimf bel iev ev write u L

h umathu v boy v cl aimf bel iev ev writeu L

Notice the way a mathematician gradually escap es out of its in situ p osition from a

through d The semantics of a mathematician is assigned a de dicto reading with resp ect

to bel iev e in a it is assigned a de re reading wrt bel iev e but is still outscop ed by every

schoolboy in b it is assigned a de re reading wrt bel iev e but at the same time a de

dicto reading wrt cl aim in c and so on If weregard de re interpretation of indenites

as a referential interpretation of indenites this prediction would b e at o dds with the

discussion in Section where Fo dor and Sag showed that referential indenites

The emb edded sub ject every schoolboy is interpreted do not takeintermediate scop es

either in situ as in a through d or out of the op erator cl aim as in e through h

This is surprising since it implies that sentence a b elow has a reading that sentence

b do esnt have ie for every schoolboy Fred claims that he left immediately Compare

this with the present theory that predicts that b oth sentences are unambiguous

a Fred claims that every scho olb oy left immediately

b Fred makes a claim that every scho olb oy left immediately

In Hendriks Flexible Montague Grammar quantifyingin for a particular NP is simulated

by successively raising the other argumenttyp e of the derived predicate that takes it

as an argument Since this is how the ob ject quantier outscop es the sub ject quantier

argument raising for an extensional verb is necessarily a blind typ eshifting rule in

the sense that b oth de re interpretations and quanticational interpretations must b e

computed by the same rule If it is in the right direction to distinguish the two kinds of

interpretations the rule must b e conditioned prop erly to accommo date this distinction

Quantier Raising Accounts

Quantier Raising is prop osed byMay as an op eration from SStructure to LF in

order to explain natural language quantication The discussion in this section is based

on May which explores three related prop osals According to May quantied NPs

undergo an autonomous syntactic op eration called Chomskyadjunction whichchanges

the structure a b elow to the structure b where x is a no de suchasSNPVPorPP

Notice that the structure b can receive a direct logical interpretation QX Y where Q

X andY are settheoretic denotations of the quantier the noun and the rest lab eled as

scope resp ectively The op eration QR is thus analogous to Montagovian quantifyingin

in the sense that it creates an abstraction However it is more syntactically restricted

since the op eration can not jump over Snode

a quantier noun

x np

b quantier noun e

x np n x n scope

But see also the discussion in fo otnote

Jong Park A Lexical Theory of Quantier Scop e

For example sentence a gives rise to two dierent structures b and c

a Every spy susp ects some Russian page

b every spy some Russian e susp ects e

s np s np s

c some Russian every spy e susp ects e

s np s np s

While these logical forms maybetaken to corresp ond to dierently scop ed readings

May noted that in the presence of the extended ECP suggested byKayne we are

forced to abandon the structure b ConsequentlyMay suggested to utilize the notions

of government and sequence according to whichthetwo NPs in the structure c are

memb ers of the same sequence since there is no intervening maximal pro jection and they

ccommand each other May prop osed the Scop e Principle such that memb ers of

sequences are free to takeonanytyp e of relative scop e relation page Later May

abandoned the extended ECP in favor of the P ath Containment Condition Pesetsky

that makes the same prediction but still maintained the Scop e Principle The

present theory and Mays theory would predict identical scop e ambiguous readings if

Mays theory could put in the same sequence the two NPs that can b e related in our

conjecture and vice versa This is not the case however since Mays theory do es not

incorp orate the extended notion of surface constituency as assumed in this pap er As a

result the two theories make dierent predictions esp ecially when surface constituents

contain no des that QR can not make NPs cross over such as the complementizer that

Consider sentence a whichMay called an instance of inverse linking In the

interest of letting every city bind the pronoun May suggested the logical form b but

immediately rejected it since a similar logical form c must b e rejected on the grounds

that the binding is into a syntactic island ie NP

a Someb o dy from every city despises it page

b every city someb o dy from e e despises it

s s np s

c whichcity someb o dy from e despises it

s s np s

d every city someb o dy from e e despises it

s np np np s

Noting that QR is not restricted to Sadjunction May prop osed the logical form d

instead in which every city remains inside NP by NPadjunction It can bind the pro

noun since it is in a ccommanding p osition over the pronoun Notice however that this

makes it necessary to haveanextrawellformedness constraint in the system since by

denition somebody from e and every city can outscop e each other one of the resulting

logical forms having an unb ound empty category e This do es not mean though that

the reading in which somebody outscop es every city is not derivable in the system since

every city can also PPadjoin as shown b elow This particular logical form is illformed

though since every city can not bind the pronoun

e despises it someb o dy every city from e

s s np pp pp

Notice that while Mays theory can derive b oth scopings it can not rely on the Scop e

Principle for quantiers inside NPs On the other hand the present theory makes use of

the same machinery for NPinternal quantiers and Sinternal quantiers alike

With this formulation it is interesting to note that Mays theory do es not allow

Hobbs Shieb ers reading either Consider whichshows a wellformed and close

The details of the extended ECP are b eyond the scop e of the present pap er

Jong Park A Lexical Theory of Quantier Scop e

but dierent structure Notice that acomp must b e NPadjoined so that it do es not bind

into an island cf d and most samp must b e Sadjoined after the complex sub ject

NP so that it do es not violate either the extended ECP or the PCC dep ending on the

version of Mays theory The Scop e Principle still allows the narrow scop e interpretation

of most samp in this structure In order to allow for Hobbs Shieb ers reading acomp

must b e able to change the relative scop e with resp ect to most samp indep endentofevery

rep of e but this is imp ossible Letting most samp VPadjoin do es not help either since

most samp will then b e outscop ed by b oth of the sub ject quantiers

most samp a comp every rep of e e saw e

s np s np np np s

This raises a question if Mays theory can actually accountforMays reading Maygives

the structure a for the reading in question May page Notice that the

ob ject NP is VPadjoined and furthermore that the mo difying NP of the ob ject complex

NP is NPadjoined to the Sadjoined sub ject NP It is not clear however if this structure

is indeed what Mays theory can derive since the prop osed NPadjunction is b etween two

NPs of an unrelated case Wewould rather exp ect that the structure b is what Mays

theory can actually derive and what is related to Mays reading Unfortunately though

b oth of these structures should b e ruled out since every lang binds into a syntactic island

cf d Since there are apparently no other ways to construct a structure for the

reading this seems to mean that Mays theory do es not account for Mays reading While

quantifyingin accounts derive b oth Hobbs Shieb ers reading and Mays reading Mays

QR accounts do not derive either one Consequently b oth accounts app ear to miss the

sub jectob ject asymmetry identied here

a every lang some stu e twodialofe study e

s np np s vp np vp

b every lang some stu e twodialofe study e

s np s np s vp np vp

Finallywe note that unlike the present prop osal b oth quantifyingin accounts and QR

accounts crucially distinguish the status of prep ositions and relative pronouns so that

the following sentences are argued to have a dierent range of readings

a Iknow someb o dy from every metrop olitan city in the States

b Iknow someb o dy who is from every metrop olitan city in the States

Conclusion

In this pap er wehave presented a novel conjecture that directly predicts when two

quanticational NP quantiers in a natural language sentence may b e scop eambiguous

In order to showhow the conjecture works wehavechosen to examine three English

constructions that allowmultiple NPs in a single sentence complex NPs containing PPs

complex NPs containing Whrelatives and transitiveattitude verbs While the claim is

that the data analysis allowed by the conjecture is b oth explanatory and descriptively

adequate the data wehave examined in this pap er are necessarily incomplete to show

this prop erly There are many other imp ortant and interesting English constructions that

are known to inuence scop eambiguous readings These include Whphrases quantier

b ound pronouns and other constructions such as complex NPs containing p ossessives

control and ditransitiveverbs and most imp ortantlyvarious standard and nonstandard

co ordination There is also an interesting relationship b etween extraction and co ordina

tion and quantier scop e that can b e veried with topicalization relativisation heavy

NP shift extrap osition and parasitic extraction rightno deraising leftno deraising

Jong Park A Lexical Theory of Quantier Scop e

and acrosstheb oard extraction among others There are also issues regarding weak

crossover phenomenon and sup eriority While Park contains an extensive discus

sion for most of these with resp ect to the prop osed framework it is evidentthatmuch

work needs to b e done in order to uncover the true nature of nonreferential quantiers

as opp osed to referential quantiers

Acknowledgements

The author would like to thank Mark Steedman for his advice and intuition He is also

grateful to Aravind Joshi Bonnie Webb er TonyKroch Janet Fo dor Anna Szab olcsi

Filipp o Beghelli Daniel Hardt Jason Eisner Matthew Stone and Nob o Komagata for

their help in clarifying the presented idea and checking the linguistic judgments and

esp ecially Jason and Matthew for kindly helping to write up some p ortion of the pap er

This of course do es not mean that they necessarily agree with the presented idea The

author is solely resp onsible for all the errors and mistakes in the pap er

App endix categoryWord SynSem

parseSynSemStack Buffer Answer

parseCat CatStack Buffer Answer

The following program or a fuller version

reduceCat Cat Cat

is available up on request from parseCatStack Buffer Answer

parse

parklinccisupennedu

reduceXY Y X

Complete Prolog Co de reduce Y XY X

op xfy v reduceXY YZ XZ

op yfx reduceYZ XY XZ

op xfx

op xfy THE LEXICON

usemodulelibrarylists dynamic category

go promptBuffer VERBS Features are suppressed

ifBuffer exit exit

interpretBuffer LFs Intransitive Verbs

outputLFs go

categoryslept ssleepXnpX

promptBuffer

nl writeQ readinBuffer Transitive Verbs

exit writeexit nl fail categorysaw sseeXYnpXnpY

categoryadmired sadmXY npXnpY

outputLFs writeLF lengthLFsL categoryvisited svisitXYnpXnpY

ifL categorystudied sstudyXYnpXnpY

writeunrecognized sentence categoryconfused sconfXYnpXnpY

uglywriteLFs categoryinterviewed

sintvXYnpXnpY

uglywriteLFs uwriteLFs categorywrote swriteXYnpXnpY

uwrite nl categorypleased spleaseXYnpXnpY

uwriteLFsN uwriteLFsN This is a computational trick to force

uwriteLFLFsN the association of iv with prep for tv

nl formatdN categorydanced sdanXYnpXnpYp

writeLF M is N uwriteLFsM categorywith p

interpretBuffer LFs Attitude Verbs

setofLF parse Buffer sLF LFs

categorythinks sthinkXSnpXssS

The Parser categorythink sthinkXSnpXssS

categorythought sthinkXSnp XssS

parseSynSem SynSem

SynSem SynSem standardSem Preposition

parseStack WordBuffer Answer

Jong Park A Lexical Theory of Quantier Scop e

categoryof nXNofXYnXNnpY Quantifiers

categoryof nXNSnXN

sSsofXYnpY qQ LFq QXNS

categoryof nXNSnXN assertzdetQ

sSsofXYnpY assertzcategoryQ

sLFqsSnpXnXN

WhRelatives probably incomplete entries assertzcategoryQ

sLFqsSnpXnXN

categorywho nXNSnXNsSnpX assertzcategoryQ

categorywhom nXNSnXNsSnpX sLFqnpYsSnpYnpXnXN

categorywhose assertzcategoryQ

nZNeveryXNofXZSnZN sLFqnpYsSnpYnpXnXN

sSnpXnXN assertzcategoryQ sLFqsSnpY

categorywhose sSsSnpYnpXnXN

nZNeveryXNofXZSnZN assertzcategoryQ sLFqsSnpY

sSnpXnXN sSsSnpYnpXnXN

categorywhom assertzcategoryQ

nZNSnZNsSnpX sLFqnpYsSnpYnpZ

sSsSnpXnpZ sSnpYsSnpYnpZnpX

categorywhom nXN

nZNSnZNsSnpX

sSsSnpXnpZ qone qtwo qthree

qfour qevery qsome

THAT Complementizer qmost qseveral qa

categorythat ssthatSsS READIN is from Jowsey

Proper Nouns There are missing entries readinWWs

getC readwordCWC restsentWCWs

categoryjohn npjohn restsent stop on CR or a lastword

categorymary npmary restsentW lastwordW

categorysusan npsusan restsentCW

categorybob npbob readwordCWC restsentWCWs

categoryjohn sSsSnpjohn lastwordW W WWs WWs

categorymary sSsSnpmary readwordCWC singlecharacterC

categorysusan sSsSnpsusan nameWC getC

categorybob sSsSnpbob readwordCWC

inwordCNewC getC

Common Nouns restwordCCsCnameWNewCCs

readwordWC

cnN Nplural LFn LF LFn X getC readwordCWC

assertzcategoryN nXLF restwordCNewCCsC inwordCNewC

assertzcategoryN getC restwordCCsC

nXLFnXLFnXL F restwordCC

assertzcategoryNplural nXLF

assertzcategoryNplural singlecharacter

nXLFnXLFnXLF singlecharacter

cnrepresentative representatives rep inwordCC C C

cnwoman women wmn inwordCL C C L is C

cnman men man inwordCCCC

cngirl girls girl inword

cnboy boys boy inword

cncompany companies com

cnsample samples sam lastword

cnstudent students stu

cnprofessor professors prof Standardize the logical form

cnletter letters let

cndean deans dean standardPhi standardPhi

cnpicture pictures pic standardthatPhi N

cnjudge judges jud standardPhiN

cnfrenchman frenchmen frn standardthinkXPhi N

cnrussian russians rsn standardPhiN

cnaspect aspects asp standardPhiPsi N

cndialect dialects dial standardPhi N standardPsi N

cnlanguage languages lan standardLFN

LF Q Md Var Phi Psi

Jong Park A Lexical Theory of Quantier Scop e

detQ variableN Var aXletXtwoXprofX

succN M standardPhiM everyXstuXadmXX

succM L standardPsiL writeXX

standardLF N twoXprofXeveryX

LF Q Phi Phi VarPsi stuXadmXXaXletX

variableN Var succN M writeXX

standardPsi M

standardPhi N Q Two professors interviewed three

students most pictures of whom pleased

variableN Var two judges

nameN L nameVar L threeXstuXmostXpicX

ofXXtwoXjudXpleaseX

succM N L is M N L XtwoXprofXintvXX

twoXjudXthreeXstuX

mostXpicXofXXpleaseX

Sample Runs

XtwoXprofXintvXX

The following shows sample outputs of twoXprofXthreeXstuX

mostXpicXofXXtwoX

the system without the degenerate quan

judXpleaseXXintvXX

tier semantics

twoXprofXtwoXjudX

Q Two representatives of three companies

threeXstuXmostXpicX

saw four samples

ofXXpleaseXXintvXX

fourXsamXthreeXcomX

twoXrepXofXXseeXX

Q Two girls think that three men danced

fourXsamXtwoXrepX

with four women

threeXcomXofXXseeXX

fourXwmnXtwoXgirlX

threeXcomXtwoXrepX

thinkXthatthreeXmanXd anX

ofXXfour XsamXseeXX

twoXrepXthreeXcomX

twoXgirlXthinkX

ofXXfourXsamXseeXX

thatfourXwmnXthreeX

manXdanXX

Q Some student studied two aspects of

twoXgirlXthinkX

every language

thatthreeXmanXfourX

everyXlanXtwoXaspX

wmnXdanXX

ofXXsomeXstuXstudyX

twoXgirlXfourXwmnX

thinkXthatthreeXmanXdanX

someXstuXeveryXlanX

twoXaspXofXXstudyX

someXstuXtwoXaspX References

everyXlanXofXXstud yX

Ajdukiewics Kazimierz Die

syntactiktische Konnexita t Studia

twoXaspXeveryXlanX

Philosophica An English ofXXsomeXstuXstudyX

translation app ears in Storrs McCall

editor Polish Logic Oxford

Q Two professors who interviewed every

University Press pp

student wrote a letter

BarHillel Yehoshua A

aXletXeveryXstuX

Quasiarithmetical Notation for Syntactic

twoXprofXintvXXwriteX

Description Language

aXletXtwoXprofX

Barwise Jon On Branching

everyXstuXintvXXwriteX

Quantiers in English Journal of

Philosophical Logic

everyXstuXtwoXprofX

Barwise Jon and Robin Co op er

intvXXaXletXwriteX

Generalized Quantiers Natural

veryXstuX Language Linguistics and Philosophy twoXprofXe

intvXXaXletXwriteX

Beghelli Filipp o The Phrase

Structure of Quantier Scope PhD

Q Two professors whom every student

thesis UCLA

admired wrote a letter

Carp enter Bob Phrase Meaning and

Categorial Grammar PhD thesis

Department of AI Universityof

The line breaks and indentations are added

for the output to t inside the pap er margin

Edinburgh

Jong Park A Lexical Theory of Quantier Scop e

Theories D Reidel Publishi ng Company Carp enter Bob Quantication and

pages Scoping A Deductive Account

manuscript

Krifka Manfred Denite NPs Arent

Quantiers Linguistic Inquiry

Co op er Robin Montagues semantic

theory and transformational syntax

PhD thesis University of Massachusetts

Lasnik Howard and Juan Uriagereka

at Amherst

A Course in GB Syntax Lectures on

Binding and Empty CategoriesMIT

DowtyDavid R Rob ert E Wall and

Press

Stanley Peters Introduction to

Montague Semantics D Reidel

May Rob ert The Grammar of

Publishin g Company

Quantication PhD thesis MIT

Fo dor Janet D and Ivan A Sag

gical Form Its May Rob ert Lo

Referential and Quanticationa l

Structure and Derivation MIT Press

Indenites Linguistics Philosophy

Montague Richard The Prop er

TreatmentofQuantication in Ordinary

English In Richmond H Thomason

Geach Paul T A Program for

editor Formal PhilosophyYale Syntax Synthese

University Press pages

Heim Irene File change semantics

Mo ore Rob ert C UnicationBased

and the familiarity theory of deniteness

Semantic Interpretation In Proceedings

In Rainer Bauerle et al editors Meaning

of the Annual Meeting of the Association

Use and the Interpretation of Language

for Computational Linguistics ACL

Berlin de Gruyter

pages

Hendriks Herman Studied Flexibility

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