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3.1.1. On the semantics of scalar adjectives
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Section 1.3.2.1 has shown that set-denoting adjectives denote sets of entities in the domain of discourse that have the property the adjectives refer to. A copular construction such as (3a) expresses that the logical subject Jan of the adjective slecht'bad' is part of the denotation set of the adjective, which is expressed in predicate calculus as in (3a'). The attributive construction in (3b) expresses that the intersection of the set denoted by the noun jongen'boy' and the set denoted by the adjective slecht contains at least one entity, which is expressed in predicate calculus as in (3b'). The semantic representations in (3) express more or less the same thing as the set-theoretical representations in Figure 2 in Section 1.3.2.1, sub I.

Example 3
 a. Jan is slecht. Jan is bad
 a'. slecht (Jan)
 b. een slechte jongen a bad boy
 b'. ∃x [(jongen (x) & slecht(x))]

Although these semantic representations are good approximations of the meanings of the examples in (3), they do not express their meanings fully. If we use (3a), we imply that Jan has a certain place on the implied scale of “goodness" in (2), as in (4a), and if we use (3b), we imply that there is at least one boy that has a certain place on the scale, as in (4b).

Example 4
 a. Scale of “goodness":
 b. Scale of “goodness":

In other word, scalar adjectives seem to have the additional property of relating entities to certain positions on the scale that they imply. If we assume that the range on the implied scale denoted by a scalar adjectives consists of a set of ordered degrees, this can easily be expressed by means of predicate calculus.
Consider the representations in (5), in which d stands for “degree", that is, a position on the implied scale. The copular construction in (3a) now has the semantic representation in (5a): this representation expresses that there is a degree d which is part of the range on the implied scale of “goodness" denoted by slecht, such that Jan occupies that position on the scale. The attributive construction (3b) can be represented as in (5b): this representation expresses that there is an entity x such that x is a boy, and that there is a degree d which is part of the range on the implied scale of “goodness", denoted by slecht, such that x occupies that position on the scale.

Example 5
 a. ∃d [SLECHT (Jan,d)]
 b. ∃x [JONGEN (x) & ∃d [SLECHT (x,d)]]

In contrast to the semantic representations in (3a') and (3b'), the representations in (5) do take into account that the examples in (3) imply a mapping of entities onto certain positions on the implied scale.