In current linguistic theory, determiners and quantifiers/numerals are generally assumed to be external to the NP-domain, and to function as the head of a projection containing the NP-domain, as in (5). Determiners and quantifiers determine the referential and/or the quantificational properties of the noun phrase.

The determiner slot D can be filled by any of the elements in Table 3.

	neuter noun	common noun	plural noun
articles	het boek the book	een boek a book	∅ boeken ∅ books
demonstrative pronouns	dit/dat boek this/that book	deze/die pen this/that pen	deze/die boeken these/those books
possessive NPs and pronouns	Jans/zijn boek Jan’s/his book	mijn moeders/haar pen my mother’s/her pen	onze boeken our books
quantifiers and numerals	elk boek each book	elke pen each pen	alle boeken all books

The assumption that articles, demonstratives and possessive pronouns are located in position D explains why these elements are in complementary distribution, since it is generally accepted that a head position of a phrase can be occupied by only one head. This claim has also led to the hypothesis that the noun phrase may contain other projections besides DP and NP identified in (5); since numerals and specific quantifiers can be combined with articles, demonstratives and possessive pronouns, it is sometimes claimed that they function as the head of a projection, commonly referred to as QP and/or NumP, embedded in DP. This hypothesis would ascribe the more articulate structure in (6) to an example such as mijn vijf broers; cf. Chapter 20 for a more accurate discussion.

Although questions about the number of projections involved are obviously of interest (see Alexiadou et al., 2007: part II, for discussion), the main point to remember here is that determiners and quantifiers/numerals are assumed to be external to the NP, which implies that they have no effect on the denotation of the (modified) noun. Their semantic contribution is restricted to the referential and/or quantificational properties of the noun phrase as a whole. Below, we will illustrate this briefly with examples in which the noun phrase acts as the subject of the clause. Before doing so, however, we need to provide some background by briefly outlining the set-theoretic treatment of the subject-predicate relation, which will be central to our discussion of the denotational properties of the NP.

Certain aspects of the meaning of a clause can be expressed by means of set theory. For the sake of illustration, suppose for a moment that a proper noun such as Jan denotes a singleton set (which in fact differs from what is commonly assumed in current formal semantics; cf. Smessaert 2014: 83). Thus, an example such as Jan loopt op straatJan is walking in the street expresses that the singleton set denoted by Jan is contained in the set denoted by the verb phrase loopt op straatwalks in the street. More generally, the subject-predicate relation in a clause can be expressed by Figure 1, where A is the set denoted by (the NP-part of) the noun phrase and B is 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 like Jan en Marie wandelen op straatJan and Marie are walking in the street, it is claimed that the complete set denoted by A, viz. {Jan, Marie}, is contained in set B, which consists of the people walking in the street. In other words, it expresses that the intersection (A ∩ B) exhausts set A so that the remainder of set A is empty: |A - (A ∩ B)| = 0; the semantic function of determiners and quantifiers/numerals is to specify the intersection A ∩ B and the remainder of A - (A ∩ B); nothing is said about the remainder of set B, i.e. B - (A ∩ B). Below we will describe this informally for some determiners and quantifiers/numerals. A more complete and formal description can be found in Chapter 20.

The definite article dethe in (7) expresses that in the domain of discourse all entities that satisfy the description of the NP-part of the subject (here: jongen) are included in the intersection A ∩ B, i.e. that A - (A ∩ B) is empty. Therefore the singular noun phrase de jongenthe boy in (7a) has approximately the same interpretation as the proper noun Jan in the discussion above; it expresses that the cardinality of A ∩ B is 1 (for this we will use the notation: |A ∩ B| = 1). The plural example in (7) is usually taken to differ from the singular example only in that it expresses that |A ∩ B| > 1 (although this claim will be slightly adapted in Section 19.1.1.1).

The meaning of a definite demonstrative pronoun like dezethis/these and diethat/these or a possessive pronoun like mijnmy is similar to that of the definite article, the only difference being that these determiners effect a partitioning of the set denoted by A; the use of these determiners asserts the claim that one of the resulting subsets is properly included in B (e.g. these boys but not those).

The semantic contribution of the indefinite articles in (8a&b) is to indicate that A ∩ B is not empty, but they imply nothing about the set A - (A ∩ B); the latter may or may not be empty (the other boys in set A may all be in school). The difference between the singular indefinite article een and the (phonetically empty) plural indefinite article ∅ is that the former expresses that |A ∩ B| = 1, while the latter expresses that the cardinality can be greater than 1. From a semantic point of view, the cardinal numerals belong to the same class as the plural indefinite article; an example like (8c) is similar in all respects to (8b), except that it expresses that |A ∩ B| = 2.

The semantic contribution of quantifiers like enkelesome, veelmany and weinigfew can be described in similar terms. The main difference is that the cardinality of the set A ∩ B is somewhat vaguer: an example such as (9a) expresses more or less the same thing as (8b), but in addition the use of enkele suggests that the cardinality of A ∩ B is lower than some implicitly assumed norm “c”: 1 < |A ∩ B| < c. The interpretation of the quantifiers veel and weinig also seems to depend on some implicitly assumed norm: veel expresses that |A ∩ B| > c' and weinig that |A ∩ B| < c''. In the case of enkele in (9a), the implicit norm c still seems more or less fixed; the cardinality of the set of boys walking in the street will never be greater than, say, eight or nine. In the case of veel and weinig, on the other hand, the implicitly assumed norm is contextually determined: a hundred visitors may count as many for a vernissage but as few for a music festival. Note also that, as in the case of indefinite articles and numerals, the examples in (9) do not imply anything about the set A - (A ∩ B).

When we combine a definite determiner and a numeral/quantifier, the meanings of the two are combined. An example such as (10a) expresses that |A ∩ B| = 2, which can be seen as the semantic contribution of the numeral tweetwo, and that A - (A ∩ B) is empty, which can be seen as the semantic contribution of the definite article de. Similarly, (10b) expresses that |A ∩ B| > c, which is the contribution of the quantifier, and that |A - (A ∩ B)| = 0, which is again the contribution of the definite article de.

Special attention must be paid to a set of expressions often referred to as predeterminers. These expressions are quantifiers that appear in a position left-adjacent to the determiners. Some examples are given in (11), where the determiners mijnmy and dethe in the determiner position are preceded by the predeterminers alall and heelwhole/all of the. Since the semantics of these predeterminers is extremely complex, we will not discuss these elements here but refer the reader to the discussion in Chapter 21.