The core meaning of the cardinal numerals in prenominal position can be easily explained by using Figure 1 from Section 15.1.2, sub IIA, repeated below. It represents the subject-predicate relation in a clause: set A represents the denotation set of the lexical part (i.e. the NP-part) of the subject, and set B represents the denotation set of the verb phrase, where A and B are both contextually determined, i.e. dependent on 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. For instance, in an example such as De jongen wandelt op straatthe boy is walking in the street, it is claimed that the set denoted by the noun jongenboy is included in set B, which consists of the people walking in the street. In other words, it is asserted that A - (A ∩ B) is empty.

The semantic function of the cardinals is to indicate the size or cardinality of the intersection of A and B. For example, in (31a) the cardinal tweetwo indicates that the cardinality of the intersection A ∩ B is 2; recall that ∅ in (31b) stands for the phonetically empty indefinite article.

The cardinals do not give any information about the remainder of set A, that is, A - (A ∩ B) may or may not be empty. Such information is usually expressed by the determiners: in addition to the information expressed by the cardinal that the cardinality of the intersection A ∩ B is 2, the definite article de in (32a) expresses that A - (A ∩ B) is empty.

In the absence of the definite determiner, it is the sentence type that provides additional information about the cardinality of A - (A ∩ B). In (31a), 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 can 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) is empty.

This interpretation of the indefinite noun phrase twee jongens does not seem to be syntactically or lexically encoded, but to result from a conversational implicature (Grice 1975). The first reason for this assumption is that the implication that A - (A ∩ B) is empty is clearly not part of the meaning of the indefinite noun phrase in non-representative clauses. In (33a), for example, the subject is interpreted as specific, i.e. at least known to the speaker, and now the implication that all boys in the domain of discourse are part of the intersection A ∩ B is missing.

An even more compelling reason is that the implication in expletive constructions such as (31a) 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. This is illustrated by the discourse chunk in (34). Since the context leaves no doubt that many students were involved in the protest, neither (34b) nor (34b') implies that the two arrested students exhaust the complete set of protesting students.

The difference between (31a) and (33a) is usually discussed in terms of the purely quantificational versus partitive reading of indefinite noun phrases (cf. De Hoop 1992): the former is supposed to express only the quantificational meaning of the cardinal, whereas the latter additionally expresses that we are dealing with only a subset of the denotation set of the NP. In the latter case, the noun phrase twee studententwo students is treated as essentially synonymous with the true partitive construction in (35a), in which the partitive van-PP explicitly refers to the superset from which the entities referred to by the complete noun phrase are taken; cf. Section 20.4 for further discussion of this construction.

However, the interpretation of the examples in (34b&b') has shown 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, since we should keep in mind that the distinction may not be semantico-syntactic in nature, but rather forced upon us by the context in which the indefinite noun phrase is used. For now, we can simply conclude that the meaning expressed by the cardinal is limited to indicating the cardinality of the intersection A ∩ B in Figure 1.