Monotonicity, Closure and the Semantics of Few
Within theories of quantification which hold that indefinites have their basic interpretations at the
semantic type of sets (e.g. Landman 2004), monotone decreasing expressions are known to pose a
particular difficulty. In the case of a monotone increasing expression such as more than three students,
existential closure (EC) yields a generalized quantifier interpretation with the correct semantics, as in (1).
But in the case of a monotone decreasing expression such as fewer than three students (as in (2)), this
approach fails: existential closure incorrectly yields an "at least" reading (in this case, allowing values
greater than three), while at the same time excluding zero as a possible value.
Two approaches to resolving this problem are current in the literature. The first could be termed
"semantic decomposition," in that a monotone decreasing expression is decomposed into a positive
component and a negative operator that can take separate scope from the remainder of the expression,
above the existential operator (McNally 1998; Landman 2004 on no). The second is to introduce an
alternate type shifting operation applying exclusively to monotone decreasing expressions (de Swart
2001; Landman 2004 on at most three). Drawing on facts relating to the semantics of few one of a
small number of lexically simple monotone decreasing expressions in English -- I provide evidence that
the first of these alternatives is the correct one, and there is no need to resort to the second.
The first point of support for a decompositional analysis of few is provided by the existence of so-
called split scope readings under intensional verbs (Jacobs 1980; de Swart 2000). As an example, (3a) is
most naturally paraphrased as in (3b), where the negative component of few takes scope over the verb
need, while the remainder of the noun phrase remains within its scope.
Beyond this, a decompositional analysis also provides an account for an apparently unrelated
phenomenon: the relationship between the pair few/a few, the only such pair in the English count noun
quantifier system. While the standard treatment is to analyze a few as an idiomatic unit distinct from few,
a few does not always function as a unit: a and few can be separated by an adverb (a very few students) or
by an adjective modifying the head noun (a lucky few students), implying that a and few in fact combine
in the syntax. A compositional semantic analysis is thus desirable.
In addressing this issue, I build on "adjectival" accounts of cardinal numbers (Krifka 1999) by
proposing that few has the semantics of a noun modifier (type e,t,e,t) rather than a determiner. This
claim is amply supported by the fact that few exhibits adjectival properties (Partee 1989), in that it has
comparative and superlative forms (4a), may be preceded by determiners (4b), may appear in predicative
position (4c), and may be conjoined with other adjectives (4d).
Making use of Link's (1983) lattice-theoretical approach to plurality, I thus propose (5a) as the formal
semantics of few. Few may then combine with a plural noun, giving (5b) as the semantics of few students
at the set (e,t) level. A generalized quantifier interpretation for a expression such as few students may
then be obtained via existential closure and the raising of negation past the existential operator, as in (6);
the latter is consistent with the preference for negation to take wide scope in examples such as (3).
Turning now to a few, I propose that a is semantically vacuous with the exception that its presence
blocks the raising of negation under existential closure, a claim which can be separately motivated by the
behavior of the overt negator not. For example, (7a) cannot be true in the case where no students came to
the party, evidence that this expression does not have an interpretation where negation scopes over the
existential operator, as in the paraphrase (7b). Thus we have (8a) as the semantics of a few students at the
set level, and (8b) as the corresponding generalized quantifier obtained via existential closure.
It can be seen that the expressions in (6) and (8b) give the correct interpretations: few is monotone
decreasing and allows zero as a possibility (`not a large number'), while a few is existential and monotone
increasing (`a not large number').
In summary, the decompositional semantic analysis of few not only allows the derivation of a
generalized quantifier with the correct semantics (the problem we began with), but also accounts for two
other seeming unrelated facts: the existence of split scope readings, and the relationship of few to a few.
Therefore in this case there is no need to posit a separate closure operation other than existential closure,
and in fact it is not clear how such an approach could account for these facts.
(1) more than 3 students= x(*student(x) & x>3)
EC(more than 3 students) = P Qx(P(x) & Q(x))( x(*student(x) & x>3)
= Qx(*student(x) & x>3 & Q(x))
`the set of sets (properties) that contain a plural individual of
cardinality greater than 3 composed of students'
(2) fewer than 3 students= x(*student(x) & x< 3)
EC(fewer than 3 students) = P Qx(P(x) & Q(x))( x(*student(x) & x<3)
= Qx(*student(x) & x<3 & Q(x))
`the set of sets (properties) that contain a plural individual of
cardinality less than 3 composed of students'
(3) a. They need few reasons to fire you
b. `to fire you, it is not the case that they need a large number of reasons'
(4) a. fewer, fewest
b. those few students; his few friends; the few advantages of his theory
c. his good qualities are few
d. The flowers are few and small but readily identifiable by their uniquely marked lip
(5) a.few= Px(¬largeC(x) & P(x)), where largeC is a contextually defined value
b.(few students) e,t =few (students)
= x(¬largeC(x) & *student(x))
(6) (few students) e,t, t= P Qx(P(x) & Q(x))( x(¬largeC(x) & *student(x))
= Q¬ x(largeC(x) & *student(x) & Q(x))
(7) a. A not large number students came to the party
b. `it is not the case that a large number of students came to the party'
(8) a. (a few students) e,t,= a ((few students) e,t)
= x(¬largeC(x) & *student(x))
b. (a few students) e,t, t= Q x(¬largeC(x) & *student(x) & Q(x))
References
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