# 14.2: Quantifiers as relations between sets

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Let us begin by asking what claim sentence (1a) makes about the world. Under what circumstances will it be true? Intuitively, it will be true in any situation in which all of the individuals that are men have the property of snoring; that is, when every member of the denotation set ((MAN)) is also a member of the denotation set ((SNORE)). But this is equivalent to saying that ((MAN)) is a subset of ((SNORE)), as indicated in (19) of Chapter 13 (page 240).

Now let us think about how this meaning is composed. We have said that the sentence All men snore expresses an assertion that the set of all men is a subset of the set of entities that snore. This interpretation is expressed in the formula in (2). Clearly the semantic contribution of men is ((MAN)), and the semantic contribution of snore is ((SNORE)). That means that the semantic contribution of all can only be the subset relation itself.

(2) ((All men snore)) = true ↔ ((MAN)) ⊆ ((SNORE))

Now it may seem odd to suggest that all really means ‘subset’, but that is what the principle of compositionality seems to lead us to. The subset relation is a relation between two sets. More abstractly, we can think of the determiner all as naming a relation between two sets, in this case the set of all men and the set of all individuals that snore.

Now let us consider sentence (1b), No women snore. Under what circumstances will this sentence be true? Intuitively, it will be true in any situation in which no individual who is a woman has the property of snoring; that is, when no individual is a member both of the denotation set ((WOMAN)) and of the denotation set ((SNORE)). But this is equivalent to saying that the intersection of ((WOMAN)) with ((SNORE)) is empty, as indicated in (19) of Chapter 13 (page 240). This interpretation is expressed in the formula in (3). By the same reasoning that we used above, the principle of compositionality leads us to the conclusion that the determiner no means ‘empty intersection’. Once again, this is a relation between two sets.

(3) ((No woman snores)) = true ↔ (((WOMAN)) ∩ ((SNORE)) = ⌀)

Sentence (1c), Some man snores, will be true in any situation in which at least one individual who is a man has the property of snoring. This is equivalent to saying that the intersection of ((MAN)) with ((SNORE)) is non-empty, as indicated in (4). The principle of compositionality leads us to the conclusion that the determiner some means ‘non-empty intersection’.

(4) ((Some man snores)) = true ↔ (((MAN)) ∩ ((SNORE)) ≠ ⌀)

The key insight which has helped semanticists understand the meaning contributions of quantifier words like all, some, and no, is that these words name relations between two sets. The table in (5) lists these and several other quantifying determiners, showing their interpretations stated as a relation between two sets. In these examples the two sets are ((STUDENT)) (the set of all students), which for convenience we will refer to as S, and ((BRILLIANT)) (the set of all brilliant individuals) which for convenience we will refer to as B.

(5) a. All students are brilliant. S ⊆ B

b. No students are brilliant. S ∩ B = ⌀

c. Some students are brilliant. |S ∩ B|≥2

d. A/Some student is brilliant. S ∩ B , ⌀; or: |S ∩ B| ≥ 1

e. Four students are brilliant. |S ∩ B| = 41

f. Most students are brilliant. |S ∩ B| > |S − B|; or: |S ∩ B| > ½|S |

g. Few students are brilliant. |S ∩ B| < some contextually defined number

h. Both students are brilliant. S ⊆ B ∧ |S | = 2

Notice that we have distinguished plural vs. singular uses of some by stating that plural some (ex. 5c) indicates an intersection with cardinality of two or more. The interpretation suggested in (h) indicates that the meaning of both includes the subset relation and the assertion that the cardinality of the first set equals two. This amounts to saying that both means ‘all two of them’. Strictly speaking, it might be more accurate to treat the information about cardinality as a presupposition, because that part of the meaning is preserved in questions (Are both students brilliant?), conditionals (If both students are brilliant, then …), etc. However, we will not pursue that issue here.

All of the examples in (5) involve relations between two sets. We might refer to quantifiers of this type as two-place quantifiers. Three-place quantifiers are also possible, i.e., quantifiers that express relations among three sets. Some examples are provided in (6).

(6) a. Half as many guests attended as were invited.
| ((GUEST)) ∩ ((ATTEND)) | = ½| ((GUEST)) ∩ ((INVITE)) |

b. In every Australian election from 1967 to 1998, more men than women voted for the Labor party.
| ((MAN)) ∩ {x: <x,l> ∈ ((VOTE_FOR)) }| > | ((WOMAN)) ∩ {x: <x,l> ∈ ((VOTE_FOR)) }|

The kinds of meanings expressed by quantifying determiners can also be expressed by adverbs. D. Lewis (1975) refers to adverbs like always, sometimes, never, etc. as “unselective quantifiers”, because they can quantify over various kinds of things. The examples in (7) show these adverbs quantifying over times: always means ‘at all times’, never means ‘at no time’, etc. The examples in (8) show these same adverbs quantifying over individual entities. If usually in (8b) were interpreted as quantifying over times, it would imply that the color of a dog’s eyes might change from one moment to the next. If sometimes in (8c) were interpreted as quantifying over times, it would imply that the sulfur content of a lump of coal might change from one moment to the next.

(7) Quantifying over times:

a. In his campaigns Napoleon always relied upon surprise and speed.2
b. Churchill usually took a nap after lunch.
c. De Gaulle sometimes scolded his aide-de-camp (= Chief of Staff).
d. George Washington never told a lie.

(8) Quantifying over individual entities:

a. A triangle always has three sides. (= ‘All triangles have three sides.’)
b. Dogs usually have brown eyes. (= ‘Most dogs have brown eyes.’)
c. Bituminous coal sometimes contains more than one percent sulfur by weight. (= ‘Some bituminous coal contains more than one percent sulfur by weight.’)
d. A rectangle never has five corners. (= ‘No rectangles have five corners.’)

In a number of languages, including English, quantifying determiners like all can optionally occur in adverbial positions, as illustrated in (9). This alternation is often referred to as quantifier float:

(9) a. All the children will go to the party.
b. The children will all go to the party.

Not all languages make use of quantifying determiners; adverbial quantifiers seem to be more common cross-linguistically. Other strategies for expressing quantifier meanings are attested as well: quantificational verb roots, verbal affixes, particles, etc. For some languages it has been claimed that the syntactic means available for expressing quantification limits the range of quantifier meanings which can be expressed.3 Most of the examples in our discussion below involve English quantifying determiners, and these have been the focus of a vast amount of study. However, we should not forget that other quantification strategies are also common.

1 Recall from Chapter 9 that numerals seem to allow two different interpretations. In light of that discussion, this sentence could mean either |S ∩ B| = 4 or |S ∩ B| ≥ 4 depending on context. For the purposes of this chapter we will ignore the ‘at least’ reading.

2 www.usafa.edu/df/dfh/docs/Harmon28.pdf

3 Baker (1995); Bittner (1995); Koenig & Michelson (2010).

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