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5.1.1.1. The core meaning of the articles
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The easiest way of explaining the core meaning of the articles is by using Figure 1 from Section 1.1.2, sub IIA, repeated below, which can be used to represent the subject-predicate relation in a clause. In this figure, A represents the denotation set of the subject NP and B the set denoted by the verb phrase. The intersection A ∩ B denotes the set of entities for which the proposition expressed by the clause is claimed to be true. In an example such as Jan wandelt op straat, for example, it is claimed that the set denoted by A, viz. {Jan}, is properly included in set B, which is constituted by people walking in the street. In other words, it expresses that A - (A ∩ B) = ∅.

Figure 1: Set-theoretic representation of the subject-predicate relation

The core function of the determiners is to specify the intersection (A ∩ B) and the remainder of set A, that is, A - (A ∩ B). The definite article de/het'the' in (6) expresses that in the domain of discourse (domain D), all entities that satisfy the description of the NP are included in the intersection A ∩ B, that is, that A - (A ∩ B) = ∅. The singular noun phrase de jongen'the boy' in (6a) has therefore approximately the same interpretation as the proper noun Jan in the discussion above; it expresses that the cardinality of A ∩ B is 1 (for which we will use the notation: |A ∩ B| = 1). The only difference between the singular and the plural example in (6) is that the latter expresses that |A ∩ B| ≥ 1.

Example 6
a. De jongen loopt op straat.
  the boy  walks  in the.street
a'. de/het Nsg: |A ∩ B| = 1 & A - (A ∩ B) = ∅
b. De jongens lopen op straat.
  the boys  walk  in the.street
b'. de Npl: |A ∩ B| ≥ 1 & A - (A ∩ B) = ∅

      The semantic contribution of the indefinite articles in (7a&b) is to indicate that A ∩ B is not empty; they do not imply anything about the set A - (A ∩ B), which may or may not be empty. The difference between the singular indefinite article een and the (phonetically empty) plural indefinite article ∅ is that the former expresses that |A ∩ B| = 1, whereas the latter expresses that |A ∩ B| ≥ 1.

Example 7
a. Er loopt een jongen op straat.
  there  walks  a boy  in the.street
  'There is a boy walking in the street.'
a'. een Nsg: |A ∩ B| = 1 & |A - (A ∩ B)| ≥ 0
b. Er lopen ∅ jongens op straat.
  there  walk  boys  in the.street
  'There are boys walking in the street.'
b'. ∅ Npl: |A ∩ B| ≥ 1 & |A - (A ∩ B)| ≥ 0

It is important to note that only parts of the meaning descriptions in the primed examples of (6) and (7) are inherently linked to the determiner: definite articles imply that A - (A ∩ B) = ∅, whereas indefinite articles do not. The claims with respect to the cardinality of the intersection A ∩ B do not come from the articles but from the number (singular versus plural) marking on the nouns: singular marking expresses that |A ∩ B| = 1, whereas plural marking expresses that |A ∩ B| ≥ 1. It is therefore not surprising that the difference between definite and indefinite noun phrases headed by a non-count like wijn'wine' is that the former refers to a contextually determined amount of wine, whereas the latter simply refers to an indeterminate amount of wine.
      The meaning that we attribute to the number marking, which is due to Farkas & De Swart (2008), may come as a surprise. First, the meaning attributed in (7a) to the singular indefinite noun phrase breaks with the tradition in formal semantics that translates the indefinite article by means of the existential operator ∃x, which implies that the article expresses that the intersection A ∩ B contains at least one member, that is, |A ∩ B| ≥ 1. Second, (7b) attributes this meaning instead to the plural marking, which seems to conflict with the fact that plural nouns are normally interpreted as expressing that the intersection A ∩ B contains more than one member, that is, |A ∩ B| > 1. Below, we will therefore motivate why we adopt the proposal by Farkas & De Swart (who actually assume that the plural marking is ambiguous and can express either |A ∩ B| ≥ 1 or |A ∩ B| > 1, but we will ignore this here).
      The traditional assumption that indefinite singular noun phrases express that |A ∩ B| ≥ 1 predicts that a speaker would use an indefinite singular noun phrase if he has no clue about the cardinality of a certain set. The proposal here, according to which the plural marking on the noun expresses that |A ∩ B| ≥ 1, on the other hand, predicts that the speaker would use an indefinite plural noun phrase in that case. That the latter prediction is correct becomes clear when we consider the questions in (): if a speaker is interested whether the addressee is a parent, that is, whether the addressee has one or more children, the typical way of asking the question would be as given in (8a), not as in (8b).

Example 8
a. Heb je kinderen?
  have  you  children
  'Do you have children?'
b. # Heb je een kind?
  have  you  a child
  'Do you have a child?'

Example (8b) is, of course, not ungrammatical but can only be used if the speaker presupposes that the cardinality of the referent set will not be larger that one: so one could ask a question like Heb je al een kind?'Do you already have a child?' if the presupposition is that under normal circumstances the addressee would be childless. For the same reason, examples such as (9a) require that the singular be used, given that this expresses that the speaker is aware of the fact that people normally have just one nose; using the plural would violate Griceʼs (1975) maxim of quantity as this would wrongly suggest that the speaker lacks this knowledge. Similarly, by opting for one of the options in (9b) may make more explicit what the speaker actually desires, a single cigarette or, e.g., a packet of cigarettes.

Example 9
a. Heb jij een mooie neus/#mooie neuzen?
  have  you  a beautiful nose/beautiful noses
  'Do you have a beautiful nose/beautiful noses?'
b. Heb je een sigaret/sigaretten voor me?
  have  you  a cigarette/cigarettes  for me
  'Do you have a cigarette/cigarettes for me?'

Another context that licenses the use of a plural indefinite noun phrase involves clauses containing the modal willen'to want'. Consider the two examples in (10): example (10a) is similar to (8a) in that it inquires whether the addressee is planning to have one or more cats as a pet; example (10a) would be infelicitous in this use and instead suggests that the speaker has a certain cat in mind that is on offer.

Example 10
a. Wil je katten?
  want  you  cats
  'Do you want cats?'
b. # Wil je een kat?
  want  you  a cat
  'Do you want a cat?'

A third case in which the speaker may use a plural indefinite noun phrase to express that he has no presupposition about the cardinality is in the case of inferences. When the speaker is visiting some people that he does not know intimately and enters a room littered with toys, he could utter something like (11a) without excluding the possibility that his hosts have only one child. For at least some speakers, using example (11b) is less felicitous in this context as it may suggest that the speaker has reason to believe that the cardinality of the set of children is one.

Example 11
a. Er wonen hier kinderen.
  there  live  here  children
  'There are children living here.'
b. # Er woont hier een kind.
  there  lives  here  a child
  'There is a child living here.'

      That the singular number marking in definite noun phrases like (6a) implies that the intersection has the cardinality 1 seems uncontroversial, which means that Farkas & De Swartʼs proposal makes it possible to assign a single meaning to the singular: |A ∩ B| = 1. That the plural marking in definite phrases like (6b) can express |A ∩ B| ≥ 1 is harder to establish. This is due to the fact discussed in Section 5.1.1.2 below that the definite article generally presupposes that the speaker and the addressee are able to indentify the referents in the referent set of the noun phrase. In the majority of cases the speaker will therefore know whether the cardinality of the referent set is one or more than on. If the former is the case, using a singular definite noun phrase will be more informative than using a plural definite noun phrase; the former will therefore be preferred by Griceʼs maxim of quantity.
      Nevertheless, there are certain contexts that show that plural definite noun phrases do not make any implication concerning the cardinality of the referent set. Picture some employee of a company responsible for dealing with custumers’ complaints. When he comes into the office in the morning, he begins by having a look at the newly arrived complaints, at least, if there are any. One morning, there is no post on his desk; he picks up the phone and asks the person who normally sorts and distributes the post the question in (12a), which sort of presupposes that there will be some new complaints but does not imply anything about the number of those complaints. In this respect, (12a) is crucially different from (12b), which implies that the referent set has the cardinality 1.

Example 12
a. Kan je me de nieuwe klachten brengen?
  can  you  me  the new complaints  bring
  'Can you bring me the new complaints?'
b. Kan je me de nieuwe klacht brengen?
  can  you  me  the new complaints  bring
  'Can you bring me the new complaint?'

The same thing can be observed in conditionals. Example (13a) is taken from a text on family law concerning divorce. Using the singular noun, as in (13b), would be distinctly odd in this context since this would imply that in all cases of a divorce there is only a single child involved; such implications are completely absent in examples such as (13a). The examples in (12) and (13) again support the proposal by Farkas & De Swart, which assigns a single meaning to the plural: |A ∩ B| ≥ 1.

Example 13
a. Als de kinderen aan één van de ouders zijn toegewezen, dan ...
  if  the children  to one of the parents  are prt.-awarded  then
  'If one of the parents is awarded the custody of the children ...'
b. # Als het kind aan één van de ouders is toegewezen, dan ...
  if  the child  to one of the parents  is prt.-awarded  then

      The semantic function of the negative article geen'no' is to indicate that the intersection of A and B is empty: A ∩ B = ∅. No claims are made about set A or set B: it may or may not be the case that domain D contains a set of boys and/or that there is a set of people who are walking in the street.

Example 14
a. Er loopt geen jongen op straat.
  there  walks  no boy  in the.street
a'. geen Nsg: A ∩ B = ∅ & |A - (A ∩ B)| ≥ 0
b. Er lopen geen jongens op straat.
  there  walk  no boys  in the.street
b'. geen Npl: A ∩ B = ∅ & |A - (A ∩ B)| ≥ 0

The distinction between singular and plural is again not related to the meaning of the article: examples like (14a&b) can be used to deny a presupposition that, respectively, |A ∩ B| = 1 or |A ∩ B| ≥ 1. If no such presupposition is present, the plural is used. Consider the situation in which Jan is in hospital with a fractured leg. He is bored stiff and therefore his friend Peter always brings him something to read when he is visiting: the number of books varies depending on their size. One day Peter enters the hospital ward empty-handed. In this case Jan will probably ask the question in (15a) and not the one in (15b), given that the latter presupposes that Peter normally brings just one book.

Example 15
a. Heb je geen boeken voor me meegenomen?
  did  you  no books  for me  prt-.taken
  'Didnʼt you bring me any books?'
b. # Heb je geen boek voor me meegenomen?
  did  you  no book  for me  prt-.taken

      The meaning contributions of the three articles can be summarized by means of the table in example (16). There are no implications concerning the cardinality of the intersection giventhat it is the role of the number marking of the noun to specify this: the singular marking expresses that |A ∩ B| = 1 and the plural marking that |A ∩ B| ≥ 1.

Example 16
The core meaning of the articles
  A B A - (A B)
definite article de/het non-empty empty
indefinite article een/∅ non-empty indeterminate
negative article geen empty indeterminate

In the following sections we will see, however, that more can be said about the precise characterization of the meaning of the articles, and it will also become clear that some uses of the articles do not fall under the general characterization of the meaning of the articles given in this section.

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References:
  • Farkas, Donka & Swart, Henriëtte de2008Formal and semantic markedness of number
  • Grice, H.P1975Logic and conversationCole, P. & Morgan, J. (eds.)Speech acts: Syntax and Semantics 3New YorkAcademic Press41-58
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