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6.1.1.2. Semantics
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The easiest way to explain the core meaning of the cardinal numerals is by using Figure 1 from Section 1.1.2, sub IIA, repeated below, to represent the subject-predicate relation in a clause. In this figure, A represents the denotation set of the subject NP and B the denotation set of the verb phrase, where A and B are both contextually determined, that is, dependent the domain of discourse (domain D). 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 included in set B, which is constituted by the 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 semantic function of the cardinal numerals is to indicate the size or cardinality of the intersection of A and B. In (14a), for example, the numeral twee'two' indicates that the cardinality of the intersection A ∩ B is 2.

Example 14
a. Er lopen twee jongens op straat.
  there  walk  two boys  in the.street
  'Two boys are walking in the street.'
b. ∅ twee Npl: |A ∩ B| = 2

Normally, the numerals do not give any information about the remainder of set A, that is, A - (A ∩ B) may or may not be empty. Information like this is usually expressed by means of the determiners: in addition to the information expressed by the numeral that the cardinality of the intersection of A and B is 2, the definite determiner de in (15a) expresses that A - (A ∩ B) is empty.

Example 15
a. De twee jongens lopen op straat.
  the two boys  walk  in the.street
  'The two boys are walking in the street.'
b. de twee Npl: |A ∩ B| = 2 & A - (A ∩ B) = ∅

In the absence of the definite determiner, it is the sentence type that provides additional information about the cardinality of A - (A ∩ B). In (14a), for example, the sentence contains the expletive er and is therefore presentative; the subject introduces a set of new entities into the domain of discourse, and from this we may conclude that there was no mention of boys in the domain of discourse before the sentence was uttered. The most plausible reading is therefore one according to which A - (A ∩ B) = ∅.
      It seems, however, that we are dealing here instead with a conversational implicature (Grice 1975) than with syntactically or lexically encoded information. The first reason for assuming this is that the implication that A - (A ∩ B) is empty is absent in non-representative clauses. In (16a), for example, the subject is interpreted as specific, that is, at least known to the speaker, and now the implication that all boys in the domain of discourse are part of the intersection of A and B is absent.

Example 16
a. twee jongens lopen op straat
  two boys  walk  in the.street
  'Two boys are walking in the street.'
b. twee Npl: |A ∩ B| = 2 & A - (A ∩ B) ≥ 0

An even more compelling reason is that the implication in expletive constructions such as (14a) that A - (A ∩ B) is empty can be cancelled if the context provides sufficient evidence that set A is not exhausted by the intersection A ∩ B. Consider for example the small discourse chunk in (17). Since the context leaves no doubt that many students were involved in the protest action, neither (17b) nor (17b') implies that the two students who were arrested exhaust the complete set of demonstrating students.

Example 17
a. Er was gisteren een grote demonstratie op de universiteit.
  there  was  yesterday  a big protest action  at the university
  'There was a big protest action at the university yesterday.'
b. Er werden twee studenten gearresteerd.
  there  were  two students  arrested
  'Two students were arrested.'
b'. Twee studenten werden gearresteerd.
  two students  were  arrested

      Normally, and also in this work, the difference between (14a) and (16a) is discussed in terms of the purely quantificational versus the partitive reading of indefinite noun phrases: the former is supposed to only express the quantificational meaning of the cardinal numeral, whereas the latter expresses in addition that we are only dealing with a subpart of the denotation set of the NP. In the latter case, the noun phrase twee studenten'two students' is treated as essentially synonymous with the true partitive construction in (18a) where the partitive van-PP explicitly refers to the superset from which the entities referred to by the complete noun phrase are taken; see 4.1.1.6 for more discussion of this construction.

Example 18
Twee van de studenten werden gearresteerd.
  two of the students  were  arrested

The data in (17b&b') show, however, that this one-to-one correspondence cannot be maintained. This does not mean that we cannot make a distinction between purely quantificational and partitive indefinite noun phrases, but that we should keep in mind that the distinction is probably not syntactic in nature, but rather forced upon us by the context in which the indefinite noun phrase is used. For the moment, we may therefore simply conclude that the meaning expressed by the numeral is restricted to indicating the cardinality of the intersection A ∩ B in Figure 1.

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References:
  • Grice, H.P1975Logic and conversationCole, P. & Morgan, J. (eds.)Speech acts: Syntax and Semantics 3New YorkAcademic Press41-58
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