Quantifier determiners like all, every, and most, are referred to as proportional qantifiers because they express the idea that a certain proportion of one class is included in some other class. Certain complex determiners like four out of (every) five are also proportional quantifiers. Quantifier determiners like no, some, four, and several, in contrast, are referred to as cardinal quantifiers because they provide information about the cardinality of the intersection of two sets.5 Several is vague; for most speakers it probably indicates a set containing more than two members, but not too much more (less than ten? less than seven?). Nevertheless, it clearly expresses cardinality rather than proportion.
The determiners many and few are ambiguous between a cardinal sense and a proportional sense. Sentence (19a) can be interpreted in a way which is not a contradiction, even though the student body at Cal Tech is a tiny fraction of the total population of America. However, this interpretation must involve the proportional senses of many and few; the cardinal senses would give rise to a contradiction. Sentence (19b) can only be interpreted as involving the cardinal senses of many and few, since the sentence does not invoke any specific set of problems or solutions from which a certain proportion could be specified.
(19) a. Few people in America have an IQ over 145, but many students at Cal Tech are in that range.
b. Today we are facing many problems, but we have few solutions.
Both the cardinal and proportional senses of many and few are vague, and this can make it tricky to distinguish the two senses in some contexts. Cardinal many probably means more than several, but how much more? Generally speaking, proportional many should probably be more than half, and proportional few should probably be less than half; but how much more, or how much less? And in certain contexts, even this tendency need not hold. In a country where 80% of the citizens normally come out to vote, we might say Few people bothered to vote this year if the turnout dropped below 60%. In a city where less than 20% of the citizens normally bother to vote in local elections, we might say Many people came to vote this year if the turnout reached 40%. So, like other vague expressions, the meanings of many and few are partly dependent on context.
Relationships expressed by cardinal quantifiers are generally symmetric, as illustrated in the examples in (20–23):6
(20) a. No honest men are lawyers. (a entails b)
b. No lawyers are honest men.
(21) a. Three senators are Vietnam War veterans. (a entails b)
b. Three Vietnam War veterans are senators.
(22) a. Some drug dealers are federal employees. (a entails b)
b. Some federal employees are drug dealers.
(23) a. Several Indo-European languages are verb-initial. (a entails b)
b. Several verb-initial languages are Indo-European.
Relationships expressed by proportional quantifiers, in contrast, are not symmetric, as illustrated in the examples in (24–26):
(24) a. All brave men are lonely. (a does not entail b)
b. All lonely men are brave.
(25) a. Most Popes are Italian. (a does not entail b)
b. Most Italians are Popes.
(26) a. Few people are Zoroastrians. (a does not entail b, in proportional sense of few)
b. Few Zoroastrians are people.
There are several distributional differences which distinguish these two classes of determiners. The best known of these has to do with existential constructions. Only cardinal quantifiers can occur as the “pivot” in the existential there construction; proportional quantifiers are ungrammatical in this environment.7 (It is important to distinguish the existential there from several other constructions involving there. Sentences like (27b–c) might be grammatical with the locative there, or with the list there as in There’s John, there’s Bill, there’s all our cousins…; but these other uses are irrelevant to the present discussion.)
(27) a. There are several/some/no/many/six unicorns in the garden.
b. * There are all/most unicorns in the garden.
c. * There is every unicorn in the garden.
This contrast may be related to the fact that proportional quantifiers seem to presuppose the existence of a contextually relevant and identifiable set.8 In order for sentence (28a) to be a sensible statement, a special context is required which specifies the relevant set of people. For example, we might be discussing a town where most people are Baptist. Similarly, if sentence (28b) is intended to be a sensible statement, a special context is required to specify the relevant set of students. For example, we might be discussing graduation requirements for a particular linguistics program. This “discourse familiarity” of the restriction set is required by proportional quantifiers, but not by cardinal quantifiers. The sentences in (29) do not require any specific context in order to be acceptable. (Of course context could be relevant in determining what the vague quantifier many means.)
(28) a. Most people attend the Baptist church.
b. All students are required to pass phonetics.
(29) a. Many people attend the Baptist church.
b. Six hundred students got grants from the National Science Foundation this year.
c. No aircraft are allowed to fly over the White House.
Discourse familiarity is of course one type of definiteness. We suggested above that the indefinite article a(n) could be analyzed as an existential quantifier, roughly synonymous with singular some. Under this analysis, a(n) would be a cardinal quantifier, because it specifies a non-empty intersection. Similarly, one way of analyzing the definite article the is to treat it as a special universal quantifier, meaning something like ‘all of them’ with plural nouns and ‘all one of them’ with singular nouns. Since all is a proportional quantifier, this analysis predicts that the should also function as a proportional quantifier. The use of the seems to trigger a presupposition that the individual or group named by the NP in which it occurs is uniquely identifiable in the context of the utterance. This presupposition might be seen as following from the general requirement of discourse familiarity for the restriction set of a proportional quantifier.9
This analysis of the articles gets some support from the observation that a(n) can, but the cannot, occur with existential there. This is exactly what we would expect if a(n) is a cardinal quantifier while the is a proportional quantifier.
(30) There is a/*the unicorn in the garden. (under existential reading)
5 Proportional quantifiers are sometimes referred to as strong qantifiers, and cardinal quantifiers are sometimes referred to as weak qantifiers.
6 This symmetry follows from the fact that cardinal quantifiers generally have meanings of the form |A∩B|=n; and the intersection function is commutative (A ∩ B = B ∩ A).
7 Milsark (1977).
8 Barwise & Cooper (1981) suggest that asserting existence is a tautology for most proportional quantifier phrases, vacuously true if the reference set is empty and necessarily true if it is not empty. It is a contradiction for proportional quantifiers like neither.
9 Kearns (2000).