We have argued that the meaning contribution of a quantifier, whether expressed by a determiner, adverb, or some other category, is best understood as a relationship between two sets. We introduced a new format for logical formulae involving quantification, the restricted quantifier notation, which is flexible enough to handle all sorts of quantifiers. This notation also makes it possible to state rules of semantic interpretation which treat quantifiers in a more uniform way, although we did not spell out the technical details of how we might do this. A very important step in the interpretation of a quantifier is determining its scope, and we discussed several contexts in which scope interactions can create ambiguous sentences.
These concepts will be important in later chapters, especially in Chapter 16 where we discuss modality. As discussed in that chapter, a very influential analysis of modality is based on the claim that modal expressions like may, must, could, etc. are really a special type of quantifier.
Kearns (2000: ch. 4) provides a clear and helpful introduction to quantification. A brief overview of this very large topic is provided in GutierrezRexach (2013), a longer overview in Szabolcsi (2015). D. Lewis (1975) is the classic work on quantifying adverbs. Barwise & Cooper (1981) is one of the foundational works on Generalized Quantifiers, and a detailed discussion is presented in Peters & Westerståhl (2006).
A. Restricted quantifier notation. Express the following sentences in restricted quantifier notation, and provide an interpretation in terms of set relations:
1. Every Roman is patriotic.
2. Some wealthy Romans are patriotic.
3. Both Romans are patriotic.
4. Caesar loves all Romans who obey him.
5. Most loyal Romans love Caesar.
B. Scope Ambiguities. Use logical notation to express the two readings for the following sentences, and state which reading seems most likely to be intended, if you can tell.
1. Some man loves every woman.
2. Many theologians do not understand this doctrine.
3. This doctrine is not understood by many theologians.
4. Two-thirds of the members did not vote for the amendment.
5. You can fool some of the people all of the time. [Note: for now you may ignore the modal can.]
6. A woman gives birth in the United States every five minutes.
7. He tries to read Plato’s Republic every year.a
a Marilyn Quayle, on the reading habits of her husband; Wall Street Journal, January 20, 1993.
Exercise A: Translate the following sentences into predicate logic, using the standard [not restricted] format for the existential and universal quantifiers, ∃ and ∀. If any sentence allows two interpretations, provide the logical formulae for both readings.
1. Solomon answered every riddle.
2. All ambitious politicians visit Paris.
3. Someone betrayed Caesar.
4. All critical systems are not working.
5. No German general supported Stalin.
6. Not every German general supported Hitler.
7. Some people believe every wild rumor.
8. Socrates inspires all sincere scholars who read Plato.
Exercise B: Translate the sentences below into logical formulae, using restricted quantifier notation.a
1. Arthur eats everything that Susan cooks.
2. Boris mistrusts most reports from Brussels.
[hint: treat from as a two-place predicate]
3. Few who know him like Arthur.
4. William sold Betsy every arrowhead that he found.
5. Twenty-one movies were directed and produced by Alfred Hitchcock.
6. Most travelers entering or leaving Australia visit Sydney.
7. No onei remembers every promise hei makes.
8. Some officials who boycotted both meetings were sacked by Reagan.
9. Jane Austen and E. M. Forster wrote six novels each.
10. Rachel met and interviewed several famous musicians.
11. Most children will not play if they are sad.
Exercise C: The bolded phrases in the sentences below can be analyzed as quantifiers. State the truth conditions for these sentences in terms of set relations.
1. More than twenty senators are guilty.
2. Between six and twelve generals are loyal.
3. Both sisters are champions.
4. The twelve apostles were Jewish.
5. Just two of the seven guides are bilingual.
6. Neither candidate is honest.
7. Fewer than five crewmen are sober.
The discontinuous determiners in the next examples express threeplace quantifier meanings:
8. More men than women snore.
9. Exactly as many Americans are lawyers as are prisoners.b
10. Fewer wrestlers than boxers are famous.
a Ex. B-C are patterned after Kearns (2000: 89–90).
b Actually the figures are only approximately equal, but there are clearly too many of both.