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JOURNAL OF SEMANTICS AN INTERNATIONAL jOURNAL FOR THE INTERDISCIPLINARY STUDY OF THE SEMANTICS OF NATURAL LANGUAGE MAN...

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JOURNAL OF SEMANTICS AN INTERNATIONAL jOURNAL FOR THE INTERDISCIPLINARY STUDY OF THE SEMANTICS OF NATURAL LANGUAGE

MANAGING EDITOR: PETER BoscH (IBM Germany} REV IEW EDITOR: TIBOR K1ss (IBM Germany} EDITOR IAL BOARD: N.AsHER (Universiry of Texas,Austin}

R. BARTSCH (Universiry of Amsterdam) J. VAN BENTHEM (Universiry of Amsterdam) B. BoGURAEV (IBM Research, Yorktown Heights) D. S. BREE (Universiry of Manchester)

S.LEVINSON (MPI Nijmegen)

S. L6BNER (Universiry of Diisseldor� SIR JOHN LYONS (Universiry of Cambridge} A. MANASTER-RAMER (Wayne State Universiry) W. MARSLEN-WILSON (MRC, Cambridge)

B. GEURTS (Universiry of Tilburg} M. HERWEG (Universiry of Hamburg)

J. McCAWLEY (Universiry of Chicago) L. M.G. NoORDMAN (Universiry ofTilburg} R.A. VANDER SANDT (Universiry of Nijmegen) T. SANFORD (Universiry of Glasgow) R. ScHA (Universiry of Amsterdam) H. ScHNELLE (Universiry of Bochum)

P. N. JoHNSON-LAIRD (Princeton Universiry} H. KAMP (Universiry of Stuttgart} E. LANG (Universiry of Wuppertal)

W. W AHLSTER (DFKI Saarbriicken) B. WEBBER (Universiry of Pennsylvania) H. ZEEVAT (Universiry of Amsterdam)

H. BaEKLE (Universiry of Regensburg} G. BROWN (Universiry of Cambridge) 0. DAH L (Universiry of Stockholm) S. C. GARROD (Universiry of Glasgow)

P. HoPPER (Carnegie Mellon Universiry) L. R. HoRN (Yale Universiry) S. IsARD (Universiry of Edinburgh)

P. A. M. SEUREN (Universiry ofNijmegen) A. VON STECHOW (Universiry ofTiibingen) M. STEEDMAN (Universiry of Pennsylvania)

EDITORIAL ADDRE SS: Journal of Semantics, IBM Germany Scientific Centre, IWBS 7ooo-75, Postfach 8oo8 8o,D-7000 Stuttgart 8o, Germany. Phone: (49-711-) 6695-559. Telefax: (49-711) 6695-500. BITNET: bosch @ds0lilog.

New Subscribers to the Journal of Semantics should apply to the Journals Subscription Department, Oxford Universiry Press, Walton Street, Oxford, OX2 6DP. For further informacion see the inside back cover. Volumes 1 - 3 c [c �a 1\ clb]] [-.(aLb) => 3 c V d [ d �c � (d � a 1\ d � b)]] [at(a) => -.3 x [x 3 a [a � c 1\ at(a) ] ]] ]cmplx(b) � ctbl(b) 1\ -. at(b)]

The operation u is assumed nor ro be sensitive wirh respect ro the mass-counr distinction in order to allow objects in D which are neither proper masses nor

Downloaded from jos.oxfordjournals.org by guest on January 1, 2011

The properties ofu and A are specified wirh help of rhe axioms ]A E9 1]-]A E9 (>] and ]Aar].7

Carola Eschenbach I I

countable objects (cf. Link ryXJ; Krifka 1yXya). Bum ( 1 yX 5 ) gave an elaborate analysis of such 'ensembles'. The inclusion of rhe subset A of D is due to the assumption of an objective atomicity criterion, i.e. a criterion that judges which objects are atomic individuals, and which are not. In the latter discussion I will need two partial functions, mapping su bsets of D on elemenrs of D. max,; is rhe partial function defined by: _

max,;(S) :=

{x,

ifx E S and 't/y E S[y�xj . un d e f'med , orI 1erwise ·

If S only comains one element, e.g. S (xJ, we get x � max,;(S). There are two ways for max,;(S) ro yield no value: S is empty or S has more than one elemenr, ' bur it does nor conrain any elemenr which has all the other elemenrs of S as parts. The generalized sum -operation is given by: =

·= ·

z E D [z { undefined, otherwise x

,

if't/

#

x

� 't/ y E S [y

#

z]]

Since sum is not defined on0 at least, it is a partial function. To assume sum to be a total function is exactly to assume that D is a complete Boolean algebra. Bur as long as we apply this function to non-empty finite subsets of D, we will always get a well-defined result. The function sum coincides with max" in the domain on which both are defined. As long as S is finite, the condition sum(S) max,. (Su) holds. Bur if S is not finite it might be the case that max" (Su) is undefined while sum(S) is defined (cf. L0nning 1 989). Since max,. and sum are operators on sets, the corresponding conditions can only be given by definition-schemata in first-order predicate logic. The symbols I will use in LPN as counterparts to these operators are 6 and o, respectively. =

[ DS6) [ DSo]

6(P) =LX jP(x) 1\'t/y [ P(y) � y � x]j o(P) =LX 't/ z [x lz � 't/ y [P(y) � z ly]]

Nouns are represented by unary predicate symbols and relational symbols in LPN, which are mapped by [.] to subsets of D and D X D (or on) respectively. In contrast to mereological attempts to represent mass nouns, I assume that (predicative) mass nouns denote sets of mass-entities, rather than mass-entities. This allows a uniform treatment of mass nouns and counr nouns (Link 1 98 J). A proper predicative count noun will have a subset of C as irs denotation, while a proper predicative mass noun has a subset of M as its denotation. Nouns which allow for both occurrences may have elements of M and C in their denotation (cf. Link 1 9X 3; Pelletier & Schubert 1 989). The objects that are referred to by group nouns are included in A and C in the same way as the other individuals. The represenration of determiners will be discussed later.


sum(S)

12

Semantics of Number

The structures presented for the plural interpretation are similar enough to the structure Link introduced in Link ( 19R 3) to discuss his suggestion for the meaning of plural with respect to nouns on this basis.

6 SE M A N T I C S O F N U MB E R : L I N K

[D*] (0181]

[*P] [181P]

= =

(sum(X)IX � [ P] [*P]\A

A

X# 0}

The plural operators are only defined on unary predicates. From chis and the observation in Section 2, the question arises as to whether it is possibk to define corresponding operators applicable to relations. We can reformulate Link's description of the operators • and 181 in our formalism as in (Sa) and (8c). I will discuss the operators on relational nouns chat correspond to *, because this operator is always introduced as che·representation of plural in approaches following Link's suggestions (e.g. L0nning 1 9R9). (R) a. b. c. d.

* /..P/..x [ctbl(x) 1\ V z � x ( at(z)=> P(z)]] *GIRL f..x [ctbl(x) 1\ V z � x [at(z)=> GIRL(z)]] 181 /..Ph [cmplx(x) 1\ V z < x [at(z) => P(z)]] 181GIRL f..x [cmplx(x) 1\ V z < x (at(z) => GIRL(z)]] =

=

=

=

The corresponding operators applicable to relational nouns should allow us co represent the phrases (9a-e), which give all possible combinations of singular and plural head nouns and singular and plural complements. The difference berween (9d) and (9e) is not syntactic or semantic but due to world knowledge. While persons of different sex may have common children, chis is not possible


There are essentially two different approaches to the semantic representation of plural based on a lattice-structured universe. The first one is presented by Link ( 1 9R3) and will be described in this section. The second approach is given by Krifka ( 1 989a) and will be oudined in the next section. Link's basic assumption is that the core meaning of a count noun is given by the meaning of the singular form of that noun. As a consequence, there is no operator in his logic which corresponds to the meaning of singular. In his analysis, nouns such as boy are mapped to predicates which have only atomic objects in their denotation. He defines two operators, applicable to such predicates, called 'the plural operator' (*) and 'the proper plural operator' (181). The plural predicate is derived from the basic predicate by applying * or 181 to the singular predicate. The semantics of* is the function that maps a set of atoms in D to its complete u -closure (i.e. a closure operator based on sum); 181 excludes the atoms from chis set, allowing only complex objects in the denotation

Carola Eschenbach

'1

3

for persons of the same sex. The semantic representation of these sentences should allow both interpretations. (9) a. b. c. d. e.

son of Peter sons ofjohn son of Peter and Ann sons ofjohn and Mary sons ofjohn and Peter

( 1 0) A.x[SON(PETER)(x)] The obvious transformation of the plural operator to the relational nouns yields ( 1 1 a), which is of use in interpreting (9b) as in ( 1 1 b). ( 1 1 ) a. * = ARAyAx [ctbl(x) 1\ 'tJ z � x [at(z) => R(y)(z)]] b. Ax [ · *SONQOHN)(x)] = Ax [ctbl(x) 1\ V z �x [at(z) => SONQOHN)(z)]] ·

In order to represent the sentences (9c-e), we have to assume an additional operator to allow for complex arguments. This operator is not j ustified by syntactical or morphological features in natural language. It has to be applied if the internal argument of the relation is specified by a plural noun phrase. Such an operator must be applicable to the basic (singular) relation in order to get the right interpretation of (9c). The obvious interpretation in this framework is ( 1 2a). The translation of (9c) is given in ( 1 2b). ( 12) a. * · = ARAyh [ctbl(y) 1\ 'tJ w � y [at(w)=> R(w)(x)]] b. Ax [* · SON(PETER$ ANN)(x)] = Ax [ctbl(PETER$ANN) 1\ 'tJ z �PETER$ ANN [at(z) => SON(z)(x)]] - h [SON(PETER)(x) 1\ SON(ANN)(x)J9 This operator is guitc strong if we consider the examples (9d, e), in which the noun is plural, and will only yield the right representation for example (9d), as in ( 1 .2c), assuming that the common children ofjohn and Mary are referred to. ( 1 2) c. Ax [*· ( · *SON)QOHN$ MARY)(x)] Ax [ctblQOHN$ MARY) 1\ V z �JOHN$ MARY [at(x) =>

·

*SON(z)(x)]J


As stated above, Link assumes that the basic meaning of the nouns is only defined on atomic objects. This restriction is meant to reflect the distributive nature of predicative nouns: each of some boys is a boy. To make the problem of this assumption as clear as possible, I take the strict position that all argument places of a relational noun such as son , member, riferee , teacher orfriend should be restricted in the same way (i.e. [SON] � A X A). With respect to these nouns, this assumption is justified since, for example, each of the sons ofjohn is a son ofjohn, and a son ofPeter and Ann is a son of either of them. Such a basic form can directly be used to represent (9a) as in ( 1 0).

1 4 ·Semantics

of Number

Ax [ · *SONQOHN)(x) 1\ ( · *SON)(MA RY)(x)] Ax [ctbl(x) 1\ V z �x [at(z) => SONQOHN)(z)] 1\ V z �x [at(z) => SON(MARY)(z)]] Ax [ctbl(x) 1\ V z �x [at(z) => SONQOHN)(z) 1\ SON(MARY)(z)]] To represent (9e) as in ( 1 3b) we would like to have an operator ** as in ( 1 3a), which cannot be the result of combining * · and · *. J..IUyh [ctbl(x) 1\ V z �x [at(z) => 3 w �y [R(w)(z)]] /\ ctbl(y) 1\ V w �y [ at(w) => 3 z �x [R(w)(z)lJ b. A.x [**SONQOHNEBPETE R)(x)] - Ax [ctbl(x) 1\ V z �x [at(z) => 3 w �JOHN$ PETER [SON(w)(z)]] 1\ ctblQOHNEBPETE R) 1\ V w �JOHNEBPETER [at(w) => 3 z �x [SON(w)(z)]]] - A.x [ctbl(x) 1\ V z �x [at(z) => SONQOHN)(z) V SON(PETER)(z)] 1\ 3 z �x [SONQOHN)(z)] 1\ 3 z �x [SON(PETER)(z)]]

( 1 3) a.

••-

·

( 14) a. b. ( 1 s) a. b.

George is a member of the committee. George is one of the members of the committee. George is an owner of this house. George is one of the owners of this house.


If we were to assume that the proper counterpart to (X) is in fact **, and * and · * arc the consequences of restricting one of the arguments to atomic objects, (e.g. • · R - A.yA.x [**R(y)(x) 1\ at(x)] = A.y/..x [ctbl(y) 1\ at(x) 1\ V w � y [at(w) => R(w)(x)]] ), we need to assume that singular has a semantic counterpart, which is only used for relational nouns. If we considered the operators which correspond to Link's proper plural operator 0 (i.e. · 0, 0 · and 00) this possibility would not exist. One might now, as one of the referees did, argue that it is not necessary in Link's approach to assume that the basic meaning of a relational noun is restricted to A X A. In this case, one might assume that the relation SON holds berween any male person and his parents. As a consequence it is necessary to state the disrributivity of this relation separately, and the operator ** is still needed to be able to represent (9b, e) appropriately. Thus, plurals would be represented by * in the case of a predicative noun and by ** in the case of a relational one. However, there are still some relational nouns to be considered which behave differently from son and the others mentioned above. While ( 1 4a) is the sentence preferred to describe the relationship berween George and the committee, ( 1 sa) seems a bit odd in contrast to ( 1 sb). Thus, ownership might be ascribed to a collection of people, from which the relation berween one of them and the property is derived.

Carola Eschenbach

1s

(3) b. The sisters entered the room. Four sisters entered the room. ( 1 6) 1-RJ..x [ctbl(x) 1\ 'V z, w �x [(at(z) 1\ at(w)=> (R(w)(z) � (w "# z))]] Assuming with Link that the basic meaning of count nouns is restricted to atoms, one would need several operators to get all possible interpretations of relational nouns and one would have problems with the proper representation of nouns like owner. Thus it appears that Link's plural operators are not the representation of the morphological feature plural. What he defines is, rather, a way of obtaining a cumulative predicate in a structured domain on the basis of a non-cumulative one. It is justified if we consider how we may evaluate a plural predicate with respect to a collection of objects, i.e. on the conceptual level of language understanding. But these operators cannot correspond to the feature plural, as long as there are collective plural noun phrases such as the sisters , the murderers qfSa muel Edward Ratchett or plural nouns without a corresponding singular form such as people and German: Leute , Eltern .

7

SEMANTICS OF NUMBER: KRIFKA

Krifka ( 1 989a) assumes that the meaning of a count noun should be represented by a complex structure that is based on the core meaning of the noun which does not depend on number. I will use GIRL to represent the core meaning of the noun girl. I ts denotation is assumed to be closed with respect to the join operation. The difference berween mass nouns and count nouns is reflected in Krifka's approach by a difference in type. While a mass noun is represented by a unary predicate on mass-entities, a count noun is mapped to a more complex structure: a relation berween individuals and rationals. Girl is represented by


The noun owner, like invent or, auth or, and murderer, might in irs plural form behave collectively with respect to the referential argument place. In Link's framework, these nouns and their behaviour with respect to number would have to be treated differently from predicative nouns and relational nouns like son and member . The integration of relational nouns is also a serious challenge to the approach ofVerkuyl ( 1 98 1 ). Since it is based on pure distributive interpretations of count nouns, phrases as sons ofPeter andjolm and owners ofthis h ouse cannot directly be integrated. It would either need revision with respect to the type of the basic meaning of the nouns or additional operators to handle number and numerals. To get the full paradigm of possible plural noun uses, we would in both frameworks have to assume an additional operator to get the reciprocal reading of(3b). This can be formulated as ( 1 6).

16 Semantics of Number

( I 7), where NE(GIRL) is a function which maps an object to the number of (atomic) girl -parts it has. ( I 7) 1-r, x [GI RL(x) 1\ NE(GI RL)(x)

=

r]

Krifka ( I 989a: 72) specifies the semantic plural -operator as depicted in ( I 8), 1 0 and girls is represented as in ( I 9). The plural-operator is not used to represent phrases like three girls (represented by (2o) ), in which he judges the plural as a purely syntactic occurrence. So this approach seems to explain why only singu lar count nouns cannot appear as bare forms in German and English.

=

Krifka does not introduce a semantic operator to represent singular. As a consequence, he has to assume that all determiners (definite article, quantifiers, possessive pronouns, etc.) have at least two different semantic representations, one applying to singular count nouns and one applying to mass nouns and plural count nouns. While plural (which is not determined by agreement) is interpreted at the noun itself, and the singular of mass nouns is not reflected at all, the representation of the singular feature belonging to a count noun is only forced by the correspond{ng determiner (c£ (2 I ) ; Krifka I 989a: 76 f£). 1 1 In a way, Krifka assumes that the difference between singular and plural should be represented by a difference in type. He sees the difference not between the referents of such phrases (as with Bartsch I 973; Hausser I 974; and Bennett I 97 5), but between the denotations of the nouns. ( 21 ) Det.

the 1 10 each

pl. mass

sg

1-X[MAX(X)] 1-Y , X[Y n X= 0]

1-Z[MAX(Z( I ))] 1-Z, X[Z( I ) n X= 0] 1-Z, X [Z( I ) � X] 1-Z, x [Z( I )(x)j

a

E

1-X, x [X(x)]

Krifka cannot assume that the plural-operator he specifies corresponds to the plural morpheme of the noun. This becomes obvious if we regard pluralia tan tum as Leute (people , folk), Eltern (parents), and Gesclzwister (siblings), which denote complex objects and do not have (at least in the German examples) a sin gular form. It seems to be most natural to assume that the meaning of plural is fixed within their lexical entries. According to ( I 9) it seems reasonable to assume the representation (22) as the lexical entry for Leute. The existential quantifier must be included in the representation, because there is no plural morpheme which could supply it.


( I 8) /-Z, x 3 r [Z(r)(x) 1\ r � I j ( I 9) AX 3 r [ GIRL(x) 1\ NE(GIRL)(x) = r 1\ r � I ] (2o) h [GIRL(x) 1\ NE(GI RL)(x) 3]

Carola Eschenbach

(22)

17

h 3 r [LEUTE(x) 1\ NE(LEU TE)(x) = r 1\ r � 1 ]

X A N A LT E R N A T I V E A P P R O A C H TO T H E S E M A N T I C S O F N U MB E R

Although Link and Kritka introduced a structure set as the domain of reference, their analysis of number is not directly based on the embedding of the referents in this domain, bur on predicates with different behaviour with respect to it. The analysis that I present here is based purely on the embedding of the referent of the noun phrase into the domain of reference. In addition, I assume that, even if number is not always interpreted semantically, singular should have a semantic counterpart, too. In this section, I will concentrate on count nouns and not discuss the question of how to define a semantic singular operator that allows a semantic interpretation of singular in mass noun occurrences. I will use the terms 'count singular' and 'count plural' to refer to the singular and plural features of count nouns. In Section 1 2 I will generalize my analysis to include mass nouns.


Based on this representation, we could not combine Leute with a numeral as in drei Leute . To allow for the representation of drei Leute in a compositional manner, Kritka has to assume that there is a number- independent representa tion of Leute in the lexicon, whose singular form is not realized in German. The same holds for Eltern and Geschwister, and there would be no explanation for the lack of the corresponding singular forms. (Morphologically, all forms would be regular if they were derived from the singular forms Leut , Elter, and Gesc lzwist or Gesc lzwister, but these singulars simply do not exist in German.) According to this, Kritka (pers. comm.) assumes that the plural morpheme has no meaning at all and associates the plural operator with the determiners (in bare plurals not occurring directly in the surface structure) which trigger plural agreement. But according to this claim, he would have to assume that determiners like t lze and n o are ambiguous in three ways (singular, plural, mass) and E as doubly ambiguous (plural and mass), e.g. the,g: /..Z [MAX(Z( 1 ))], tlze pi: AZ [MAX(/..x 3 r [Z(r)(x)])], t lzemass: AX [MAX(X)] . So this assumption is not consistent with the description of the determiners given in Krifka ( 1 989a). I am not convinced by Krifka's analysis, which leaves the semantics of number not to the noun but to the determiners. And even his weak interpretation of plural (including r � 1 instead of r > 1 in the representation) proves inadequate for pluralia tantum. If the plural predicates include the atoms in their denotation and if it were a purely pragmatic choice (as he argues) that we use the singular form when we refer to atomic individuals, it should always be possible to use these pluralia tantum to refer to atomic individuals which is not the case.

Ig

Semantics of Number

I do not assume that the semantic core of a count noun is restricted to the atomic part of the universe as Link ( 1 98 3) did for (distributive) count nouns and L0nning ( 1 989) did more or less implicitly. I assume, rather, that the core meaning of all nouns are closed with respect to u as described in (2 3) and (24) (c£ Allen 1 980: S 54; Scha 1 9X 1 ; Link 1 9X 3 with respect to mass nouns; Krifka 1 9X9a). Krifka ( 1 9X9a, b) suggested that verbs should be analysed in a corres ponding way. (2 3) predicative: GIRL(e, t) [APJ P(a) A P(b) � P(aE9b)

As for the problems of Link's and Krifka's approach to the semantics of plural, it is obvious that the main role of number is to specify the complexity of the referential argument of the noun (i.e. the referent of the noun phrase if it refers). Thus the semantic core of number should be given by a unary predicate as in (2 5 ). (2 5 ) Count plural: cmplx Count singular: at

As we have seen above, at and cmplx involve countability of the argument, which is a result of my restriction to count nouns. To fit this into a compositional approach to semantics which takes care of the morphological structure, one can specifY the number operators, which are applied to the numberless representation of the noun phrase, as in (26). The combination of the number operators with relations is carried out by functional composition.13 at and cmplx are not necessarily realized as special morphemes attached to a noun. They may, as we will see later on, be included in the meaning of determiners, quantifiers, or nouns (e.g. pluralia tantum). My main claim is that wherever we interpret count singular or plural at all, it should be done by using these operators. The operators defined here restrict the denotations of the noun to special parts of the domain of reference. (26) Count plural: ct-pl :- A.PA.x [cmplx(x) A P(x)] Count singular: ct-sg :- A.Ph [at(x) A P(x)] This leads to translations from English to LPN as given in (27). Whether a phrase like the sons ofPe ter an d Ann refers to their common sons or to the sum ofPeter's sons and Ann's sons is not a matter of different readings of the phrase, but only a question of the kinds of models in which this phrase is evaluated. Based on the analysis given, it is possible to have lexical entries of Leu te ,


(24) relational: SON(e, (e, t)) 12 [ARJ R(a)(b) A R(c)(d) � R(aE9c)(bE!1d)

Carola Eschenbach 1 9

NL expression singular form

plural form

girl(s) group(s) son(s) son(s) of Peter son(s) of P'ercr and Ann

Ax [cmplx(x) 1\ GIRL(x)] Ax lcmplx(x) 1\ GROUP(x)] 1-yh [cmplx(x) 1\ SO N (y)(x)] Ax lcmplx(x) 1\ SON(P)(x)] h [cmplx(x) 1\ SON(PEB A)(x)]

h [at(x) 1\ G I RL(x)] Ax [at(x) 1\ GROUP(x)] 1-yh [at(x) 1\ SON(y)(x)] Ax [at(x) 1\ SON(P)(x)] h [at(x) 1\ SON(PEB A)(x)]

To describe rhe reciprocal use of relational nouns, I will use an operation rec , which maps binary relations to predicates (i.e. an operation of type ((e, (e, t)), (c, t))). It is defined by [Dr•c]. The result of applying rec to the relation SISTER is given in (2X)_I4 [Dr•c] rec : /.RA.z [cmplx(z) 1\ V x , y < z [x l y � R(x)(y)]] (2X) S I STER •c - Az [cmplx(z) 1\ V x, y < z [x L y � SISTER(x)(y)]] -

'

The possible interpretations of relational nouns that are nor anti -symmetric are given in (29). The plural form of these nouns is ambiguous between the (2 t)) NL expression

singular form

plural form

sisrer(s)

/.yh [at(x) 1\ S I STE R(y)(x) ]

1-yh[cmplx(x) 1\ S ISTE H. (y)(x) ] Ax [SISTEw•c(x) ] h [cmplx(x) 1\ SI STER(P)(x) ]

sisrer(s) of Peter Ax [at(x) 1\ S ISTER(P)(x)]

relational and rhe reciprocal reading. The only nouns I know that do not fulfil this strict condition in the reciprocal occurrence are nouns like cousin and neighbour. For example, if we have three people, two of whom live together and the third is their neighbour, then we might refer to them as the ne��hbours. Bur I think that this should not lead to a weaker interpretation of the reciprocal use, but be seen as due to the fact that live together specifies a stronger relation than ne��hbour. The need for the strong reciprocal reading is obvious in that the neighbours cannot be used to refer to the inhabitants of a city, even if there is a chain with respect to the neighbour relation between each pair of inhabitants. As pointed out by a referee, the twins in (3o) docs not obey the strict reciprocal interpretacion. But it seems to me that in this case twin is used predicatively. I


Eltern , and Ceschwister which already include rhc meaning of plural in rhc semantic representation.


would not be surprised if, of some twin-pairs, only one attended the meeting, or if triplets, etc. were expected to come, too. (3o) The twins gathered in the big stadium for their annual twin -festival.

[Drec 1 ] rec : = /.R/..z [:3 x [x < z] 1\ 'V x, y < z [x l y => R(x)(y)JJ [Drec2] rec : = f.R/..z['t/3 x, y < z [x l y => R(x)(y)ll

Though some languages have a more elaborate number system, I will not go further into their analysis. The approach presented here can be extended to describe other number categories. The only additional category that may be useful in the analysis of English and German is the dual. The expressions both , either, neither, and pair may be seen as a lexicalized version of this category, which appears in some other languages as a morphological category (e.g. Greek; cf. Humboldt 1 X27). If we reconsider what was said in Section 4, we see that expressions that agree in number with a noun or noun phrase usually take the referential argument of the noun to fill the external argument position. The approaches of Verkuyl, Krifka, and Link do not provide a basis for explaining the number feature of verbs. Although the subject-verb agreement has nearly become a pure syntactic restriction in languages such as German and English, its semantic basis is due to the double specification of the complexity of the referent of the subject noun phrase. This background is still obvious in coordination (29a), the treatment of group nouns in British English (29b), and pseudo-quantifiers like the German eine Menge or a lot (29c) (cf. Hoeksema 1 9X 3 ; Barker 1 992; Pollard & Sag 1 99 1 ).


We might assume that the operation rec represents the semantic core of a lexical rule of reciprocals that is applicable to all relational entries, i.e. not only to relational nouns but also to transitive verbs and prepositions. 1 5 It is not necessary to restrict it to those relations which allow for symmetry. Anti-sym metric relations are excluded because they do not allow for a proper clique. But we could also mark anti-symmetric relations in the lexicon to avoid using the operation in these cases. On the other hand, this operator is meant to describe how reciprocal uses of relational nouns are interpreted. Whether it has to be assumed as being applied to relational nouns on the semantic level mainly depends on the general framework of meaning composition assumed. The reciprocal use of relational nouns is restricted to complex objects (i.e. in the plural form). A similar restriction applies to reciprocal forms of verbs. I regard this as a consequence of the semantics of reciprocals: the quantification has existential impact, i.e. is not allowed to yield truth because of emptiness of the domain. We can reformulate [Drec] as [Drec l J or [Drecz], where 't/3 is a general quantifier with existential impact.


(3 1 ) a. John and Mary walk home. Tea and coffee i s in the kitchen. The committee is old. b. The committee are old. c. Eine Menge Kinder spielen im Ho( Eine Menge Sekt ist getrunken worden. A lot of champagne has been [A lot of children play in the yard. drunk.]

9 THE M E A N I N G O F THE D E F I N ITE ARTICLE Some natural language determiners can occur in front of both mass nouns and count nouns. It seems reasonable to represent them on the semantic level as a single operator on nouns. One of these determiners is the definite article (c( Lobner 1 9X s: 2Xo). Lobner described the behaviour of the definite article as follows: The children refers ro the entire complex object ro which children applies the child to the entire obj ect to which c/,ifd applies (which is necessarily only one child); and the snow to the entire object to which snow applies. (Li:ibner 1 yX5: 2X2).

We have already seen that Krifka ( 1 9X9a) had to assume two or three different representations of the definite article, one which combines with singular count nouns and one which combines with plural nouns and mass nouns. Link ( 1 9X 3, 1 99 1 ) assumes four different semantic operators to represent the definite article: singular (l), plural (o), proper plural (o"), and mass (J.A.). In Link ( 1 99 1 ) he defined l and o " as restrictions of o. L0nning ( 1 9X9) discusses Link's o operator and its consequences with respect to the model theory. On the semantic level, they associate this operator with the generalized sum operation. Thus it corresponds in the main respects to the operator o as defined here. There are mainly two problems with this analysis. The function sum allows for sum(S) � S. It will always be the case that P"(o(P)) holds (if o(P) refers), bur in this approach it is not possible to handle predicates which do not have the cumulative reference properry (c( Krifka 1 9X9a: 7 4 f(). The second problem with this analysis is that the plural defi n ite article is not


As a consequence, it does not matter whether we assume that a number feature which occurs as a consequence of agreement (with referring noun phrases) is represented or not: representing it semantically yields redundancy, bur no conflict. Though the determiner of a noun phrase influences the status of the referential argument of the noun, it may determine whether number should be interpreted or not. This will be elaborated in Section 1 1 , · where I discuss quantificational determiners.

22 Semantics

of Number

applied to one of the plural predicates *P or 0P, but to the singular predicate P. This does not in any way correspond to the facts in languages like English, where there is only one definite article, which is not sensitive to countability and number, and the nouns are the expressions which bear the number feature.

(32) [b(P)] = max,.([P])

Assuming that we represent the definite article by 0 and the number distinction as described above, the representation of some definite noun phrases are listed in ( 3 3). The evaluation in the last two columns is with respect to the models M I and M2. M I = (((a, b, d, e, f, g, h, j . . .) u, ( a, b, d, e, f, g, h, j), u), [.] M 1) with [JOHN]M 1 = j, [BOY]M 1 = {a, b, a u b), [SON]M 1 = {U. a), U, b), U, a u b)), [SISTER] M 1 = ( U, e), U. h), (h, e), (e, h), (f, g), (g, fjju and from this [SISTERrec] M , = {e u h, fu g) M2 = (((a, d, e, f, j . . .) u, ( a, d, e, f, j), u), [.] M2) with [JOHN] M2 - j, [BOY] M 2 - {a), [SON] M 2 = (U, a)), [SISTER] M2 = {(d, e), (e, d), U, fjju and from this [SISTERrec]M2 = (d u e) . (3 3) Expression

LPN

all

the boy

b(ct-sg(BOY)) b(ct-pl(UOY)) b(ct-pl(SISTEH'«)) b(ct-sg(SONUOHN))) b(ct-pl(SONUOHN))) b(ct-sg(SISTERUOHN))) b(ct-pl(SISTEI�UOHN)))

maxd[ BOY] n A) max5 ([uov] n (C:\A)) max5 ( SISTEw••] n (C\A)) max., ( SON] ([JOHN] ) n A) max6 ([SON] ([JOHN] ) n (C\A)) maxs ([SISTER] ([JOHN]) n A) max,. ([SISTER] ([JOHN] ) n (C\A))

the boys the sistl'rs the so11 of John rhe so11s of Johll the· sister of John the sisters of John

models

[

undef a aub

undef

undef d u e· undef a aub

undef

undef cuh

undef

If we want to describe definite mass noun phrases in this framework, we have primarily two possibilities. Either we assume that the domain of masses is structured by the same relation �; (as I did in the description of the LPN strucmres), or we assume that the ordering relation in max., is a parameter


Regarding the closure properties of noun denotations as described above, the most natural way to get the 'entire object' interpretation relative to a predicate in a semi- lattice structure is to use the maximum function relative to the underlying ordering relation �; (c( Krifka I 989a: 7 5 , who also used max., but defined it on the basis of sum, and Kadmon I 990). Correspondingly, I will use the operator 0 to represent the definite article. The logical type of this operator is ((e, t), e) (i.e. it maps predicates to entities); its model theoretic interpretation is given in (32).1 6

Carola Eschenbach

23

which is contextually specified: the sortal properties of the elements of the set determines which ordering relation is used. 1 7

10 I N D E F I N ITES A N D NUMERALS

It has been widely recognized that only singular count nouns allow the indefinite singular article while plural count nouns and singular mass nouns allow for bare forms. If a noun (like apple) can occur as a mass noun and as a count noun, an occurrence of the indefinite article indicates that it should be interpreted as a count noun, while a bare form indicates that it should be interpreted as a mass noun. Krifka's suggestions on the meaning of count nouns give an explanation for this difference but get into difficulties with all unspecific determiners, as I showed above. Because the indefinite article a ( n ) only allows singular nouns, I assume that the singular feature of the noun in this context is due to agreement with the article and not interpreted. The basis of the indefinite singular article is at (as in (3 s) ) i.e. the meaning of singular in connection with count nouns. This explains why mass nouns do not occur with the indefinite article and nouns which allow for count and mass occurrences are interpreted as count occurrences in connection with it.'8 ,

(3 5) a (n )

/.N/.x !at(x) 1\ N(x)]

If we assumed that the effect of bare forms is semantically determined, we would have to assume that there is an empty determiner with special semantic effects as in ( 36). This assumption is quite problematic because there is no syntactical or morphological counterpart to this operator (c( Lobner 1 9H6).

(36)

E

/.Nh [-.at(x) 1\ N(x)J

It is also not clear in which contexts this restriction on bare occurrences really holds. Regarding bare nouns in prepositional phrases like out of town/bed , at


I n English (and German), indefinites can appear as bare forms as i n (pa, b), or with a determiner from a certain class, which includes the indefinite articles a ( n ) and some (pc-f), numerals (pg, h), and a few more. (34) a. Mary puts water into the soup. b. Bill likes books . c. John owns a donkey . d. Some man entered the pub. e . Mary puts some water into the soup. f Some book belongs to me. g. One book belongs to me. h. Three boys entered the pub.


(37) some

t.N/..x [N(x)J

The combination of a numeral with a count noun usually yields a stronger restriction on the complexity of the object referred to. It has become quite common to analyse them as generalized quantifiers or to map them directly to integers or rationals (cf Krifka 1 989a). I will analyse numerals as operators with behaviour similar to the number operators (38) (cf Verkuyl 1 98 1 ; Bunt 1 98 5). (38) one three

t.N/..x [ctbl(x) 1\ quantity(x) - 1 1\ N(x)] I.NI.x [ctbl(x) 1\ quantity(x) - 3 1\ N(x)]

The function quantity, which maps entities to natural numbers, can be defined as in (39). 1t has some similarity to Krifka's NE function, and to attain a more elaborate analysis of various numeral constructions (e.g. measure and classifier constructions) one should also include the possibility of modifYing it with respect to various criteria of counting or measuring (cf Eschenbach 1 99 1 ). The main difference between my approach and Krifka's is that he assumed NE to be a part of the noun meaning, while I include quantity in the semantics of the numerals. (39) [quantity] :-

{ C --- IN

x ..... l {a E A ! a �; xJ I


home/sclzool!work , in bed!sclzool/church!prison , in part , in coordinating structures like kitchen and bath , week after week , day in, day out , or in complex phrases such as in time of war, in point/case of, sort ofthing/person , kind ofsin/situation , it is not clear whether these bare nouns should be regarded as mass nouns. It is rather a question of whether the bare nouns in these contexts refer to entities at all. I think that the restrictions on bare forms have to be analysed with respect to their referential behaviour and discourse function, but such an analysis is beyond the scope of my discussion. Allen ( 1 980) gave an explanation of the effect of bare forms as referring expressions on a pragmatic basis: 'If his listeners do not already know the countability of the NP reference, the speaker must make it known to them' (Allen 1 980: 542). This rule is only applied in connection to indefinite noun phrases. Accordingly, using a bare singular form signals that the referent is not countable. The main difference between the indefinite noun phrases with the determiners a ( n ), some or without any determiner corresponds to the extent to which these phrases are used referentially, and I assume that this should be handled on a level different from semantics. So I take (3 5) as the semantic representation of a (n) and (37) as the semantic representation of some.


Numerals can also occur in partitive constructions as three '?[the boys . Assuming (4o) as the semantic counterpart of the partitive of, we get regular inter pretations of such phrases as in (4 1 ). (4o) of (4 1 ) '?[the boys three ofthe boys

1-.y/..x [x :::;; y] 1-. [x :::;; o(ct-pl(BOY) )] Ax [ctbl(x) 1\ quantity(x) �

I I

3 1\

X :::;; o(ct-pl(BOY))]

Q U A N T I F I E RS

(.p )

N no

every each hoth all each of' both ,1· all 'l

def NP

+

+ + + +

smg. counr

pl. counr

mass

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

Due to their fle xibility in use, I present two lexical entries of all and both . This ambiguity is due to different logical types (((e, t), ((e, t), t)) and (e, ((e, t), r))). In both cases the first entry can be derived by functional composirion of the


To conclude my remarks on number, I will give a brief sketch of how quantifiers can be handled in my framework. I will concentrate on each , every , all , both , and no , and restrict myself to the analysis of their occurrence as determiners of nouns and in combination with definite noun phrases (both and all can appear either before or after a definite noun phrase; cf all the man , weal/ , they both ). (42) describes the differences in disrribution of these quantifiers. The first column indicates whether the quantifier can be combined (with bare) nouns and the second column indicates whether it can be combined with definite noun phrases. Columns 3-5 describe their restrictions with respect to the structural basis. This table shows that the resemblance of plural count nouns and mass nouns is greater than the resemblance of singular count nouns and mass nouns. There seems to be no determiner that excludes the plural count nouns alone. This justifies the assumption that the number of mass nouns should be handled differently from the singular of count nouns.

2.6 Semantics of Number

(43) no each , every each of all

all of

A.NA.P V y, x [(N(y) 1\ x �y) � --. P(x)] A.NA.P V x [at(x) 1\ N(x) � P(x)] A.yA.P V x [at(x) 1\ x < y � P(x)] A.NA.P [P(b(N)) 1\ V3 x < b(N)[P(x)]] ( � A.NA.P [3 x [x < b(N)] /\ V x �b(N)[P(x)]]) A.yA.P [P(y) 1\ V3 x < y [P(x)]] ( A.yA.P (3 x [x < yj /\ V x � y [P(x)]]) A.NA.P [ctbl(b(N )) 1\ quantity(b(N)) = 2 1\ V x < b(N) [P(x)]] A.yA.P [ctbl(y) 1\ quantity(y) = 2 1\ V x < y [P(x)]] =

both both (of)

Since I am not a native speaker of English, I am not aware of any clearly semantic difference berween each and every . In the Oxford Advanced Learner's Dictionary of Current English ( 1 974), the difference is described as: 'When every is used . . ., attention is directed to the whole; when each is used, attention is directed to the unit or individual.' If this is correct, one could assume another representation of every , which lies in berween each and each of and shows a great resemblance to all , too. (44) every

A.NA.P V x [at(x) 1\ x < b (N) � P(x)]

In this cse, both turns out to be a more specific version of every as well as all . This might explain the fact that neither o f them i s used if there are only rwo (atomic) individuals in the range of the quantifier. However, I do not . assume the given analysis of the English quantifiers to be complete. A much finer analysis has to be made in order to caprure all the differences berween their uses (cf Vendler 1 967).


second entry with b. Correspondingly, the representation of each of can be regularly derived from the partitive of and each In contrst to no and all , the determiners each , every and both determine the number of the following noun, so I assume that the number of the noun is a result of agreement in these cases and will not be interpreted. The different behaviour of the quantifiers can be seen as a reflection of their semantic structure (43). Each and every quantity over the atomic domain of N, which is reflected as the restriction to singular count nouns. The representations of all (of) include the general quantifier with existential impact, and will yield no truth-value if the respective domain is empty. Consequently, they are not applicable to atomic individuals (i.e. entities that have no proper part). All N makes a statement about the N and quantifiers over the (proper) parts of the N; thus all is only applicable to objects with proper parts, not allowing singular count nouns. This representation takes care of the ambivalent behaviour of all , which can i n many cases replace the definite article. Though I did not include a special operator to describe the dual, my description of both includes ctbl(.) 1\ quantity(.) - 2, which ought to be replaced by a basic predicate dual.

Carola Eschenbach 12

2.7

W H A T I S C O M M O N TO M A S S A N D S I N G U LAR COUNT N O U N S ?

at(x) cr- sg: -.3 y � x [at(y)] mass: sg � cr-sg V mass: at(x) V -.3 y � x [at(y)] - at(x) V (-.at(x) 1\ 3 y < x [at(y)]) - at(x) V -.3 y < x [at(y)] - -.3 y < x [at(y)] -.

Consequently, the approach described here is open to the assumption that singular is ambiguous between ct-sg and mass as well as to the assumption that singular is unambiguous but underdetermined.

1 3 CONCLUSION I have described a semantic treannent o fnumber which is based on the assump tion that number always specifies which properties the referent of a noun phrase has with respect to the structure of the domain of referents. Singular and plural of count nouns are both represented as operators on the core meaning of rhe noun, which is assumed to be indifferent with respect to the complexity of the referents. The basis for the representation is given by assuming a basic pre dicate at, which holds for the proper individual atoms in the domain of refer ents. While this predicate directly corresponds to the restriction that singular imposes on count nouns, the representation of plural is more complex. This corresponds to the observation that in many languages plural is marked with respect to singular. Taking an objective criterion of atomicity for granted, the analysis outlined in this article presents a simplified view of the phenomena of number and numerals. It is neither capable of handling units of measurement such as litre , and classifiers like head in three head ofcattle , nor can it systematically cope with


As we have seen, the behaviour of plural count nouns and mass nouns is much more similar than the behaviour of singular count nouns and mass nouns. Therefore it seems quire strange that in languages such as German and English mass nouns are syntactically singulars (i.e. agree with singular forms of verbs). Bur in the presented framework we can recognize what is common to the referents of the syntactically singular noun phrases (general singular): they do not allow for proper atomic (individual) parts. The difference lies in how this condition is fulfilled: count singulars do not allow for any proper parts and mass nouns do not allow any (even improper) atomic individual parts.

2B

Semantics of Number

count noun occurrences of mass nouns like three wines . But this is another story, which has to be told another time. CAROLA ESCHENBACH FB Informatik

Received: 3 1 - 1 0-9 1 Revised version received: 28-7-92

Universitiit Hamburg Bodenstedtstr. 1 6 D-2000 Hamburg 50 Germany

Acknowledgements

N OT E S 1

2

�

C( Lohner ( 1en von Gegenstanden , Part 1 , Narr, Tubingen, X4-97. Landman, F. ( 1 9X9), 'Groups I & II', Lin,l!uistics and Philosophy, I l , 5 59-Ms , 7.23 -44. Langendoen, D. T. ( 1 97X), 'The logic of reciprocity', Linguistic Inquiry, 9, 1 77-97· Link, G. { 1 -67, Stanford. Also in P. Gaerdenfors (ed.),

B


Hausser, R. { 1 974), 'Symax and semantics of plural', Papers from the 1oth Regional Meeting of the Chicago Linguistic Society. 234-47· Hoeksema, J. ( 1 .g - the function h E Ebs such that h(s) - [ P ] M.>.g for all s E S, where b is the type of /3 [ · P] M.>.g - [ P ]M.,.g(s). The interpretation of a discourse marker is the value it has in the state of evaluation. Quantification over the values of discourse markers is mediated by quantification over these states (not over ordinary variable assignments). The intension operator abstracts over states; the extension operator applies an intension to the state of evaluation. The principle of A. -conversion (for extensional type theory) (34) (a) [A.v .p](o) equals [o!v]P if all free variables in 0 are free for v in p.

Reinhard B lumer

47

abstracting over states and to close the argument term intensionally has the following pleasant consequences for ), -conversion in OIL:

c5) equals 1 · b /v J P if all free variables in c5 are free for v in p. Note that no condition on discourse markers in c5 need be obtained. Although their interpretation is state-dependent, rhe fact rhar ·c5 is intensionally closed

(34) (b) j ).v .IJ](

·

and hence stare-independent is sufficient to validate the equivalence (34b). The following example demonstrates how this equivalence can be used for providing a dynamic binding relation-i.e. a binding relation where a discourse quantifier binds a discou rse marker outside the syntactic scope of the quantifier:

We see that rhe ), -term ), p. 3d jQ (d) 1\ ·p] in fact is dynamic in rhe indicated sense: indirecrly (via ), -conversion) rhe embedded existential quantifier binds a discourse marker rhar occurs in the argument expression of the A -term, i.e. a discourse marker outside the syntactic scope of the existential quantifier. We arl' now ready for a systematic introduction of the notational conventions that will facilitate rhe representation of the information change potential of sentences or pieces of discourse. We will use ·a as short for the type (s, a) and t as short for ( ·r, r)-rhe type of rhe information change potential. In the simplest case, rhe information change potential of a sentence is introduced by means of rhe dynamic operator 1: (36) 1� - ), p. j� 1\ 'p], where � is an expression of type r and p is a variable for propositions (type ·r). The dynamic operator 1 maps rhe static semantic value of an expression of type t into irs dynamic counterpart. Formally, 1 can be viewed as a type shifting operator which maps expression of type r into expressions denoting sets of propositions (1: r I-+ t). The converse of the dynamic operator 1 is rhe static operator !. Given rhe information change potential of a sentence, rhe static operator retrieves irs truth-conditional import, simply by applying ir to a tautology. (37) lque Franco A llemand de grammain· transjormationelle , verbs and e-theory', Natural Lan,J!uaxc and Niemeyer, Tiibingen. Lit�J!uistic Tl11wy, 6, 2.9 1-3 )2.. l3innick, Robert I. ( 1 99 1 ), Time and tlte V£•rb: A Carlson, Gregory N. ( 1 9Xo). R£ji·rellce to Kinds Guide to Tense and Aspect , Oxford Uni in English , Garland Publishing, Inc., New York!London. versiry Press (Canada), Oxford. l3oons,Jean-Paul ( 1 974), 'Acceptabilite, inter- Chomsky, Noam ( 19) s). The L

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