Forthcoming in Massimiliano Carrara, Friederike Moltmann and Alexandra Arapinis (eds.), Pluralityand Unity: Philosophy, Logic and Semantics, Oxford University Press

Quantifiers, Determiners and Plural Constructions

Byeong-uk Yi

Abstract: This paper presents analyses of natural language quantifiers and determiners. In doing

so, the paper pays special attention to plural determiners, determiners that can combines with plural

noun phrases (e.g., all, some, any, the, most), and argues that Generalized Quantifier Theory gives

clearly incorrect accounts of those determiners by assuming the traditional bias against plural

constructions. The paper gives a sketch of a recent approach to plural constructions, the pluralist

approach, that regards them as peers of their singular cousins that have autonomous semantic

functions, and presents analyses of plural determiners that result from taking the approach. The

paper also presents analyses of Bach-Peters sentences that use polyadic quantifiers or determiners,

and a treatment of donkey anaphora based on the view that indefinite noun phrases by themselves

have no quantificational component.

Keywords: quantifier; determiner; plural construction; plural logic; Bach-Peters sentence; donkey

anaphora; Generalized Quantifier Theory; conservativity thesis; denoting phrase; indefinite noun

phrase

Contents

1. Generalized quantifier theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2. Determiners and plural constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1. Conservativity thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2. GQT analyses of plural determiners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3. Bias against plurals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3. Taking plurals seriously . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4. Regimented plural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5. Plural determiners: plural language analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.1. Plural definite descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.2. Some and any . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.3. Most . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.4. All . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.5. Conservativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6. The pluralist GQT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

7. Quantifiers and determiners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

7.1. The Russellian GQT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

7.2. Bach-Peters sentences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7.3. Donkey anaphora . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Appendix: is one of and include . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

1 Among denoting phrases he includes everything, nothing and something, which he says are“the most primitive of denoting phrases” (1905, 480).

2 In contemporary linguistics, determiners are taken to form a closed word class. SeeBloomberg (1933, 203–5), where he introduces the notion of determiner. He says, “determiners aredefined by the fact that certain types of noun expressions (such as house or big house) are alwaysaccompanied by a determiner (as, this house, a big house)” (ibid., 203). It is notable that heimmediately adds: “This habit of using certain noun expressions always with a determiner, ispeculiar to some languages, such as the modern Germanic and Romance. Many languages have notthis habit; in Latin, for instance, domus ‘house’ requires no attribute and is used indifferently where[sic whether] we say the house or a house” (ibid., 203). The ‘habit’ has much to do with a strongdeterminer system that employs articles, but most languages do not have articles, as Lyons (1999,xv) says. Languages without articles include Old English, Latin, Chinese, Japanese and Korean.Some issues about such languages are discussed in my (2012a). For an account of the Englishdeterminer system, see, e.g., Quirk et al. (1985, 253–265).

3 See, e.g., Barwise and Cooper (1981) and Keenan and Stavi (1986).

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In the modern development of logic, the standard universal and existential quantifiers (‘’ and ‘’)

are introduced in regimented (or symbolic) languages. They are considered refinements of some

natural language expressions: everything, something, every, a, some etc. Russell discusses a group

of related expressions, “all, every, a, any, some and the” (1903/1937). He calls the phrase resulting

from a count noun preceded by “one of the above six words or some synonym of one of them” a

“denoting phrase” (ibid., 55f).1 The six words belong to a larger group of expressions called

determiners in contemporary linguistics, which also include his, two, most and many others.2

Generalized Quantifier Theory (in short, GQT), developed in the last quarter of the 20th century,

gives a semantic account of a larger group of expressions that include the so-called denoting phrases,

viz., those that result from combining determiners with common noun phrases. The theory takes the

expressions in this larger group to be quantifiers in a liberal sense, and call them generalized

quantifiers.3 In this paper, I discuss analyses of determiners and quantifiers with special attention

to those involving plural constructions (in short, plurals).

Many determiners of natural languages combine with plural noun phrases, yielding

4 See my (1998; 1999b; 2002; 2005; 2006; 2012b; 2013). In my (2012b), I discuss some ofthe issues about GQT that relates to plurals.

5 For the theory, see Russell (1905; 1919/1920) and Whitehead and Russell(1910–13/1925–27). The theory, on which denoting phrases are ‘incomplete symbols’, departs fromhis earlier theory, which assigns semantic function (‘denoting’) to the phrases. See Russell(1903/1937, Chapter 5) for the earlier theory.

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‘generalized quantifiers’ involving plurals: ‘some boys’, ‘any boys’, ‘all boys’, ‘most residents of

London’, ‘the scientists who discovered the structure of DNA’ etc. (Call such determiners plural

determiners.) But GQT pays little attention to plurals and, as we shall see, gives flagrantly incorrect

accounts of many determiners commonly discussed in logic: some, all etc. I begin this paper by

presenting GQT and discussing some problems of the theory that result from ignoring intricacies of

plurals (§§1-2). And I present an approach to solving the problems that takes plurals seriously. In

doing so, I give a sketch of the treatment of plurals I have presented in other works (§§3-4),4 present

analyses of plural determiners in first-order extensions of elementary languages that include

refinements of natural language plurals, languages I call (basic) plural languages (§5), and formulate

an improvement of GQT, the pluralist GQT, that incorporates the plural language analyses (§6).

I also discuss some issues about determiners and quantifiers that are independent of treatment

of plurals (§7). First, I formulate an alternative to GQT that we might consider a descendent of

Russell’s later theory of denoting phrases, which includes his accounts of definite and indefinite

descriptions.5 I call the Russellian alternative the Russellian GQT (in short, RGQT), and formulate

its pluralist cousin, the pluralist RGQT, which is to RGQT what the pluralist GQT is to GQT (§7.1).

Then I discuss two constructions that pose serious challenges to most contemporary accounts of

quantifiers and determiners, including GQT and RGQT (and their pluralist cousins): (a) Bach-Peters

sentences (e.g., ‘The pilot who shot at it hit the Mig that chased him’), and (b) donkey sentences

(e.g., ‘If Smith owns a donkey, he beats it’). I present an analysis of Bach-Peters sentences that

results from liberalizing the GQT and RGQT notions of determiner and quantifier (§7.2). I also give

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a sketch of my approach to treating donkey sentences (§7.3). The treatment is based on an account

of indefinite noun phrases that departs radically from traditional accounts thereof. On the account

I propose, indefinite noun phrases (e.g., ‘a donkey’) have no components with quantificational

meaning, although sentences featuring them (e.g., ‘A donkey is drawing a cart’, ‘A donkey is a

mammal’) might attain universal or existential meaning in some linguistic or extralinguistic contexts.

1. Generalized Quantifier Theory

Determiners include, among others, every, a, any, some, all, two, most and the. Combining them

with nouns (or noun phrases) yields a special group of expressions called generalized quantifiers in

GQT, such as the following:

(1) a. every {boy, funny boy, boy who lives in London}

b. a {boy, funny boy, boy who lives in London}

c. any {boy, funny boy, boys, boys of the same age}

d. some {boys, funny boys, boys of the same age}

e. all {boys, funny boys, boys of the same age}

f. two {boys, funny boys, boys of the same age}

g. most {boys, funny boys}

h. the {boy, funny boy, boys, funny boys, boys of the same age}

GQT regards the so-called generalized quantifiers as proper constituents of the sentences featuring

them, and assigns them semantic values. In this respect, GQT follows Montague (1973) and departs

from Russell (1905), which presents his later theory of denoting phrases. This theory (which

6 On this principle, the so-called denoting phrases do not denote, as Kaplan once put it, forthey have no semantic function.

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comprehends his theory of descriptions) is based on “the principle of the theory of denoting”:

“denoting phrases never have any meaning in themselves, but . . . every proposition in whose verbal

expression they occur has a meaning” (ibid., 480). On this principle, denoting phrases are

‘incomplete symbols’: they are not syntactic constituents on the proper analysis, just as ‘he put’ and

‘put it’ are not syntactic constituents of ‘He put it on.’6 On the Russellian theory that extends this

principle to all determiners, one might regard determiners as two-place second-order predicates and

sentences featuring generalized quantifiers as having the following form:

(2) Q(A, B)

where Q is the second-order predicate amounting to a determiner, and A and B one-place first-order

predicates. Consider, for example, a sentence featuring the definite description ‘the boy’:

(3) a. The boy is funny.

This would be analyzed as follows:

(3) b. Qthe(is.a.boy, is.funny)

where ‘Qthe’ is the second-order predicate amounting to the (in the singular), and ‘is.a.boy’ and

‘is.funny’ are elementary language predicates amounting to ‘is a boy’ and ‘is funny’, respectively.

Now, one can analyze ‘Qthe’ in terms of logical expressions of elementary languages:

7 The ‘’ symbol followed by two or more variables (e.g., ‘X’ and ‘Y’), separated by commas,forms complex multi-place predicates.

8 I call expressions for functions functors. On GQT, determiners are third-order functors;they combine with (first-order) predicates to yield second-order predicates.

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(3) c. Qthe =df X,Y x[y(y = x Xy) Yx].7

Applying this analysis to (3b) yields a second-order logical equivalent of Russell’s elementary

language analysis of (3a):

(3) d. x[y(y = x is.a.boy(y)) is.funny(x)].

This amounts to ‘There is one and only one boy and he is funny.’

On GQT, sentences featuring generalized quantifiers have the following form:

(4) D(X)(Y)

where D is the one-place third-order functor amounting to a determiner.8 On this analysis, a

determiner takes a common noun phrase (or predicate), X, to yield the generalized quantifier, D(X),

which is a second-order predicate that can take a predicate, Y, to yield a sentence. For example, (3a)

can be analyzed as follows:

(3) e. Dthe(is.a.boy)(is.funny)

where ‘Dthe’ is the functor we can analyze as follows:

9 See, e.g., Keenan and Stavi (1986, 277). Applying this analysis to (3e) yields (3d).

10 See also Tarski (1986), who formulates conditions for logicality.

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(3) f. Dthe =df X Y x[y(y = x Xy) Yx].9

Unlike (3b) or (3d), (3e) has a constituent amounting to ‘the boy’: the compound second-order

predicate ‘Dthe(is.a.boy)’. And GQT assigns this predicate a semantic value, a second-order

property. If there is one and only one boy, the property is one that is instantiated by any property of

that boy; otherwise, it is a non-instantiated property. For ‘Dthe’ signifies a function that takes

properties (e.g., being a boy) to yield second-order properties. In general, on GQT, determiners

signify one-place third-order functions, and generalized quantifiers signify one-place second-order

properties.

GQT applies this analysis to a wide range of expressions, what the theory considers

quantifiers in a liberal sense. Most of them (e.g., every boy, the boy, his boy) are non-logical

expressions, although some of those are led by logical determiners (e.g., every, the). This liberal

notion of quantifier results from successive liberalizations of the notion of quantifier.

Mostowski (1952) proposes a generalization of the notion of quantifier, one on which

“generalized quantifiers” (quantifiers on his broad notion) are one-place second-order functions that

satisfy a certain invariance condition, a condition meant to capture the idea that quantifiers must be

logical (ibid., 12f).10 What he calls (generalized) quantifiers might be considered second-order

properties of a special kind, and his generalization amounts to the proposal to take second-order

predicates of a special kind to be quantifiers in a general sense. On this proposal, quantifiers include

second-order predicates amounting to cardinal numerals: ‘(there are) two’, ‘(there are) uncountably

many’ etc. Some of them (e.g., ‘uncountably many’), unlike the, cannot be defined in elementary

11 See also Keisler (1971).

12 So ‘MxFx’, where ‘F’ amounts to ‘is funny’, is true if a majority of things are funny.Rescher assumes that ‘most’ is interchangeable with ‘a majority (of)’. This assumption, I think, isincorrect; I think ‘most’ is usually used to mean nearly all, not more than half or a majority.Westerståhl (1985) holds that it is ambiguous and has two different readings: nearly all and morethan half. I doubt it can mean the latter. (Incidentally, Yiannis Moschovakis, in personalconversation, suggests that one possible reading of ‘most’ or ‘nearly all’ with regard to an infinitenumber of things is to take it to mean co-finite.) The subsequent discussion in this paper does notdepend on whether or not ‘most’ is interchangeable with ‘a majority’.

13 See my (2012b, 180f) for an account of Kaplan’s proof. See also Kaplan (1966b), Barwiseand Cooper (1981, 214f) and Almog (1997).

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languages, as Mostowski shows.11 Inspired by his work, Rescher (1962) introduces a quantifier, ‘M’,

related to an expression commonly used in a natural language: the English most. The quantifier

amounts to the use of most in (5):

(5) Most are funny

which he takes to be true if a majority of the things in question are funny.12 While pointing out that

‘M’ satisfies Mostowski’s conditions for quantifiers, he holds that it cannot be defined in elementary

languages. This was proved by Kaplan (1966a).13

Barwise and Cooper (1981) propose a further liberalization of the notion of quantifier. As

Rescher (1962) discusses, most can also be used as a determiner combining with noun phrases, as

in (6):

(6) Most boys are funny.

Barwise and Cooper argue that the most in (6) does not qualify as a quantifier in Mostowski’s sense

14 For their proof of this, see Barwise and Cooper (ibid., 214f).

15 One can generalize Mostowski’s notion in another way. Removing the monadicitycondition yields a notion of (generalized) quantifier that allows polyadic quantifiers. See Lindström(1966). Using this notion, one can formulate the Russellian cousin of GQT. See §6.2.

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because one cannot analyze it even in extensions of elementary languages resulting from adding

Rescher’s ‘M’14 (ibid., 161f). Moreover, they liberalize the notion of quantifier by removing

Mostowski’s invariance condition. On their notion, quantifiers need not be logical expressions, and

natural language determiners, including many that are usually considered quantifiers (e.g., every,

some), are not quantifiers. For example, some boys and most boys are quantifiers resulting from

combining the determiners some and most with boys.15

While liberalizing the notion of quantifier, Barwise and Cooper (1981) focus on a rather

small number of determiners. Keenan and Stavi (1986) list and discuss a wide range of determiners

found in English, including many that cannot be considered logical or mathematical (e.g., his), and

they argue that all natural language determiners have an interesting feature, conservativity.

They define conservativity as a feature of functions (ibid., 275):

(A) Definition of Conservativity:

A one-place third-order function F is conservative iff for all first-order properties X

and Y, Y has F(X) iff x [Xx Yx] has F(X).

And we can say that determiners are conservative if they signify conservative functions:

(A) A determiner D is conservative iff D(X)(Y) is logically equivalent to D(X)(x [Xx

Yx]).

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This yields a simple test for conservativity for natural language determiners. Consider the following

pairs of sentences:

(7) a. Some boys are funny.

b. Some boys are boys and are funny.

(8) a. All boys are funny.

b. All boys are boys and are funny.

(9) a. The boy is funny.

b. The boy is a boy and is funny.

The sentences in each pair are logically equivalent, so the determiners some, all and the (in the

singular) are conservative.

Generalizing these findings, Keenan and Stavi hold that all determiners of English are

conservative and, moreover, that this is a universal feature of natural languages:

(B) Conservativity Thesis: All natural language determiners are conservative.

They say (a) “all English dets [sic determiners] satisfy the Conservativity Universal” (ibid., 253) and

(b) “Extensional determiners in all languages are always interpreted by conservative functions”

(ibid., 260). As is explicit in (b), they restrict the conservativity thesis to extensional determiners,

for they restrict GQT to those determiners (ibid., 257). Determiners are said to be extensional if they

preserve truth values under substitution of extensionally equivalent predicates (ibid., 257). We can

16 It is necessary, however, to have a suitable notion of extensionality that is applicable toplural determiners. See the last two paragraphs of §2.2.

17 Although some determiners (e.g., much) can combine with mass nouns, I discuss onlydeterminers combining with count nouns. While discussing determiners that can combine with bothmass and count nouns (e.g., all), I do not discuss their combination with mass nouns.

18 See however the use of a and every in ‘One in a hundred Americans is now in prison’ and‘One in every hundred Americans is now in prison.’

19 Some plural determiners (e.g., any) can also combine with singular count noun phraseswhile others (e.g., some, all, two and most) cannot. We can call the former neutral, and the latterexclusively plural.

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see that some English determiners are not extensional. For example, ‘surprisingly many’ is not

extensional; ‘Surprisingly many doctors of the club attended the meeting’ and ‘Surprisingly many

lawyers of the club attended the meeting’ might have different truth values while every doctor of the

club is a lawyer of the club and vice versa. In this paper, however, I focus on extensional

determiners. So I leave the extensionality restriction implicit in stating the thesis.16

2. Determiners and Plural Constructions

I think GQT has serious problems in coping with plurals. Among English determiners that can

combine with count noun phrases,17 some (e.g., a, every) cannot combine with plural noun phrases

(i.e., noun phrases in the plural form),18 but many can. Call the former determiners singular, and the

latter plural. For example, some, any, all, two, the and most are plural determiners. They can

combine with boys, for example, to yield generalized quantifiers: ‘some boys’, ‘any boys’, ‘all boys’,

‘two boys’, ‘the boys’ and ‘most boys’.19 Now, I think GQT gives clearly incorrect accounts of

plural determiners. To see this, it is useful to consider how plural determiners interact with

collective predicates: ‘cooperate’, ‘surround Bob’, ‘lift Bob’, ‘be siblings’, ‘win a Wimbledon

20 Most collective predicates have distributive homonyms. The distributive homonyms ofthe collective ‘lift Bob’ and ‘be siblings’, for example, are logically equivalent to ‘each lift Bob’ and‘be siblings of some ones’, respectively. When I discuss ‘lift Bob’, for example, the collectivepredicate is meant unless noted otherwise. For its distributive cousin, I will often adddisambiguating phrases in square predicates, as in ‘[each] lift Bob’ or ‘be siblings [of some ones].’(Note that the distributive ‘[each] lift Bob’ is what I call the plural expansion of the collective ‘liftBob’. See Def. 2 in §4.)

21 In my (2012b, 189), I give a case against the conservativity thesis, using (11a) below as anexample, and use a plural language analysis of most to explain both the breakdown and partialvalidity of the thesis. Zuber (2004) argues that a group of Polish determiners are non-conservative.They amount to the determiners resulting from filling the first argument place of the Polish for‘Apart from — only . . .’ with a proper name. (He argues that they are genuine determiners althoughtheir English counterparts, like only, are pseudo-determiners.)

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doubles title’, ‘write Principia Mathematica’, ‘discover the structure of DNA’ etc.20

2.1. Conservativity thesis

Considering collective predicates, it is straightforward to see counterexamples to the conservativity

thesis.21 Consider the following pairs of sentences:

(10) a. Some of the boys surrounding Bob are funny.

b. Some of the boys surrounding Bob are boys surrounding Bob and are funny.

(11) a. Most of the boys surrounding Bob are funny.

b. Most of the boys surrounding Bob are boys surrounding Bob and are funny.

These sentences involve determiners: some of the and most of the. (The latter figures in Keenan and

22 Ibid., p. 255. Also listed under the same heading are ‘half the’ and ‘more than two thirdsof the’ (ibid., 255). Similarly, Barwise and Cooper give examples featuring ‘more than half thepeople’, ‘more than half of John’s arrows’ and ‘most of John’s arrows’, and analyzes the first interms of the determiner ‘more than half the’ (1981, 160f). These compound determiners are alsonon-conservative.

23 Richard Zuber objects that it is wrong to assume that the determiner heading thegeneralized quantifier in, e.g., (10a) is ‘some of the’, and argues that one can and must instead takeit to be ‘some of the . . . (that are) surrounding Bob’ (private correspondence). But this analysis doesnot help to defend the conservativity thesis. First, we can see that this phrase is non-conservativeby combining it with the collective ‘boys surrounding Carol’; ‘Some of the boys surrounding Carol(that are) surrounding Bob are funny’ is not logically equivalent to ‘Some of the boys surrounding

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Stavi’s list of proportional determiners;22 the former must also belong to the same category.) But

(10a) and (10b) are not logically equivalent, nor are (11a) and (11b). Suppose that four boys are

surrounding a piano, named Bob, and that no other boys do. Suppose also that only three of the four

boys are funny but that those three fall short of surrounding Bob. Then (10a) and (11a) are true but

(10b) and (11b) are not. So some of the and most of the are not conservative. Similarly, any of the

is non-conservative: ‘Any of the boys surrounding Bob are funny’, for example, does not logically

imply ‘Any of the boys surrounding Bob are boys surrounding Bob and are funny.’

Interestingly, Keenan and Stavi discuss compound determiners closely related to some of the,

most of the and any of the, namely, the italicized determiners in (a)–(c) (1986, 286f):

(a) At least two of the ten delegates voted for Smith.

(b) Not one of John’s twenty cats is inoculated.

(c) Most of the twenty liberal delegates voted for Smith.

They assert that these determiners “are easily seen to be conservative” (ibid., 286f). But none of

them are conservative, which we can easily see by adding surrounding Smith to the noun phrases

following the determiners: delegates, cats and liberal delegates.23

Carol (that are) surrounding Bob are boys surrounding Carol and are funny.’ Second, those who takethe phrase to figure as a determiner in (10a) cannot deny that ‘some of the’ can figure as a determinerin other sentences, such as ‘Some of the boys are funny.’ If so, we can replace the italicized ‘boys’in this sentence to get (10a) (or its homonym), which is not logically equivalent to (10b) (or itshomonym). This shows that ‘some of the’ is also non-conservative.

24 See, e.g., Barwise and Cooper (1981, 169) and Keenan (2002, 628).

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2.2. GQT analyses of plural determiners

We can also see that GQT gives incorrect analyses of many commonly discussed determiners. GQT

analyzes the function signified by some, Dsome, as follows:

(12) Dsome =df X Y x[Xx Yx].24

On this analysis, ‘Some boys are funny’, for example, is logically equivalent to ‘There is at least one

boy who is funny.’ Although this is correct, it does not mean that (12) is a correct analysis of some.

We can see that the analysis is incorrect by considering sentences involving collective predicates,

such as (13):

(13) Some boys are surrounding Bob.

On (12), this sentence is logically equivalent to ‘There is at least one boy surrounding Bob.’ But this

is not correct; (13) is true if four boys (and no others) are surrounding Bob while none of them

suffices by himself to surround the piano. Similarly, GQT yields incorrect analyses of two, three,

infinitely many etc. For example, ‘Two boys lifted Bob’ is logically equivalent to ‘There are two

boys who each lifted Bob’ on the GQT analysis, but they are not — two boys might join forces to

lift Bob while neither of them can alone lift the piano.

25 See, e.g., Keenan (2002, 628).

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It is the same with the GQT analysis of all:

(14) Dall =df X Y x[Xx Yx].25

This takes all to be just a variant of the singular every. But it is not, as we can see by considering,

e.g., (15a) and (15b):

(15) a. All the king's men couldn't put Humpty together again.

b. All boys in the club are friends.

Although one might use these sentences to mean ‘Every one of the King’s men was incapable of

putting Humpty together again’ and ‘Every boy in the club is a friend [of someone, of mine etc.]’,

respectively, they also have collective readings: ‘All of the kings men [taken together] were

incapable of putting Humpty together again’ and ‘All of the boys in the club are friends [of each

other].’ (15a), on this reading, cannot be paraphrased by its GQT analysis, which is logically

equivalent to ‘Every man of the king was incapable of putting Humpty together again.’ The king’s

men might be able to put Humpty together by joining forces while none of them can alone do so.

Similarly, GQT yields an incorrect analysis of (15b) on the collective reading.

The GQT analysis of the, (3f), also clearly fails for its use with plural noun phrases that yields

plural definite descriptions, such as the italicized in (16a) and (16b):

(16) a. The boys in the club are funny.

b. The boys surrounding Bob are funny.

26 See Russell (1908), Whitehead and Russell (1910–13/1925–27) and Russell (1919/1920,Chapter 17) for the no-class theory. See my (2013) for a critical discussion of the theory.

27 Accordingly, their extensions, {x: x is two students} and {x: x is three students}, are thesame (both are empty).

15

The analysis has the same problem as Russell’s. Applying either analysis to (16a), for example,

yields the incorrect result that the sentence implies ‘There is one and only one boy in the club.’ So

Russell distinguishes two uses of the: “the in the singular” and “the in the plural” (1919/1920, 167).

One might take GQT to presuppose the same distinction in presenting (3f), as I do in marking the

with the prime in (3f). While Russell gives a different analysis for the plural the that conforms to

his principle of denoting (the no-class theory),26 however, proponents of GQT have yet to offer an

analysis thereof. Can one give an adequate GQT analysis of plural definite descriptions?

I think the answer is no. Consider the following sentences:

(17) a. The boys in the club are two students.

b. The boys in the club are three students.

GQT rests on a framework for analyzing language that would take the italicized predicates as

extensionally equivalent. Anything denoted by one of the predicates is denoted by the other, and vice

versa, for there is nothing (no one thing) that is two students and nothing that is three students.27 So

on GQT, which takes the as an extensional determiner, (17a) and (17b) must have the same truth

value. Surely, however, the sentences might have different truth values. If the club has only two

boys, (17a) is true while (17b) is false.

This means that one cannot give an adequate GQT analysis of the plural the. It is the same

with the other plural determiners discussed above. Although proponents of GQT might propose an

alternative to (12) for some, for example, they cannot give a GQT analysis that can explain the

28Some might attempt to resolve the problem by combining GQT with an approach to plurals(the plurality approach) on which a typical plural term (e.g., ‘the boys in the club’) refers to acomposite object (e.g., a set of boys, a sum or aggregate of boys) while the plural predicate ‘be twostudents’, for example, denotes any composite object with two students as components. See, e.g.,Link (1983; 1987). But the approach to plurals has serious problems partly because it implies thatthere is one object that is also many. See my criticisms of this view in my (1998, §§1–2; 1999a;1999b, §2; 2005, §2; 2014). See also the discussion of the plurality approach in §3.

16

logical disparity between ‘Some boys in the club are two students’ and ‘Some boys in the club are

three students.’28

Some might object that the plural the turns out not to be extensional and thus lie beyond the

scope of GQT. Those who do so would have to draw the same conclusion about most plural

determiners, including all those discussed in this section. I think this is correct on the traditional

notion of extensional equivalence. But it is wrong to lump some, all, most etc. with surprisingly

many. While ‘be two students’ and ‘be three students’ are extensionally equivalent in the traditional

sense (there is nothing denoted by either of the predicates, for no one thing is two or three students),

we can formulate a straightforward generalization of the notion on which they are not. Say that

predicates and are p-extensionally equivalent (‘p’ for plural), if denotes any things (taken

together) that does, and vice versa. Then the predicates in question are not p-extensionally

equivalent. One of them, ‘be two students’, denotes Adam and Kevin (taken together), for they are

two students, but the other does not. So it is straightforward to formulate a notion of extensionality

suitable for plural determiners: plural determiners are p-extensional if they preserve truth values

under substitution of p-extensional predicates. Now, we can see that surprisingly many, unlike all

the other plural determiners discussed above, is not even p-extensional. ‘Surprisingly many doctors

of the club attended the meeting’ and ‘Surprisingly many lawyers of the club attended the meeting’

might differ in truth value while any things that are doctors of the club are lawyers of the club, and

vice versa.

Because the plural the in (17a) and (17b), for example, is p-extensional, noting that ‘be two

17

students’ and ‘be three students’ signify different properties does not suffice to explain the logical

difference between the sentences. To explain this, it is necessary to invoke the fact that the

predicates are not p-extensionally equivalent (albeit extensionally equivalent), while noting that the

adequate notion of extensional equivalence for such predicates is that of p-extensional equivalence,

not the traditional notion of extensional equivalence. To do so, it is necessary to take plurals

seriously. By failing to do so, we have seen, GQT fares miserably with plural determiners.

2.3. Bias against plurals

GQT inherits the longstanding bias against plurals: plurals are merely devices for abbreviating

singular constructions (in short, singulars). It assumes that natural language determiners can be

taken to interact only with singulars, as in ‘A boy is funny’, ‘Every boy is funny’, ‘Many a boy is

funny’ etc. That is, they might all be considered singular determiners. How about plural

determiners? GQT takes them to derive from singular determiners: some, all and many, for example,

are mere variants of a, every and many a, respectively, and ‘Many boys are funny’, for example,

derives from ‘Many a boy is funny.’ But many plural determiners of English have no singular

cousins from which they might be taken to derive: three, most etc. GQT takes such determiners to

derive from underlying singular determiners of which the language happens to display only plural

variants. So Barwise and Cooper, for example, assume that one can take ‘Most of John’s arrows hit

the target’ and ‘Most people voted for Carter’, for example, to feature a determiner that one can

“abstract out” (1981, 161) as follows:

Most x such that (x) satisfy (x), or (most )x(x). (Ibid., 161)

18

This is a singular determiner that is to the plural most what many a is to many. For elementary

language variables (e.g., ‘x’) are singular variables, variables amounting to singular pronouns (e.g.,

it) as used anaphorically, and (x) and (x) cannot stand in for results of combining predicates with

plural pronouns (e.g., they): ‘They are boys’, ‘They are funny’, ‘They are John’s arrows’ etc.

But it is wrong to take plural determiners to derive from their singular cousins. Although

some sentences featuring them (e.g., ‘Many boys are funny’) have logical equivalents featuring their

singular cousins (e.g., ‘Many a boy is funny’), these singular determiners are merely emasculated

versions of plural determiners. Plural determiners can help to express what their singular cousins

cannot.

To see this, consider the following sentences:

(18) a. Some boys are surrounding Bob. (= (13); collective)

b. Some boys are [each] surrounding Bob. (distributive)

c. A boy is surrounding Bob.

(19) a. Some boys in the club are friends [of each other]. (collective)

b. Some boys in the club are friends [of some ones]. (distributive)

c. A boy in the club is a friend [of someone].

Although (18b) and (19b) seem to say, in a sense, essentially the same things as (18c) and (19c),

respectively, this does not mean that some is just a stylistic variant of a. (18a) cannot be taken to

derive from (18c), nor can (19a) be taken to derive from (19c). (18a) is not even logically equivalent

to (18c); four boys, for example, might gather to surround Bob while no one of them surrounds it.

Likewise with (19a) and (19c), which have substantial logical disparity.

29 By this, I mean the predicate whose plural form is ‘are boys’ while its (third person)singular form is ‘is a boy’. It is, on my analysis, logically equivalent to ‘be one or more boys’.

19

The singular a, we have seen, does not help to formulate some sentences that the plural some

helps to formulate: (18a), (19a) etc. If it does not, why are some sentences featuring some (e.g.,

(18b) and (19b)) logically equivalent to their singular cousins? We can explain this by attending to

the special character of the predicates they involve: ‘be boy(s)’,29 ‘be friends [of some ones]’ etc.

To formulate the explanation, it is useful to take sentences featuring determiners to involve

two predicates, rather than a noun phrase and a predicate. For example, (18b) might be taken to

involve ‘be boy(s)’ and ‘be [each] surrounding Bob’, which take the plural forms ‘are boys’ and ‘are

[each] surrounding Bob’:

(18) b. Some things that are boys are [each] surrounding Bob.

And (18c) might be taken to involve singular forms of the same predicates:

(18) c. A thing that is a boy is surrounding Bob.

Now, we can analyze the predicate ‘be boy(s)’ in terms of its singular form, ‘is a boy’:

Some things are boys if and only if every one of them is a boy.

Similarly, we can analyze ‘be friends [of some ones]’ in terms of ‘is a friend [of someone]’:

Some things are friends [of some ones] if and only if every one of them is a friend [of

30 See Def. 2 in §4 for a precise definition of plural expansions.

31 See the last paragraph of §4.

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someone].

Call plural predicates subject to this type of analysis plural expansions.30 So ‘be boy(s)’ and ‘be

friends [of some ones]’ are plural expansions of ‘is a boy’ and ‘is a friend [of someone]’,

respectively. Similarly, the distributive predicate ‘be [each] surrounding Bob’ figuring in (18b) is

the plural expansion of ‘is surrounding Bob’, and it is logically equivalent to ‘be such that every one

of them is surrounding Bob’.

Both predicates figuring in (18b), we have seen, can be analyzed as plural expansions.

Applying the analyses to the sentence yields (18b):

(18) b. Some things every one of which is a boy are such that every one of them is

surrounding Bob.

And we can see that this is logically equivalent to (18c), the singular cousin of (18b).31 This

explains the logical equivalence between (18b) and (18c) without invoking the assumption that the

plural some derives from the singular a. While this assumption leads to the incorrect conclusion that

(18a), for example, is also logically equivalent to (18c), the above explanation helps to explain why

it is not. One cannot apply the explanation to this case because one of the predicates figuring in

(18a), the collective ‘be surrounding Bob’, cannot be analyzed as the plural expansion of its singular

form. For some boys might be surrounding Bob while none of them is surrounding it. This also

means that the predicate is not even logically equivalent to the plural expansion of ‘is surrounding

Bob’ (i.e., the distributive ‘be [each] surrounding Bob’). And this explains why (18a) and (18c) are

32 For both two boys and two girls might be friends [of each other] while the four youthstaken together are not friends [of each other].

33 What I call elementary languages are often called first-order languages. I avoid thisterminology because it suggests contrasts only with higher-order languages. The regimented plurallanguages presented in §4 are first-order extensions of elementary languages; they have no higher-order variables, quantifiers or predicates.

34 Some natural languages (e.g., Chinese, Japanese, Korean) have neither singulars norplurals, because they have no grammatical number system. This does not mean that those languagesare like the usual symbolic languages in having no counterparts of plurals or that they have noexpressions for talking about many things (as such). In languages without a grammatical numbersystem, count nouns (e.g., the Korean ‘so’, which means cows) do not take singular or plural forms,and denote one or more things of a given kind (e.g., any one or more cows). See my (preprint, §2.3).

21

not logically equivalent. They are not logically equivalent unless (18a) is logically equivalent to

(18b), but these are not logically equivalent because the collective ‘be surrounding Bob’ is not

logically equivalent to its distributive homonym. Similarly, (19a) is not logically equivalent to (19c),

because the collective ‘be friends [of each other]’ cannot be considered a plural expansion because

it is not logically equivalent to its distributive homonym, ‘be friends [of some ones]’.32

3. Taking Plurals Seriously

Plurals are as prevalent in natural languages as singulars. In this respect, natural languages contrast

with the usual symbolic languages, such as elementary languages33 and their higher-order extensions.

These are singular languages, languages with no counterparts of natural language plurals, for they

result from regimenting singular fragments of natural languages.34 But it is commonly thought that

the lack of plurals in those languages results in no deficiency in their expressive power, for it is

assumed that plurals are more or less devices for abbreviating their singular cousins. This is the

traditional view of plurals that underlies most contemporary accounts of them. But I think the view

merely embodies a bias against plurals that sets a major stumbling block to a proper account of their

35 See, e.g., my (1999b; 2002; 2005; 2006).

22

logic and meaning. So I reject the traditional view and propose an alternative approach that departs

radically from the tradition that one can trace back to Aristotle through Frege. Plurals, in my view,

are fundamental linguistic devices that enrich our expressive power and help to extend the limits of

our thoughts. They belong to basic linguistic categories that complement those to which their

singular cousins belong, and they have a distinct semantic function: plurals are by and large devices

for talking about many things (as such), whereas singulars are more or less devices for talking about

one thing (‘at a time’). Call this view pluralism or the pluralist view of plurals. Let me give a sketch

of the view, which I have presented in other works.35

Those who assume the traditional view of plurals attempt to explain their logic and meaning

by reducing them to singulars. In doing so, they propose a large variety of accounts by offering

different reduction schemes. Some might assume that plurals are mere stylistic variants of their

singular cousins, as proponents of GQT do, but the assumption cannot withstand scrutiny. For

example, plural determiners (e.g., some), we have seen, cannot be considered mere variants of their

singular cousins. So, many accounts of plurals are based on sophisticated reduction schemes. To

explain my pluralist view, it is useful to begin by contrasting it with two prominent approaches taken

in such accounts: (a) the plurality approach and (b) the predicate approach.

We can distinguish these approaches by considering how they attempt to deal with plural

terms: ‘they’, ‘Tom and Dick’, ‘the boys in the club’, ‘the boys surrounding Bob’ etc. On the

plurality approach, typical plural terms refer to a plurality or plural object, one object that in some

sense comprehends many components: a set or class, an aggregate or agglomeration, a fusion or

mereological sum, a totality or plurality etc. Accounts taking this approach regard ‘Tom and Dick’,

for example, as referring to a set or class of which both Tom and Dick are members, a sum or fusion

of which they are parts etc. By contrast, proponents of the predicate approach deny that plural terms

36 The view has obvious difficulties in coping with, e.g., ‘the boys surrounding Bob’, whichis not equivalent to ‘the boys each of whom is surrounding Bob.’

37 Plural predicates contrast with singular predicates, which can admit only singular terms.In my view, all English predicates (which one must clearly distinguish from their singular forms) areplural predicates while the usual symbolic languages (e.g., elementary languages) have no pluralpredicates.

23

refer to one object, compound or not, and hold that they must be analyzed in terms of underlying

predicates. On their view, ‘Tom and Dick’ and ‘the boys in the club’, for example, are disguised

predicates, devices for abbreviating what one can say with ‘is Tom or Dick’ and ‘is a boy in the

club’, respectively.36

I agree with proponents of the predicate approach in denying the core assumption of the

plurality approach, namely, that non-vacuous plural terms must refer to one object (or a plurality).

But I do not think plural terms are disguised predicates. I think they, like their singular cousins, are

referential expressions. While the singular ‘Tom’, for example, refers to one object, viz., Tom, the

plural ‘Tom and Dick’, for example, refers to two objects, viz., Tom and Dick (taken together). Note

that this does not mean that the term denotes (or is true of) Tom and also Dick as the predicate ‘is

Tom or Dick’ does. Nor does it mean that the term refers to one object comprehending them both

as the singular terms ‘the set {Tom, Dick}’ and ‘the fusion of Tom and Dick’ do.

I give a matching account of the semantic function of (one-place) plural predicates,

predicates that can combine with plural terms: ‘cooperate’, ‘be two’, ‘be two boys’, ‘be surrounding

Bob’, ‘be boy(s)’ etc.37 These predicates, like elementary language predicates, have the function of

denoting. Unlike elementary language predicates, however, plural predicates might denote many

things as such (or taken together). While the elementary language predicate ‘is.a.boy’ (which

amounts roughly to the singular form of the English predicate ‘be boy(s)) denotes any one boy ‘at

a time’, the plural ‘be two boys’ denotes any two boys ‘at a time’. Using these notions of reference

and denotation, we can state truth conditions of plural predications, sentences resulting from

38 The asterisk is used to mark an ill-formed sentence.

24

combining plural predicates with plural terms. Consider, e.g., (20):

(20) Tom and Dick are two boys.

We can state its truth condition as follows:

(20) is true if and only if the predicate ‘be two boys’ denotes the things that (taken together)

are referred to by the term ‘Tom and Dick’.

So (20) is true because Tom and Dick, being two boys, are denoted by the plural predicate, just as

the elementary language sentence ‘is.a.boy(t)’ (which amounts to ‘Tom is a boy’) is true because

Tom is denoted by ‘is.a.boy’. And ‘Tom is two boys’ (where ‘be two boys’ takes its singular form)

is false, because ‘be two boys’ does not denote Tom, although it denotes Tom and Dick (taken

together).

Now, it is useful to note that there are different kinds of plural predicates. Some plural

predicates cannot combine with singular terms at all. For example, ‘cooperate’ and ‘be people’ are

probably such predicates; neither ‘*Tom is people’ nor ‘*Tom are people’ is well-formed,38 and

‘Tom cooperates [with others]’, though well-formed, involves a different (if homonymous) predicate.

Call such predicates exclusively plural. Not all plural predicates are exclusively plural. Some one-

place plural predicates can combine with singular terms as well. For example, ‘be two boys’

combines with ‘Tom’ (while taking the singular form) to yield ‘Tom is two boys’ and figures in the

negation ‘Tom is not two boys.’ Call such plural predicates neutral predicates. Although some

neutral predicates (e.g., ‘be two’, ‘be two boys’) do not denote any one thing, many do. ‘Be one or

39 The two predicates ‘be boy(s)’ and ‘be one or more boys’ are p-extensional equivalents.

40 In my (1999b, §2), which argues against the thesis, I formulate two versions of it: theprinciple of singularity and the principle of singular instantiation. The thesis is implicit in, e.g.,Aristotle (1963) and Frege (1884; 1892/1979). Russell (1903/1937, 132) comes close to making itexplicit. For discussions of the implicit appeal to the thesis by Frege and Russell, see my (2013;2014, §5). I think Plato argues against the thesis in one of his works, his (1982).

25

more boys’ denotes Tom, and it is the same with the plural expansion ‘be boy(s)’, which is why

‘Tom is a boy’ (which features its singular form) is true.39 And ‘be surrounding Bob’ would denote

Tom if he could stretch his body thin to surround Bob by himself.

Although the above discussion focuses on one-place plural predicates, it is straightforward

to generalize it for multi-place plural predicates. Argument places of predicates are plural if they

admit plural terms; plural argument places are neutral if they admit singular terms as well,

exclusively plural otherwise. Predicates are plural (neutral or exclusively plural), if they have at

least one plural (neutral or exclusively plural) argument place. So ‘be surrounding’, ‘write’,

‘outnumber’, for example, are two-place neutral predicates, and ‘include’ is a two-place exclusively

plural predicate (its first-argument place is exclusively plural). Now, ‘The boys in the club are

surrounding Bob’, for example, is true if the predicate ‘be surrounding’ denotes what ‘the boys in

the club’ refers to (as such) in relation to Bob. Note, in passing, that the semantic predicates used

above, ‘refer to’, ‘denote’ and ‘denote . . . in relation to’ are themselves plural predicates (their

second argument places are neutral), and that I use them as collective on the second argument places,

as is made clear by ‘as such’, ‘taken together’ etc.

Some might object to analyzing (20), for example, as a plural predication. They might argue

that genuine plural predicates cannot figure in a proper analysis of apparent plural predications

because such predicates could not signify any properties. For any subject of a property, any thing

or things that instantiate a property (‘at a time’), they assume, must be some one thing.40 This is a

longstanding view entrenched in the standard conception of attributes (i.e., properties and relations)

26

that I think one can trace back to Aristotle, and it has a strong alliance with the traditional view of

plurals, giving rise to the bias against plurals while in turn being buttressed by it. With its alliance

with the biased view of plurals, I think the standard conception of attributes stifles natural accounts

of plurals. I reject the conception and propose a liberal conception of attributes.

On this conception, there are attributes of a special kind that violate the constraint imposed

by the standard conception:

(a) properties instantiated by many things (as such).

(b) attributes with argument places that can admit many things (as such).

For example, being two boys is a property instantiated by two things (e.g., Tom and Dick) as such,

and this requires its argument place to be capable of admitting the two things as such. Likewise with

being two (which is instantiated by any two things as such) and collaborating. And surrounding is

a two-place relation the first argument of which can admit many things as such, and the relation

relates the many things (taken together) to something (e.g., Bob) if they surround it. Likewise with

lifting, writing etc. By contrast, standard attributes (e.g., being a boy, being a friend of), which

conform to the constraint of the standard conception, have no argument place that can admit many

things (as such), and cannot be instantiated by many thing (as such), although they might be

instantiated by any one of them.

Say that argument places of attributes are plural if they can admit many things as such, and

singular if they cannot. And call attributes with a plural argument place plural, and those with only

singular places singular. Then standard attributes are singular while the nonstandard ones discussed

above, for example, are plural. And the liberal conception of attributes that I propose embraces

plural attributes as well as singular ones. Call it the plural conception of attributes.

41 And there is a good reason not to do so. Plurals cannot be paraphrased into singularregimented languages, for they are not reducible to singulars. See the arguments for this in my(1998, §§1–2; 1999b, §2; 2005, §2).

27

On this conception (20), for example, involves a predicate signifying a property, for plural

predicates signify plural attributes and one-place plural predicates signify plural properties. So ‘be

two boys’ signifies a plural property, being two boys, and denotes, e.g., Tom and Dick (as such)

because these (as such) instantiate the property. Similarly, ‘be surrounding’ signifies a plural relation

although its elementary language analogue ‘is.surrounding’ signifies a singular property. And we

can formulate the truth conditions of plural predications in terms of the signification relation:

(20) is true if and only if the things that (taken together) are referred to by the term ‘Tom and

Dick’ instantiate the property signified by the predicate ‘be two boys’.

Now, can one explain the logic of plurals on the pluralist view? Some might argue that one

cannot. Like singulars, to be sure, plurals have logical relations to one another (and to their singular

cousins). For example, (20) together with ‘Tom and Dick are surrounding Bob’ implies ‘There are

two boys surrounding Bob’, just as ‘Tom is a boy’ together with ‘Tom is surrounding Bob’ implies

‘There is a boy surrounding Bob.’ Some might hold that one cannot explain such logical relations

without paraphrasing plurals into regimented languages, and that this would require taking plurals

to have singular equivalents. For the regimented languages developed with modern logic are

singular languages and have no counterparts of plurals.

Once we reject the traditional view of plurals to uphold their syntactic and semantic

autonomy, however, we can see no good reason to attempt to explain their logic by paraphrasing

them into singular languages.41 Instead, we can develop extensions of the usual regimented

languages by adding counterparts of natural language plurals. Then we can formulate the logic of

42 See, e.g., my (1999b; 2002; 2005; 2006).

43 Or ‘everything (or anything) is such that . . .’ and ‘there is something such that . . . ’.

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the extended languages by taking advantage of their syntactic clarity and simplicity, and account for

logical relations among natural language plurals by giving natural paraphrases of them into those

regimented languages. This is the approach I take to analyzing natural language plurals in order to

explain their logic, and the approach leads to a natural account of the logic and meaning of plurals.

In earlier works, I have given accounts of the logic and meaning of basic plural constructions

of natural languages by formulating basic regimented plural languages, first-order extensions of

elementary languages that include plural cousins of elementary language variables, predicates and

quantifiers.42 Those plural languages help to formulate proper analyses of plural determiners. I give

a sketch of basic plural languages in the next section, §4, and use those languages to formulate

analyses of plural determiners in §5.

4. Regimented Plural Languages

The usual regimented languages are singular languages with no counterparts of plurals. Elementary

languages have constants, variables, predicates and quantifiers, and all of these can be considered

refinements of singular constructions of natural languages: (singular) proper nouns, singular

pronouns (e.g., he, she, it) used anaphorically, singular forms of natural language predicates (e.g.,

‘is a boy’), and the singular quantifiers everything (or anything) and something.43 As refinements

of singular forms of natural language predicates, elementary language predicates (e.g., ‘is.a.boy’,

‘is.identical.with’) differ from natural language predicates in having no plural argument places. The

usual higher-order languages are built on elementary languages and inherit their singular character.

Although they extend elementary languages by including higher-order variables, predicates,

44 Like elementary languages, they are first-order languages in having no higher-orderexpressions.

45 Compare they in the sentence with it in ‘A piano was lifted by some boys that are nowsurrounding it.’ Other natural language counterparts of variables include the he in ‘He who hesitatesis lost’ and the ‘generic’ or ‘indefinite’ uses of the personal pronouns you, one etc. (‘These days youhave to be careful with your money’, ‘One must be careful about one’s investments’). See, e.g.,Quirk et al. (1985, §§6.20–62, §6.56 & §19.51). Among anaphoric pronouns, Evans (1977; 1980)

29

quantifiers etc., these are not counterparts of natural language plurals, and they inherit the singular

character of their elementary language bases. For example, second-order variables, which can

replace elementary language predicates, have only singular argument places.

In studying the logic and semantics of natural language plurals, then, it is useful to formulate

(regimented) plural languages, regimented languages that contain counterparts (or refinements) of

natural language plurals. We can do so by extending the usual, singular regimented languages by

adding to the languages plural cousins of their singular constructions. By extending elementary

languages in this way, we can obtain first-order plural languages that contain basic plural

constructions of natural languages (as well as their singular cousins).44 Those languages have plural

cousins of elementary language variables, predicates and quantifiers as well:

(i) (first-order) plural variables: ‘xs’, ‘ys’ etc.;

(ii) (first-order) plural predicates: ‘C’ (or ‘cooperate’), ‘H’ (or ‘is.one.of’) etc.;

(iii) (first-order) plural quantifiers: ‘’ (the universal) and ‘’ (the existential).

These are refinements of basic plural constructions of natural languages.

As plural cousins of singular variables (e.g., ‘x’), which amount to singular pronouns (e.g.,

it), plural variables are refinements of plural pronouns (e.g., they) as used anaphorically, as in ‘Some

boys lifted a piano that they are now surrounding.’45 The plural quantifiers, which bind plural

distinguishes those corresponding to bound variables from what he calls “E-type pronouns” (1980,337). An example he gives for the latter is they in ‘Few M.P.s came to the party but they had a goodtime’ (ibid., 338). This is a useful distinction, but I think he makes some incorrect applications ofand generalizations about the distinction. See note 96.

46 Its second argument place is plural, although its first argument place is singular.

47This does not mean that one cannot introduce exclusively plural predicates into plurallanguages. One might do so while taking, e.g., ‘Tom cooperates’ and ‘Tom is one of Tom’ to be ill-

30

variables, are plural cousins of the standard quantifiers ‘’ and ‘’. They result from refining the

plural quantifier phrases ‘Any things are such that . . .’ and ‘There are some things such that . . .’,

just as the standard ‘’ and ‘’ result from refining the singular quantifier phrases ‘Anything is such

that . . .’ and ‘There is something such that . . . .’ Plural predicates are refinements of natural

language predicates (e.g., ‘be boy(s)’, ‘cooperate’) that can combine with plural terms (e.g., ‘they’),

often taking plural forms. So they have plural argument places, those that can admit plural terms

(e.g., ‘xs’). For example, ‘C’ (which amounts to ‘cooperate’) is a one-place plural predicate —

‘C(xs)’ is well-formed; ‘H’ (which amounts to ‘is one of’)46 is a two-place plural predicate — ‘H(x,

ys)’ (which amounts to ‘It is one of them’) is well-formed.

Note that while singular variables of elementary languages relate to any one thing, plural

variables relate to any one or more things. Accordingly, the plural quantifiers amount to ‘Any one

or more things are such that . . .’ and ‘There are some one or more things such that . . . .’ But these

are equivalent to the simpler ‘Any things are such that . . .’ and ‘There are some things such that .

. .’, respectively. So ‘Cicero is a Roman and Tully is a Roman’ (which implies ‘Cicero and Tully

are Romans’) implies ‘Some things are Roman’, but not ‘Two or more things are Roman.’ Note also

that plural language predicates are neutral predicates. They are refinements of plural predicates, not

of their plural forms, and can combine with singular terms (e.g., singular constants or variables) as

well. Specifically, all their plural argument places are neutral. So ‘C(x)’ and ‘H(x, y)’, as well as

‘C(xs)’ and ‘H(x, ys)’, are well-formed.47

formed. But I think one might take the former sentence to be well-formed (albeit false), and thelatter to be true.

48 The superscript ‘N’ is for neutral, for plural expansions are neutral predicates.

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Among plural language predicates, ‘H’ is worthy of special notice. Being a two-place

predicate, it signifies a two-place plural relation, one that holds between an object and any objects

that include it. Like the singular identity predicate of elementary languages, ‘=’, it is a logical

predicate. And we can use it to define two other logical predicates, ‘’ (or ‘be.the.same.as’ or

‘be.the.same.things.as’) and ‘’ (or ‘be.among’ or ‘be.some.of’):

Def. 1 (Plural Identity and ‘be.some.of’):

(a) xs ys df z(zHxs zHys).

(b) xs ys df z(zHxs zHys).

(‘xHxs’, ‘xs ys’ and ‘xs ys’, for example, are informal versions of ‘H(x, xs)’, ‘(xs, ys)’ and

‘(xs, ys)’, respectively.) The plural identity predicate ‘’ (which figures in, e.g., ‘The authors of

Principia Mathematica are Russell and Whitehead’) is the plural cousin of the singular identity

predicate, ‘=’; and ‘’ amounts to ‘be among’ or ‘be some of’, of which ‘include’ is the converse.

And we can use ‘H’ to define plural expansions:

Def. 2 (Plural Expansion):

N(xs) df y[yxs (y)], where is a predicate.48

For example, ‘is.a.boyN’ amounts to ‘be boy(s)’, which can be analyzed as ‘be such that every one

of them is a boy’. And both ‘is.surrounding.BobN’ and ‘be.surrounding.BobN’ amount to ‘be [each]

49 Ben-Yami (2009, 219–223) objects that plural languages deviate from natural languagesby introducing the predicate ‘H’ or ‘’. See the Appendix for a reply to this objection.

50 Plural logic is a conservative extension of elementary logic, but the logic is notaxiomatizable. For model-theoretic characterizations of first-order plural logic, see my (2002; 2006).For a partial axiomatization thereof, see my (1999b; 2002; 2006).

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surrounding Bob’ (or ‘be such that every one of them is surrounding Bob’).49

We can characterize the logic of plural languages to explain the logic of natural language

plurals. Call the logic of (first-order) plural languages (first-order) plural logic.50 Logical truths of

plural languages include all the logical truths of elementary languages. They also include logical

truths that pertain to plural language logical expressions: plural quantifiers and the predicate ‘H’.

Logical truths pertaining to plural quantifiers are straightforward, drawing parallels with those

pertaining to their singular cousins. Basic logical truths pertaining to ‘H’ include the following:

[P1] y yHxs. (Given any things, there is something that is one of them.)

[P2] xHy x = y. (Something is one of something if and only if the former is the latter.)

[P3] Substitutivity of Plural Identity: [xs ys (xs)] (ys). (If some things are the

same as some things and the former are so-and-so, the latter are also so-and-so.)

[P4] Plural Comprehension: y xsy[yHxs ], where ‘xs’ does not occur free

in . (If there is something that is so-and-so, there are some things of which

something is one if and only if it is so-and-so.)

[P3] is the plural cousin of the principle of substitutivity of identity familiar in elementary logic.

[P4] is comparable to the comprehension principles for classes and properties, but differs from them

in being restricted by the antecedent, y . We cannot strengthen it by removing this restriction

because the consequent implies the antecedent (for [P1] is a logical truth).

51 We may disregard the complexity of ‘is.surrounding.Bob’ for the present purpose. Seenote 52. Instead of the singular predicates ‘is.a.boy’ and ‘is.surrounding.Bob’, we can use theirplural expansions to paraphrase (18c):

(21) c. x[is.a.boyN(x) is.surrounding.BobN(x)].

Using [P1], it is straightforward to see that this is logically equivalent to (21b); N(x) and (x) arelogically equivalent, although (xs) and N(xs) are not.

52 By adding the -operator to elementary languages, we can get an analysis of‘is.surrounding.Bob’, which figures in (21b): ‘x is.surrounding(x, b)’. Applying this analysis to(21a) yields a logical equivalent of ‘xs[y(yxs is.a.boy(y)) y(yxs is.surrounding(y, b))]’,and this is logically equivalent to the result of applying the analysis to (21b).

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Using the logic of plurals, we can explain the logical equivalence between, e.g., (18b) and

(18c) as suggested in §2.2. We can paraphrase the sentences as follows:

(21) a. xs[is.a.boyN(xs) is.surrounding.BobN(xs)].

b. x[is.a.boy(x) is.surrounding.Bob(x)].51

The analysis of plural expansions, Def. 2, unpacks (21a) as follows:

(21) a. xs[y(yxs is.a.boy(y)) y(yxs is.surrounding.Bob(y))].

And we can derive the equivalence between (21b) and (21a) from the logical truths [P1] and [P2].52

5. Plural Determiners: Plural Language Analysis

In this section, I present plural language analyses of prominent plural determiners: the plural the,

some, any, most and all. To formulate the analyses, it is useful to employ a device for forming

53 We may take ‘is.a.boyN(xs)’ to abbreviate ‘x[xxs is.a.boy(x)].’

54 See the last paragraph of §5.1.

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complex predicates, which are available in neither elementary nor basic plural languages. So I state

the analyses using modest extensions of basic plural languages, those resulting from adding the -

operator. Such languages have paraphrases of, e.g., ‘is a funny boy’, ‘be boys surrounding Bob’ and

‘be boys who lifted Bob’: ‘x [is.a.boy(x) is.funny(x)]’, ‘xs [is.a.boyN(xs) be.surrounding(xs,

b)]’ and ‘xs [is.a.boyN(xs) lifted(xs, b)]’. Applying the analyses to English sentences often yields

plural language paraphrases with -operators. The resulting sentences, however, have

straightforward logical equivalents in basic plural languages. For example, applying the analysis of

some (Def. 4(b) in §5.2) to ‘Some boys lifted Bob’ yields ‘xs[is.a.boyN(xs) xs lifted(xs, b) (xs)]’,

which is logically equivalent to ‘xs[is.a.boyN(xs) lifted(xs, b)].’53 Moreover, it is straightforward

to reformulate the analyses in basic plural languages by formulating them in relation to open

sentences (e.g., ‘[is.a.boy(x) is.funny(x)]’) instead of complex predicates.54

5.1. Plural definite descriptions

To analyze plural definite descriptions, it is necessary to distinguish two kinds thereof:

(a) ‘the boys (in the club)’, ‘the residents of London’ etc.

(b) ‘the boys who lifted Bob’, ‘the boys surrounding a piano’, ‘the scientists who

discovered the DNA structure’, ‘the British who wrote Principia Mathematica’ etc.

There is a significant logical difference between the two kinds. To see this, compare the following:

55 Similarly, ‘the boys in the club’ can be taken to involve ‘is a boy in the club’ (moreprecisely, the plural expansion of the conjunction of the singular predicates underlying the predicates‘be boy(s)’ and ‘be in the club’).

56 Both are plural terms and combine only with plural predicates. We can define ‘the’ interms of ‘the’, for ‘the boys’, for example, can be taken to abbreviate ‘the things of whichsomething is one if and only if it is a boy’.

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(22) a. Tom is one of the boys (in the club) if and only if he is a boy (in the club).

b. Any things that are some of the boys (in the club) are boys (in the club).

(23) a. Tom is one of the boys who lifted Bob if and only if he is a boy who lifted

Bob.

b. Any things that are some of the boys who lifted Bob are boys who lifted Bob.

(22a) and (22b) are logical truths, but (23a) and (23b) are not. I think this difference stems from the

different constructions involved in them. One can analyze ‘the boys’ as involving the plural

expansion of ‘is a boy’ (more precisely, the singular predicate underlying both the predicate ‘be

boy(s)’ and its singular form).55 It is not the same with ‘the boys who lifted Bob’, which involves

the collective predicate ‘be boys who lifted Bob’. This predicate figures with its full semantic profile

in the description. In ‘the boys (in the club)’, by contrast, the plural expansion ‘be boy(s) (in the

club)’ does not figure with its full force, but bears only the semantic profile of its singular cousin.

In addition to the singular the (i.e., the), then, I think English has two homonymous

determiners that yield different kinds of plural definite descriptions: (i) the, which takes plural

expansions; and (ii) the, which takes any plural predicates. We can introduce their plural language

counterparts: ‘the’ and ‘the’. Then ‘the(is.a.boyN)’ and ‘the(xs [is.a.boyN(xs) lifted(xs,

b)])’, for example, paraphrase ‘the boys’ and ‘the boys who lifted Bob’, respectively.56 And we can

analyze both determiners in plural languages:

57 We can replace (y) with N(y); they are logically equivalent.

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Def. 3 (Plural definite descriptions): Let be a predicate and and plural predicates.

Then the following hold:

(a) (the(N)) df xs{y[yxs (y)] (xs)}.57

(b) (the()) df xs{ys[ys xs (ys)] (xs)}.

Applying this analysis to ‘Tom is one of the boys’ and ‘they are some of the boys’ yields the

following paraphrases:

(24) a. xs{y[yxs is.a.boy(y)] xs tHxs (xs)}.

b. xs{y[yxs is.a.boy(y)] xs (ys, xs) (xs)}, where the free ‘ys’ matches

‘they’.

Although these are not sentences of basic plural languages, they have straightforward logical

equivalents in the languages; ‘xs tHxs (xs)’ and ‘xs (ys, xs) (xs)’ are logically equivalent to

‘tHxs’ and ‘(ys, xs)’, respectively. Similarly, applying the analysis to ‘Tom is one of the boys who

lifted Bob’ and ‘they are some of the boys who lifted Bob’ yields logical equivalents of basic plural

language sentences:

(25) a. xs{ys[ys xs (is.a.boyN(ys) lifted(ys, b))] tHxs}.

b. xs{ys[ys xs (is.a.boyN(ys) lifted(ys, b))] ys xs} (where the free

‘ys’ matches ‘they’).

Now, we can use the above analyses to explain the logical difference noted above. Using

58 (25a) does not imply (25b), either; boys other that Tom (e.g., Dick and Harry) might havelifted Bob while Tom (who is a boy) also lifted it.

59 Thanks are due to Thomas McKay for discussions a while ago about issues discussed inthis paragraph and the next.

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plural logic (e.g., [P1]–[P5]), it is straightforward to see that (24a) is logically equivalent to

‘is.a.boy(t)’, which paraphrases ‘Tom is a boy.’ Similarly, we can see that (24b) is logically

equivalent to ‘is.a.boyN(ys)’, which paraphrases ‘They are boys.’ This shows that (22a) and (22b)

are logical truths. But the same does not hold for (23a) and (23b), which involve a plural definite

description of the second kind. (25a) does not imply the plural language paraphrase of ‘Tom is a boy

who lifted Bob’:

(25) a. is.a.boy(t) lifted(t, b).

Tom and another boy might have lifted Bob together while no other boys (i.e., boy or boys) lifted

it. In that case, (25a) would be true but (25a) false.58 Similarly, (25b) does not imply the matching

plural language paraphrase of ‘They are boys who lifted Bob’:

(25) b. is.a.boyN(ys) lifted(ys, b) (where ‘ys’ matches ‘they’).

Suppose that two boys (Tom and Dick) lifted Bob together with another boy while no other boys

(e.g., Tom and Dick) lifted it alone or together. In that case, (25b) is true for the two boys, but (25b)

is not. This explains why (23a) and (23b) are not logical truths while (22a) and (22b) are.

Now, some might object to distinguishing two versions of the plural the and attempt to give

a uniform analysis that explains the logical difference between, e.g., (22a) and (23a).59 McKay

(2006, 166–9) offers a uniform analysis:

60 This results from modifying the analysis given by Sharvy (1980). See example (16c) inMcKay (2006, 168). McKay also presents an alternative analysis, which he prefers; see example(16b) in his (ibid., 167). On this analysis, (the*())) is “semantically anomalous” (neither true norfalse) if the first two conjuncts in 3(c) (i.e., (ys) and ys[(ys) ys xs]) are not jointly satisfiable.The objection to 3(c) given below applies to this analysis as well.

61 In symbols, xsys[(xs) (ys) ([xs@ys])], where ‘@’ amounts to the term-connective ‘and’ figuring in ‘Tom and Dick cooperated’ (in my view, it figures in ‘Tom and Dickare boys’ as well). We can use the plural ‘the’ to define ‘@’: [xs@ys] df the(x [xHxs xHys]N).In basic plural languages, we can define it contextually. See my (2006, 244).

62 For definite descriptions of the first kind, where plural expansions figure, 3(a) and 3(c) areequivalent.

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Def. 3(c). (the*())) df xs{(xs) ys[(ys) ys xs] (xs)}.60

On this analysis, ‘the boys’ and ‘the boys who lifted Bob’ contain the same definite description

operator, the one amounting to ‘the*’, but they are logically different because of a logical difference

between the predicates figuring in them. Say that a plural predicate is cumulative if any things that

denotes and any other things the predicate denotes (taken together) are denoted by the predicate.61

Then the predicate ‘be boy(s)’ is necessarily cumulative (for it is a plural expansion), while ‘be boys

who lifted Bob’ is not: four boys might have never worked together to lift Bob while two of them

once worked together to lift Bob and the other two did so as well. This, on the account, explains

why (22a) is a logical truth while (23a) is not.

But Def. 3(c) leads to an incorrect analysis of definite descriptions of the second kind.62

Consider a sentence involving ‘the boys who lifted Bob’:

(26) The boys who lifted Bob are funny.

63 Instead of the usual ‘in a situation’, I use ‘on a situation’ to avoid the suggestion that asentence must be in a situation to be true (or false) on the situation.

64 The plural definite description ‘the so-and-so’s’ is proper if there are some things that arethe so-and-so’s, improper otherwise.

65 McKay discusses a related example: ‘the students who are at least three in number’. Heholds that this description is “peculiar” (albeit proper) on situations where there are more than threestudents (ibid., 170). I think it strikes one as “peculiar” because it is improper in most situations,all those except that in which there are exactly three students (in such situations, ‘the three students’would refer to the same students). Note that the problem with his analysis of the runs through hisanalysis of the so-called proportional determiners, most and all; although he does not directly analyzethem with the, his analyses of them (ibid., 69ff) presupposes the analysis of the: “when Q is aproportional quantifier”, he says, “[QX:AX] has an interpretation only when some things are

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Note that this is false on the situation63 described above, on McKay’s analysis, for the description

is improper: there are no things that are the boys who lifted Bob.64 But consider a somewhat

different situation: Bob was once lifted by two funny boys (Tom and Dick), once more by two other

funny boys (Harry and Ted), and yet once more by the four boys all together. On this situation, the

predicate ‘be boys who lifted Bob’ is cumulative, albeit contingently. So, on McKay’s analysis, the

description in (26) is proper (it refers to the four boys), and the sentence is true. I do not think this

is correct. The description is improper on the situation. To see this, it is useful to compare (26) with

(27):

(27) The boys who are either the first two boys (Tom and Dick) or the second two boys

(Harry and Ted) or all of the four boys (Tom, Dick, Harry and Ted) are funny.

On the analysis, the description in (27) is a proper description referring to the four boys (as such) and

the sentence is true on the given situation. This is not correct. The description is improper on the

situation. It is the same with the description in (26), ‘the boys who lifted Bob’, for it is equivalent

to the description in (27) on the situation.65 So I do not think one can give a uniform analysis of the

identifiable as ‘the As’” (ibid., 62). See also note 72.

66 We can apply the same strategy to argue against Sharvy’s analysis of singular definitedescriptions, the analysis on which ‘the table’, for example, is equivalent to ‘the table of which anyother table is a part’ (1980, 610). He considers the situation in which his room has a large tablejoined by two smaller tables (and no others), and holds that ‘the table in my room’ (as he said), onthe situation, would indeed refer to the largest table (for if he said ‘Put the bread on the table’ on thesituation, he would be referring to that table). I disagree. Let A be the largest table, and B and C thesmaller tables. Then ‘the table in Sharvy’s room’ and ‘the table that is identical with A or B or C’must be equivalent on the situation. But I do not think the latter denotes A simply because B and Care its parts. Neither does the former.

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plural the.66

McKay argues that Def. 3(b) leads to an incorrect analysis because ‘the students surrounding

Bob’ refers to all of the students involved in surrounding Bob on a situation in which both the male

students and the female students by themselves suffice to surround Bob (private conversation). I

think the apparent plausibility of this view of the description is due to the ambiguity of the

description. On one reading, it is a description of the second kind and is equivalent to ‘the students

who are either males surrounding Bob or females surrounding Bob or males and females

surrounding Bob’. On this reading, it is improper on the situation in question. But it has another

reading, for one might use ‘surrounding Bob’ as a distributive predicate that means being involved

in surrounding Bob (or helping to surround Bob). The description on this reading, on the situation,

refers to all the male and female students, taken together, for it is a description of the first kind.

Distinguishing the two readings, we can see that Def. 3(b) yields a correct analysis of the description

on one reading while Def. 3(a) yields a correct analysis of the description on another reading.

Let me conclude the discussion of plural definite descriptions by noting that we can

reformulate my analysis thereof in basic plural languages, without using the -operator. The

operators ‘the’ and ‘the’ amount to the definite description operators ‘< ... : — >’ and ‘I’

67 See, e.g., my (1999b, 179; 2006, 244f).

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introduced in my other works.67 While the former combine with predicates, the latter combine with

sentences (together with variables) as follows:

(a) <x: >, where is a sentence (open or closed).

(b) Ixs , where is a sentence (open or closed).

Using these operators, we can obtain counterparts of the results of combining the plural the with

complex predicates. For example, ‘<x: is.a.boy(x) is.funny(x)>’ and ‘Ixs[is.a.boyN(xs) lifted(xs,

b)]’ amount to ‘the funny boys’ and ‘the boys who lifted Bob’, respectively. And we can give

contextual definitions of the operators in basic plural languages; see, e.g., my (2006, §4). Similarly,

we can formulate the analyses of other plural determiners given below in basic plural languages. But

I formulate them in languages with the -operator for convenience of exposition.

5.2. Some and any

Consider a few sentences featuring some:

(28) a. Some boys {are funny, cooperated, lifted Bob}.

b. Some of them {are funny, cooperated, lifted Bob}.

c. Some of the boys {are funny, cooperated, lifted Bob}.

d. Some of the boys who lifted Bob {are funny, cooperated, lifted Bob}.

It is straightforward to analyze the some in (28a) in terms of the plural existential quantifier ‘’:

68 It is necessary to modify this analysis of some to account for donkey sentences. See §7.2(especially, Def. 4(a)*).

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Def. 4(a). (some()) df xs[(xs) (xs)], where and are plural predicates.68

And we can analyze the some of in (28b) in terms of the quantifier and the predicate ‘’, and some

of the in terms of some of and the plural the.

Plural languages have two logical expressions one might take to figure in the quantificational

some of in (28b): (a) the quantifier ‘’ that amounts to some things, and (b) the predicate ‘’ that

amounts to be some of. Accordingly, we can use the predicate ‘’ and the some in (28a) to analyze

the quantificational some of:

Def. 4(b). (some.of(xs)) df (some(ys (ys, xs))), where is a plural predicate.

The some of the in (28c) and (28d) results from combining the quantificational some of with the

plural the, and this is used homonymously in (28c) and (28d), as we have seen. So we need to

distinguish two versions of some of the: some of the and some of the. These figure in (28c) and

(28d), respectively. We can define both in plural languages:

Def. 4(c). Let be a predicate and and plural predicates. Then the following hold:

(i) (some.of.the(N)) df xs (some.of(xs)) (the(N)).

(ii) (some.of.the()) df xs (some.of(xs)) (the()).

On this analysis, ‘Some of the boys (who lifted Bob) cooperated’, for example, amounts to ‘The boys

(who lifted Bob) are such that some of them cooperated.’

69 The plain any, which is a neutral determiner, might also be taken to have the singular andplural concord versions, but I ignore its singular concord version, which combines with singularforms of nouns (or predicates), as in ‘any student’.

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The universal any figures in sentences that draw parallels with (28a)–(28d):

(29) a. Any boys {can be funny, can cooperate, can lift Bob}.

b. Any of them {can be funny, can cooperate, can lift Bob}.

c. Any of the boys {can be funny, can cooperate, can lift Bob}.

d. Any of the boys who lifted Bob {can be funny, can cooperate}.

The analysis of any draws parallels with that of some except that any of has two versions: one that

takes singular concord because it amounts to any one of, and one that takes plural concord because

it amounts to any ones among.69 We can analyze any and the plural concord versions of the

italicized phrases in (29b)–(29d) as follows:

Def. 5. Let be a predicate and and plural predicates. Then the following hold:

(a) (any()) df xs[(xs) (xs)].

(b) (any.of(xs)) df ys[ys xs (ys)].

(c) (any.of.the(N)) df xs (any.of(xs)) (the(N)).

(d) (any.of.the()) df xs (any.of(xs)) (the()).

Using ‘H’ instead of ‘’, we can modify Def. 5(b)–(d) for the singular concord versions of the same

phrases.

It is notable that the some in (28a) is not interchangeable with some of the, nor is the any in

(29a) interchangeable with any of the. Although (some.of.the(N)) is logically equivalent to

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(some(N)), (some.of.the()) is not. To see this, consider (30a) and (30b):

(30) a. Some boys who lifted Bob are funny.

b. Some of the boys who lifted Bob are funny.

Neither (30a) nor (30b) implies the other. (30b) is false while (30a) is true, if some funny boys lifted

Bob while some other boys (e.g., unfunny ones) also lifted it; (30a) is false while (30b) is true, if four

boys of whom only two are funny lifted Bob together while no other boys lifted the piano (together

or alone). Similarly, (any.of.the()) is not logically equivalent to (any()). ‘Any boys who

lifted Bob are surrounding it’ does not imply ‘Any of the boys who lifted Bob are surrounding it.’

Moreover, (any.of.the(N)) is not logically equivalent to (any(N)), either. If there are no boys,

then ‘Any boys are funny’ is vacuously true but ‘Any of the boys are funny’ is false.

Ben-Yami (2009) gives an account of quantifiers that does not distinguish some of the, any

of the, most of the, all of the etc. from their plain cousins: some, any, most, all etc. He takes

sentences featuring either of the two kinds of determiners to have “the form ‘q S are P’ (or ‘q of S

are P’)”, where ‘q’ is for a plain determiner (“quantifier”), and he holds that “the general term ‘S’

following the quantifier ‘q’ (of ‘q of’) is logically not a predicate but a referring expression” (ibid.,

225). In (7a), ‘Some boys are funny’, for example, he takes ‘boys’ to figure not as a predicate but

as a plural term, one that, like ‘the boys’, refers to all of the boys (in question), taken together. And

he characterizes the truth conditions of sentences of the form as follows:

Suppose the quantified noun phrase ‘q A’ governs sentence S. Then S is true iff a definite

noun phrase, designating q of the particulars ‘A’ designates can be substituted for ‘q A’,

generating a true sentence. (Ibid., 227)

70 It gives correct truth conditions for some of the plain some sentences (e.g., (7a)), for theyare logically equivalent to their some of the cousins. Consider, e.g., (7a). If the sentence is true, thenthe definite description ‘the funny boys’ refers to some of the things that ‘(the) boys’ refers to and‘The funny boys are funny’ is true; and if there is any such noun phrase, the sentence is true.

71 While Ben-Yami’s truth condition for existential (or “particular”) sentences, (4*), isdesigned for the some of the sentences, the existential quantifier introduction rule he formulates(“Particular Introduction”) is designed for the plain some sentences (ibid., 228). So the inferencerule is inadequate for the some of the sentences, for (30a), for example, does not imply (30b). Forexample, ‘The funny boys are boys who lifted Bob, and the funny boys are funny’ implies (30a), butnot (30b). This means that the inference rule is not sound on his semantic characterization.

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We can formulate the instance of this characterization for some as follows:

4*. (some()) is true if and only if there is a definite noun phrase that refers to some

of the things that refers to such that () is true.

This characterization fails for plain some sentences (e.g., (30b)), because it fails to distinguish them

from the some of the sentences.70 Consider the situation discussed above that falsifies (30a) while

verifying (30b). On the situation, in which four boys lifted Bob while only two of them are funny,

(a) the definite description ‘the funny boys that the boys who lifted Bob include’ refers to some of

those that ‘(the) boys who lifted Bob’ refers to (viz., the four boys who lifted Bob), and (b) ‘The

funny boys that the boys who lifted Bob include are funny’ is true (for two of those four boys are

funny). As we have noted, however, (30a) is not true on this situation.71

5.3. Most

Consider sentences involving most that draw parallels with (28a)–(28d):

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(31) a. Most boys {are funny, cooperated to lift Bob}.

b. Most of them {are funny, cooperated to lift Bob}.

c. Most of the boys {are funny, cooperated to lift Bob}.

d. Most of the boys who lifted Bob {are funny, cooperated}.

We can give analyses of (31b)–(31d) that draw parallels with the analyses of (28b)–(28d). We can

then identify the most in (31a) with the most of the in (31c).

While basic plural languages have the plural existential quantifier and resources for defining

the logical predicate ‘’, they are not required to have a counterpart of most or most of. But we can

add to the languages a predicate amounting to most of, the two-place plural ‘be.most.of’, as the

counterpart of the English predicate be most of that figures in ‘These are most of the books I have.’

By combining the predicate with the plural existential quantifier, we can analyze the quantificational

most of as follows:

Def. 6(a). (most.of(xs)) df ys[be.most.of(ys, xs) (ys)], where is a plural predicate.

Invoking this analysis, it is straightforward to analyze the rest:

Def. 6(b). Let be a predicate and and plural predicates. Then the following hold:

(i) (most.of.the(N)) df xs (most.of(xs)) (the(N)).

(ii) (most.of.the()) df xs (most.of(xs)) (the()).

(iii) (most(N)) df (most.of.the(N)).

This analysis takes the most in (31a) to amount to most of the, for it takes plural expansions

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(e.g., ‘be boy(s)’). Some might object that it can combine directly with collective predicates as well,

as in the following:

(31) a. Most boys who lifted Bob are funny.

But ‘be boys who lifted Bob’ has two versions: the plural expansion of ‘is a boy who lifted Bob’ that

amounts roughly to ‘be boys who each lifted Bob’, and the collective predicate akin to ‘boys who

lifted Bob together’. It is not clear that the version of (31a) that involves the second is well-formed.

I think adding ‘together’ to ‘boys who lifted Bob’ in the sentence yields an oddity, a sentence that

is ill-formed or has a dubious syntactic status:

(31) a. Most boys who lifted Bob together are funny.

Still some might take the most in (31a) to have a cousin that combines directly with any plural

predicates, most*, and analyze it as follows:

Def. 6(c). (most*()) df (most.of.the()).

The analysis of most and its ilk diverges from that of some (or any) and its ilk, because there

is significant logical difference between the former and the latter. Unlike most, some can clearly

combine directly with collective as well as distributive predicates:

(30) a. Some boys who lifted Bob are funny (where ‘lifted Bob’ is used collectively).

a. Some boys who lifted Bob together are funny.

72 McKay, who does not distinguish two versions of the plural the (see §5.2), analyzes mostand most of the so that ‘Most boys are funny’ and ‘Most of the boys are funny’ are “semanticallyanomalous” (so neither true nor false) if there are no boys (2006, 69ff & 166). On this analysis, ‘Ifmost (of the) boys are funny, then some boys are funny’ fails to be a logical truth.

73 One cannot consider all a variant of any. ‘All of the natural numbers include an oddnumber’ is true, but ‘Any of the natural numbers include an odd number’ is false.

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These sentences are not logically equivalent to (30b) and (30b), respectively:

(30) b. Some of the boys who lifted Bob are funny (where ‘lifted Bob’ is used

collectively).

b. Some of the boys who lifted Bob together are funny.

Neither (30a) nor (30b) implies the other, as we have seen, and the same holds for (30a) and (30b).

So one cannot take some to abbreviate some of the, although one might take most to abbreviate most

of the (or one of its two versions).72

5.4. All

Like some and most, all is a plural determiner. And it figures with the same range of constructions:

(32) a. All boys {are funny, cooperated to lift Bob}.

b. All of them {are funny, cooperated to lift Bob}.

c. All of the boys {are funny, cooperated to lift Bob}.

d. All of the boys who lifted Bob {are funny, cooperated to lift Bob}.

We can analyze these sentences in the same way as their cousins involving most.73

74 The definition of ‘be.all.of’ is logically equivalent to that of ‘’. I think the two predicatesdiffer in presupposition. The use of all presupposes that the first conjunct ‘xs ys’ holds.

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In plural languages, we can introduce a two-place plural predicate, ‘be.all.of’, that amounts

to the English ‘be all of’ that figures in, e.g., ‘These are all of the books I have.’ And we can define

‘be.all.of’, like ‘’, using only logical expressions of basic plural languages:

Def. 7. be.all.of(xs, ys) df xs ys ys xs.74

For example, some things are all of my books if and only if any one of my books is one of them

while any one of them is one of my books. Using the predicate ‘be.all.of’, we can analyze the

italicized phrases in (32a)–(32d):

Def. 8. Let be a predicate and and plural predicates. Then the following hold:

(a) (all.of(xs)) df ys[be.all.of(ys, xs) (ys)].

(b) (all.of.the(N)) df xs (all.of(xs)) (the(N)).

(c) (all.of.the()) df xs (all.of(xs)) (the()).

(d) (all(N)) =df (all.of.the(N)).

This analysis takes ‘all boys’, for example, to abbreviate ‘all of the boys’, which is often

shortened to ‘all the boys’. Applying the analysis to (8a), ‘All boys are funny’, yields:

(8) a. xs ys[be.all.of(ys,xs) is.funnyN(ys)] (the(is.a.boyN)).

This is logically equivalent to a basic plural language sentence:

75 (33b) does not imply ‘There is a boy who lifted Bob’ although it implies ‘There are boyswho lifted Bob’; and (33b) is false while (33b) is true, if two unfunny boys lifted Bob by joiningforces while the only boy who lifted Bob alone is funny.

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(8) a. xs{y[yxs is.a.boy(y)] ys[be.all.of(ys,xs) is.funnyN(ys)]}.

On the analysis, (8a) is not logically equivalent to ‘Every boy is funny’, for (8a) implies ‘y

is.a.boy(y).’ Still, it has a singular logical equivalent, ‘There is a boy and every boy is funny’, whose

elementary language paraphrase is logically equivalent to (8a). But it is not the same with, e.g.,

(33a) and (33b):

(33) a. All of the boys (in the club) lifted Bob [by joining forces].

b. All of the boys who lifted Bob are funny.

Unlike (8a), these sentences involve predicates that cannot be considered plural expansions: the

collective ‘lifted Bob’ or ‘be boys who lifted Bob’. So they are not logically equivalent to their

singular cousins:

(33) a. There is a boy (in the club) and every boy (in the club) lifted Bob.

b. There is a boy who lifted Bob and every boy who lifted Bob is funny.

Neither (33a) nor (33a) implies the other. (33a) is false while (33a) is true, if the boys (in the club)

lifted Bob by joining forces while none of them lifted it alone. And (33a) is false while (33a) is true,

if there are exactly two boys (in the club) and they did not lift Bob together, although both of them

lifted it alone. Similarly, neither (33b) nor (33b) implies the other.75 The above analyses of all of

the preserve and help to explain all these logical relations.

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5.5. Conservativity

Distinguishing two versions of the plural the, we can refine the observation of the violation of

conservativity made in §2.1. One of the two versions of the, the, yields non-conservative

determiners by combining with some of, most of, any of etc. For example, some of the is not

conservative. Consider, e.g., the following:

(10) a. Some of the boys surrounding Bob are funny.

b. Some of the boys surrounding Bob are boys surrounding Bob and are funny.

(34) a. Some of the boys who lifted Bob are funny.

b. Some of the boys who lifted Bob are boys who lifted Bob and are funny.

(10a) does not imply (10b), as we have seen; nor does (34a) imply (34b). But this does not mean that

some of the is also non-conservative. For example, ‘Some of the boys are funny’ and ‘Some of the

boys lifted Bob (together)’ are logically equivalent to ‘Some of the boys are boys and are funny’ and

‘Some of the boys are boys and lifted Bob (together)’, respectively. Similarly, most of the and any

of the are conservative but most of the and any of the are not.

This disparity stems from the logical difference between the two versions of the noted in

§5.1. For example, ‘Some of the boys lifted Bob’ and ‘Some of the boys are boys and lifted Bob’

are logically equivalent because (22b), ‘Any things that are some of the boys are boys’, is a logical

truth. By contrast, (34a) and (34b) are not logically equivalent because (23b), ‘Any things that are

some of the boys who lifted Bob are boys who lifted Bob’, is not a logical truth. It is straightforward

to turn counterexamples to the thesis that (23b) is a logical truth into those to the thesis that (34a)

76 The cousins of (22b) with most and any (i.e., ‘Any things that are most of the boys areboys’ and ‘Any things that are any of the boys are boys’) are logical truths while those of (23b) withmost and any are not.

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and (34b) are logically equivalent. Similarly, the logical difference between the two versions of the

explains the disparity in conservativity between the two versions of most of the and any of the.76

Now, although it helps to yield non-conservative determiners, the (like the) is a

conservative determiner. ‘The boys who lifted Bob {are funny, are surrounding Bob}’ is logically

equivalent to ‘The boys who lifted Bob are boys who lifted Bob and {are funny, are surrounding

Bob}’, for ‘Any things that are the boys who lifted Bob are boys who lifted Bob’ is a logical truth.

Similarly, all of the (like all of the) is conservative because ‘Any things that are all of the boys

who lifted Bob are boys who lifted Bob’ (like ‘Any things that are all of the boys are boys’) is a

logical truth. The above analyses of the determiners yield explanations of these facts as well.

6. The Pluralist GQT

GQT fares miserably in coping with plural determiners, we have seen, because it rests on the bias

against plurals. Those who shed the bias to take the pluralist approach can formulate an

improvement of GQT that does justice to the plural character of those determiners, the pluralist

GQT. This theory agrees with GQT about singular determiners (e.g., a, every, the singular the), but

differs from GQT with regard to plural determiners. In the pluralist GQT, these determiners are

analyzed in plural languages.

GQT (i.e., the standard GQT) is a singularist theory formulated in the usual, singular higher-

order languages (i.e., higher-order extensions of elementary languages). By contrast, the pluralist

GQT is formulated in higher-order plural languages (i.e., higher-order extensions of first-order

plural languages). In addition to first-order plural variables, predicates and quantifiers, these

77 I say predicates or functors are closed if they have no free variables, open otherwise. Openpredicates (e.g., ‘x loves(x, y)’) cannot be taken to signify attributes, just as pronouns or freevariables cannot be taken to be names of objects. Likewise with open functors (e.g., ‘x y [loves(x,y) loves(y, z)]’). But one can take open predicates and functors to signify attributes (e.g.,properties) and functions relative to assignments to free variables figuring in them.

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languages have:

(a) plural predicates and functors, which have argument places that admit first-order

plural predicates;

(b) plural predicate variables, variables that can replace first-order plural predicates: ‘P’,

‘Q’ etc.; and

(c) second-order plural quantifiers, quantifiers binding plural predicate variables: ‘2’

and ‘2’.

Plural predicate variables range over plural attributes (e.g., being boy(s), lifting Bob); second-order

plural quantifiers yield generalizations over plural attributes; and (closed) plural predicates or

functors signify higher-order attributes or functions pertaining to plural attributes.77 In addition, the

languages for the pluralist GQT, like those for GQT, have -operators. In addition to first-order -

operators (which figure in ‘x [is.a.boy(x) is.funny(x)]’ and ‘xs [is.a.boyN(xs) lifted(xs, b)]’),

they have higher-order -operators, which yield higher-order predicates or functors. For example,

‘Q x[is.a.boy(x) Qx]’ is a predicate that signifies the second-order property of being a singular

property instantiated by a boy, and ‘Q xs[is.a.boyN(xs) Qxs]’ one that signifies the second-order

property of being a plural property instantiated by some boys (as such); ‘P Q x[Px Qx]’

signifies a third-order function that takes a first-order property, P, to yield the second-order property

of being a property instantiated by something instantiating P; and ‘P Q xs[Pxs Qxs]’ signifies

a function that takes a first-order plural property, P, to yield the second-order property of being a

78 Note that some of, which does not take a common noun or predicate but a term (e.g., aplural pronoun), is not a determiner and cannot be taken to signify a third-order function. But itmight be taken to signify a plural second-order function that takes some things (as such) to yield asecond-order plural property. I use ‘Dsome.of’ as an expression signifying such a function. Note thatit can be analyzed in terms of ‘Dsome’ and the predicate ‘’:

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plural property instantiated by some things instantiating P.

GQT regards determiners as expressions for third-order singular functions, analyzing a (or

at least one), for example, as ‘P Q x[P(x) Q(x)]’. It is the same with the pluralist GQT. While

GQT takes all determiners to be germane to singular properties, however, the pluralist GQT takes

plural determiners to be germane to plural properties. The theory regards them as expressions for

third-order plural functions, functions that take plural properties to yield second-order plural

properties. It takes the some in (28a), for example, to amount to ‘P Q xs[P(xs) Q(xs)]’.

We can obtain pluralist GQT analyses of plural determiners from the analyses given in §5

(call these basic plural language analyses). Let ‘P’ and ‘Q’ be monadic plural predicate variables,

which range over (first-order) plural properties; ‘P’ a monadic singular predicate variable, which

ranges over (first-order) singular properties; and ‘PN’ abbreviate the plural expansion of ‘P’ (i.e.,

‘Q [Q(xs) y(yHxs Py)]’). Then we can formulate the analyses of the two versions of the

plural the as third-order functors (‘Dthe’ and ‘Dthe’):

Def. 3. (a) Dthe =df PN Q Q(the(PN)).

(b) Dthe =df P Q Q(the(P)).

And we can analyze some and its cousins as functors:

Def. 4. (a) Dsome =df P Q Q(some(P)).

(b) Dsome.of =df xs Q Q(some.of(xs)).78

Def. 4(b) Dsome.of =df xs Q Q(some(ys (ys, xs))).

79 We can see that these are equivalent to Def. 4(c) and 4(d), respectively, by applying Def.3 and 4(b).

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(c) Dsome.of.the =df PN Q Q(some.of.the(PN)).

(d) Dsome.of.the =df P Q Q(some.of.the(P)).

Similarly, we can turn the basic plural language analyses of any, most, all and their cousins into

pluralist GQT analyses.

In analyzing, e.g., some of the, however, we can improve on the pluralist GQT analyses that

result from importing the final products of the basic plural language analyses. As is made clear in

these analyses, some of the, for example, results from combining some of and the. This connection

is lost (or hidden) in Def. 4(c). We can restore and clarify the connection by relating ‘Dsome.of.the’ to

‘Dsome.of’ and ‘Dthe’ by reformulating components of the basic plural language analysis of

‘some.of.the’ in terms of these expressions. To do so, it is useful to note that ‘Some of the boys

are funny’, for example, amounts to ‘The boys are such that some of them are funny.’ So we can

formulate improvements of Def. 4(c)–(d) as follows:

Def. 4. (c) Dsome.of.the =df PN Q Dthe(PN) (xs Dsome.of(xs) (Q)).

(d) Dsome.of.the =df P Q Dthe(P) (xs Dsome.of(xs) (Q)).79

Similarly, we can formulate pluralist GQT analyses of most and its cousins as follows:

Def. 6. (a) Dmost.of =df xs Q Q(most.of(xs)).

(b) Dmost.of.the =df PN Q Dthe(PN) (xs Dmost.of(xs) (Q)).

80 One cannot apply this analysis of quantifiers to some of them, most of us etc., which resultsfrom combining some of, most of etc. with pronouns.

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(c) Dmost.of.the =df P Q Dthe(P) (xs Dmost.of(xs) (Q)).

(d) Dmost =df Dmost.of.the.

It is straightforward to give parallel analyses of any, all and their cousins.

7. Quantifiers and Determiners

We can formulate an improvement of GQT (the pluralist GQT), we have seen, by analyzing plural

determiners in higher-order plural languages. In this section, I will discuss some issues in analyzing

natural language determiners and quantifiers that are independent of treatments of plurals. I

distinguish the GQT approach to analyzing determiners from the Russellian approach, and formulate

alternatives to GQT and its pluralist cousins that can be taken to generalize Russell’s analyses of

denoting phrases (§7.1). I also discuss two prominent constructions that give rise to serious

challenges to most contemporary analyses of quantifiers and determiners: the so-called Bach-Peters

sentences and donkey anaphora (§§7.2–7.3).

7.1. The Russellian GQT

GQT and its pluralist cousin sharply distinguish (natural language) quantifiers from determiners.

Both theories regard determiners as third-order functors and take quantifiers to result from

combining determiners with noun phrases (e.g., boy, boys).80 This approach to analyzing quantifiers

and determiners, as noted in §1, results from liberalizing Mostowski’s conception of quantifiers as

81 Lindström (1966) proposes this notion. Polyadic quantifiers are often called Lindströmquantifiers.

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monadic second-order predicates of a special kind. While the GQT notion of quantifier retains the

monadicity condition, one can reach a different generalization of Mostowski’s notion by removing

the condition. Removing the condition yields a notion of quantifier that allows polyadic quantifiers,

quantifiers that combine with two or more predicates.81 One might then analyze (6), ‘Most boys are

funny’, for example, as involving a dyadic quantifier amounting to most, ‘Qmost’ (which is called the

Rescher quantifier):

(6) Qmost(is.a.boy, is.funny).

This yields an alternative to GQT that generalizes the Russellian analysis of the singular the

presented in §1 (see (3b)), a theory we can call the Russellian GQT (in short, RGQT). On this

theory, determiners are binary quantifiers, those that combine with two predicates (e.g., ‘is a boy’

and ‘is funny’). While Russell’s analysis of the singular the results from adding an analysis of the

corresponding binary quantifier in terms of standard quantifiers (as in (3c)), RGQT (like GQT) does

not takes all quantifiers to be analyzable in terms of these. For example, ‘Qmost’ is not so analyzable

and yet is considered a quantifier (in the general sense).

Like GQT, the Russellian GQT is a singularist theory. The theory analyzes (6), for example,

with a quantifier, ‘Qmost’, that can combine only with singular predicates. So it has the same

problems as GQT in dealing with plural determiners. For example, ‘Qmost(is.a.boy, is.a.brother)’ is

not a correct analysis of ‘Most boys are brothers [of each other]’, because it amounts to ‘Most of the

boys are brothers [of some ones]’, which involves the distributive predicate ‘be brothers [of some

ones]’. In higher-order plural languages, however, it is straightforward to formulate a pluralist

82 I think neither GQT-style analyses nor Russellian ones are superior to the others for allnatural language determiners or quantifiers: GQT-style analyses (singularist or pluralist) might bemore plausible for some determiners or quantifiers, while others are more amenable to Russelliananalyses (or some other analyses). The main advantage of GQT-style analyses is that they conformto the intuition that in ‘All boys are funny’, for example, all forms a syntactic constituent with boys.But English has other uses of all: ‘They are all happy’, ‘They all ran fast’ etc. Moreover, in otherlanguages (e.g., Korean), natural translations of ‘All boys are funny’ might place the counterpartsof all between common nouns and predicates, as in ‘The boys all ran fast.’ (The Korean counterparthas no definite article.) Likewise with, e.g., ‘Many boys are coming.’ Those counterparts of all (ormany) might be considered adverbs (which amount to functors yielding compound predicates), butthey might alternatively be analyzed as Russellian quantifiers. In any case, it is less plausible toregard them as GQT determiners. (Although the English all in ‘The boys all ran fast’ might beconsidered a postpositive determiner, Korean has a counterpart of many figuring between noun andpredicate phrases that one cannot regard even as a postpositive determiner because it is clearly anadverb. Consider also typically in ‘Koalas typically inhabit open eucalypt woodlands.’)

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theory that analyzes plural determiners as binary quantifiers combining with plural predicates. For

example, we can introduce such a quantifier amounting to most, ‘Qmost’, and analyze it as follows:

Def. 6(a). Qmost =df P Q Q(most(P)).

The resulting theory is to the Russellian GQT what the pluralist GQT is to GQT. Call it the pluralist

Russellian GQT (in short, the pluralist RGQT).

How do the two types of theories compare with each other? Are GQT and its pluralist cousin

superior to RGQT and its pluralist cousin, respectively, or vice versa? One cannot decide these

questions on logical grounds alone, for one can use a Russellian quantifier (singular or plural) to

define the corresponding GQT determiner, and vice versa. One might still ask whether GQT (or the

pluralist GQT) or its Russellian cousin gives better accounts of other linguistic features (e.g., syntax)

of natural language determiners or quantifiers. I do not aim to decide this issue in this paper, where

the main aims are to highlight problems plaguing both GQT and its Russellian cousin in coping with

plurals, and to explain ways to improve on them on the pluralist framework.82

83 Although it is rarely discussed in analyses of quantifiers, I think the construction involvedin, e.g., ‘No citizen is deemed guilty unless proven guilty’ raises interesting problems in analyzingno and negation. I leave it for another occasion to discuss the problems and proposed solutions.

84 The indices indicate the antecedent and postecedent of the pronouns him and it.

85 Bach (1970) and McCawley (1970, 176f) formulate such sentences to challenge populartheories of pronouns. The example given above is from Karttunen (1971). For other discussions ofBach-Peters sentences, see, e.g., Hintikka and Saarinen (1975), Evans (1977, 528ff), McCawley(1981, 182ff) and May (1985, 19ff).

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It is notable, however, that some constructions that do not necessarily involve plurals pose

serious challenges to most contemporary accounts of quantifiers and determiners, including GQT

and its Russellian cousin:

(35) The pilot who shot at it hit the Mig that chased him. (Karttunen, 1961)

(36) If Smith owns a donkey, he beats it. (Geach 1964, 128f)

These sentences challenge attempts to analyze singular determiners. They are germane to the

pluralist GQT and its Russellian cousin as well, for (a) the pluralist theories inherit the analyses of

singular determiners from their singularist cousins, and (b) the sentences have plural cousins that

directly challenge attempts to analyze plural determiners. In the rest of this section (§7.2 and §7.3),

I give a sketch of the approaches I propose for analyzing the constructions figuring in the sentences.83

7.2. Bach-Peters sentences

(35) has two nominal phrases involving the so-called crossing coreference: ‘The piloti who shot at

itj’ and ‘the Migj that chased himi’.84 (Such sentences are called Bach-Peters sentences.)85 The latter

has an anaphoric pronoun, him, that relates (or ‘refers’) to the former as its antecedent, the former

86That is, they abbreviate ‘x,y [is.a.pilot(x) shot.at(x, y)]’, ‘y,x [is.a.Mig(y) chased(x,y)]’ and ‘x,y hit(x, y)’, respectively.

87In languages without the -operator forming complex predicates, we can take the quantifier‘Qthe–the’ and determiner ‘Dthe–the’ to simultaneously bind two variables of open formulas tantamount

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a cataphoric one, it, that relates to the latter as its postecedent. So neither phrase can be taken to

have the wider scope, under which the other falls, and one cannot apply the GQT or RGQT analysis

of the to (35). Applying either analysis to its first occurrence in the sentence bars the cataphoric it

from relating to its postecedent; applying the analysis to the second dislodges the anaphoric he.

We can resolve this problem by analyzing (35) as involving a single quantifier or determiner

amounting to the combination of the two occurrences of the. It is straightforward to refine RGQT

to give such an analysis. The theory is based on a liberal notion of quantifier that allows polyadic

quantifiers. Those who accept this notion might take (35) to involve a triadic quantifier, ‘Qthe–the’,

that combines with three two-place predicates:

(35) Qthe–the(R, S, T)

where ‘R’, ‘S’ and ‘T’ are two-place predicates amounting to ‘... is a pilot who shot at –’, ‘– is a Mig

that chases ...’ and ‘... hit –’, respectively.86 The GQT notions of quantifier and determiner are more

restrictive. They do not allow a functor tantamount to the triadic quantifier to serve as a determiner,

for they include the condition that determiners must be unary functors, which take just one predicate.

But one might liberalize the GQT notions by removing this condition to allow, e.g., binary functors

that combine with two predicates to yield second-order predicates. One can then take the double the

construction in (35) to involve a single binary functor, ‘Dthe–the’, and analyze the sentence as follows:

(35) Dthe–the(R, S)(T).87

to two-place predicates. We can then formulate (35) and (35) as follows:

(35*) Qthe–thexy(Rxy, Syx, Txy).(35**) Dthe–thexy(Rxy, Syx)(Txy).

(‘Rxy’, ‘Syx’ and ‘Txy’ abbreviate ‘[is.a.pilot(x) shot.at(x, y)]’, ‘[is.a.Mig(y) chased(x, y)]’ and‘hit(x, y)’, respectively.)

88 That is, the pilot and the Mig are the same pilot and Mig as any pilot and any Mig,respectively, if the latter pilot shot at the latter Mig while the Mig chased the pilot.

89 So it agrees with the Russellian analysis on double the constructions that do not involve‘crossing’. See discussions of (37a) and (37b) below.

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Furthermore, it is straightforward to analyze the quantifier and determiner using logical

expressions of elementary languages (with higher-order -operators):

Def. 9. (a) Qthe–the =df X,Y,Z xy[zw(Xzw Ywz z = x w = y) Zxy].

(b) Dthe–the =df X,Y Z xy[zw(Xzw Ywz z = x w = y) Zxy].

(Here ‘X’, ‘Y’ and ‘Z’ are second-order two-place variables.) On this analysis, (35) is true if and only

if (a) a pilot shot at a Mig that chased him, (b) no other pilot and Mig were in the same relation,88

and (c) the pilot hit the Mig. The analysis generalizes the Russellian analysis of the (as formulated

in GQT or RGQT) to the double the construction without reducing it to the single the construction.89

Karttunen (1971) gives a different analysis of Bach-Peters sentences. She reduces the double

the construction to two single the constructions by regarding one or the other of the two pronouns

in the sentence (e.g., it) as what Geach calls a “pronoun of laziness” (1962, 124f), a mere

abbreviation of its antecedent or postecedent (e.g., ‘the Mig that chased him’). This yields two

possible readings of (35):

90 Evans (1977, 528–30) takes the same approach but gives only one reading for (35), (37a).

91 To see this, modify clause (c) in the description of the situation given in the next paragraphso that only one of the two Migs are hit. See also Karttunen (1971).

92 I.e, ‘The brother is such that he and the [i.e., his] sister are sitting next to each other’ and‘The sister is such that the [i.e., her] brother and she are sitting next to each other.’

93 On this reading, the sentence is true on a situation in which the relevant people are threepairs of people (two brothers, two sisters, and a brother and sister) while the people in the third pair(the brother and sister) are sitting next to each other. On readings on which ‘the brother’ or ‘thesister’ takes the wider scope, however, the sentence must be false on the same situation; if ‘thebrother’, for example, takes the wider scope, it cannot be taken to have an implicit pronoun relatingto ‘the sister’ and must be taken to be interchangeable with ‘the brother of someone’, a descriptionimproper on the situation.

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(37) a. The piloti who shot at the Mig that chased himi hit the Mig that chased himi.

b. The pilot who shot at itj hit the Migj that chased the pilot who shot at itj.90

These sentences involve no ‘crossing coreference’, and we can take the italicized phrases to take the

wider scope. We can then apply the Russellian analysis to the phrases, and see that the sentences

are not logically equivalent.91 So Karttunen concludes that (35) is ambiguous.

I agree that the sentence might be taken to result from (37a) or (37b) by pronominalization,

viz., by replacing the phrases in boldface with pronouns. But I think (35) has another reading, the

natural one on which neither definite description has the wider scope. Consider, e.g., ‘The brother

and the sister are sitting next to each other.’ Although one might take this to abbreviate sentences

where ‘the brother’ or ‘the sister’ has the wider scope,92 it has another reading, one on which neither

phrase falls under the scope of the other. Similarly, ‘The brother is sitting next to the sister’, I think,

has a reading on which neither description has the wider scope.93 It is the same with (35). We can

capture this reading, which I think is more natural, with liberalized notions of quantifiers and

determiners. On my reading, which is analyzed in Def. 8, (35) is equivalent to neither (37a) nor

94 By using branching quantifiers and applying game-theoretic semantics, Hintikka andSaarinen (1975) propose an analysis of Bach-Peters sentences on which neither of the descriptionsin the sentences take the wider scope. But their analysis is not equivalent to mine. On their analysis,(35) is true as long as there are a pilot and a Mig such that (a) the Mig is the only one that chasedthe pilot, (b) the pilot is the only one who shot at the Mig, and (c) the pilot hit the Mig. Theyconcede that this consequence of the analysis is “a little surprising”, but argue that “this is howthings ought to be” (ibid., 5). Although their analysis might relate to one possible reading of thesentence, I see no reason to rule out the stronger and more natural reading. It is notable that their

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(37b). (35), on this reading, implies both (37a) and (37b). But these, even taken together, do not

imply (35). To see this, consider a situation involving two pilots (P1 and P2) and two Migs (M1 and

M2) in which the following hold:

(a) M1 chased P1 and P2, while M2 chased only P2;

(b) P1 shot only at M1, while P2 shot at both M1 and M2; and

(c) P1 and P2 hit M1 and M2, respectively.

On this situation, (37a) is true (for P1 hit M1) and so is (37b) (for P2 hit M2). But (35), on my

reading, is false on the situation (for two different pilots shot at Migs chasing them).

Although my reading of (35) is not equivalent to either of Karttunen’s readings, my analysis

of the double the construction is useful for analyzing these readings as well. To see this, note that

we can use the analysis to degenerate double the constructions, namely, those that do not involve

‘crossing’, such as ‘The pilot hit the Mig that chased him.’ We can apply the analysis to this

sentence by taking ‘the pilot’ to involve the degenerate two-place predicate ‘xy pilot(x)’, and the

resulting analysis is logically equivalent to the result of applying the Russellian analysis by taking

‘the pilot’ to take the wider scope. Similarly, we can apply the double the analysis to (37a) and (37b)

without taking one of the descriptions in them to take the wider scope. By doing so, we can get

logical equivalents of the Russellian analyses of the sentences.94

analysis (unlike mine) diverges from the Russellian analysis on degenerate double the constructions.See also McCawley’s discussion of the analysis (1981, 455f).

95 Geach discusses (36) and a closely related sentence, ‘Every man who owns a donkey beatsit’, which he attributes to medieval works (1964, 128f & 116–8), but he does not discuss the donkeyanaphora they involve. The anaphora is discussed by Frege (1892/1960, 71f). See note 101.

96 In addition, Evans concludes that the pronoun it in (36), unlike that in (36), is an E-typepronoun (see note 45). As discussed below, however, the argument has an obvious problem, as heseems to concede (see the statement quoted below). He also argues that ‘Socrates owned a dog andit bit Socrates’ involves an E-type pronoun by holding that the sentence implies ‘Socrates owned justone dog’ (ibid., 343). I disagree. ‘Once upon a time, there was a smart boy living in a beautifulvillage, and one day he followed his father to go to a nearby town’ does not imply ‘There was exactlyone smart boy living in a beautiful village’ or ‘There was exactly one smart boy living in that

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7.3. Donkey anaphora

(36), ‘If Smith owns a donkey, he beats it’, is one of the so-called donkey sentences, sentences

involving a special kind of anaphora, the so-called donkey anaphora.95 The anaphoric pronoun it

in the sentence crosses the boundary between the two clauses of the sentence to take the indefinite

nominal ‘a donkey’ as its antecedent, and the sentence is interchangeable with the result of replacing

the article a in the nominal with any:

(36) If Smith owns any donkey, he beats it.

So (36) is logically equivalent to the manifestly universal sentence ‘Smith beats any donkey he

owns.’ But ‘a donkey’ and ‘any donkey’, it seems, involve the existential and universal quantifiers,

respectively. If so, how can (36) be interchangeable with (36)?

In (36), as in (36), the anaphora crosses the boundary between the two clauses of the

sentence. So it is necessary to take the antecedent, ‘a donkey’, to take the widest scope covering

both clauses. But Evans (1980, 342) argues that one cannot do so.96 If the phrase takes the widest

village.’ And one can say, with no contradiction, ‘I have a copy of Euclid and you, Tom, can haveit. And I have another copy and I can give it to you, Dick, if you like.’ (In response to a similarexample given by Geach, ‘Socrates kicked a dog and it bit him, and then Socrates kicked another dogand it did not bite him’, he holds that it implies ‘There was a time at which Socrates kicked exactlyone dog’ [ibid., 343]. But one cannot make the same response to my example.)

97 See also Russell (1919/1920, Chapter 16).

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scope in (36), the sentence must be interchangeable with an existential sentence (‘There is a donkey

that Smith, if he owns it, beats’), he argues, because the article a signifies existence. Here he

assumes the prevalent view propounded by Russell (1905)97 that relates a to the existential quantifier.

But the article cannot always be taken to signify existence. Consider (38a) and (38b):

(38) a. A whale lives in Lake Ontario.

b. A whale is a mammal.

The italicized a in (38b), unlike that in (38a), cannot be taken to signify existence; (38b) is not

interchangeable with ‘There is a whale that is a mammal’, but with ‘Any whale is a mammal.’ Evans

seems to concede. He suggests, in a footnote, that the article is ambiguous; he says, “the

interpretation of a-expressions is unclear, and we may be forced to recognize that they are sometimes

used as equivalent to any” (1980, 343). If so, one might take ‘a donkey’ in (36) to be used

equivalently with ‘any donkey’ while taking the wider scope. This would explain why (36) is

interchangeable with the universal sentence (36), where ‘any donkey’ takes the wider scope.

But one cannot apply the same strategy to analyze closely related sentences involving the

donkey anaphora, such as (39):

(39) If Smith owns donkeys of the same age, he places them in the same stable.

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This is interchangeable with universal sentences, such as (40a) and (40b):

(40) a. If Smith owns any donkeys of the same age, he places them in the same

stable.

b. Smith places in the same stable any donkeys of the same age that he owns.

But one cannot attribute the universality of (39) to a determiner with quantificational meaning. The

antecedent of them in the sentence (‘donkeys of the same age’) is a bare nominal, a common noun

phrase not led by a determiner. Moreover, sentences featuring bare plural nominals (in short, bare

plurals), like those featuring singular indefinites (e.g., ‘a donkey’), have a universal/existential

flexibility:

(41) a. Donkeys (of the same age) can be found in the island.

b. Donkeys are mammals.

c. One must yield to donkeys drawing a cart.

(41a) is interchangeable with ‘There are donkeys (of the same age) that can be found in the island’,

and (41b) and (41c) with ‘Any donkeys are mammals’ and ‘One must yield to any donkeys drawing

a cart’, respectively. But one cannot attribute the flexibility of sentences featuring bare plurals to

determiner ambiguity, for the nominals have no determiners leading them.

Where, then, can we locate quantificational content (the existential or universal meaning)

of such sentences: (41a), (41b) etc.? I do not think the sentences have any word or phrase signifying

existence or universality. Neither the bare nominals nor the predicates combining with them in the

sentences have components with quantificational meaning. If so, how can the sentences be given

98 Predicates with wider extensions are in general more likely to favor the universal reading.Note that compound predicates figuring in donkey sentences have very wide extensions.

99 The subscript i indicates that them takes the nominal filling ‘...’ as the antecedent.

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the universal or existential reading? The answer, I think, is that the linguistic and extra-linguistic

contexts help to determine their quantificational contents. The existential or universal reading might

be the natural or predominant one considering linguistic or extra-linguistic factors. While it is

natural to take, e.g., (41a) to be existential, it would be natural to take ‘Donkeys live on the Earth’

(as said by Martians) to make a universal statement, and the universal reading is predominant for

(41b). Although the predicates in the sentences have no components with quantificational meaning,

they might favor the universal or existential reading in relation to nominals combining with them

unless the nominals themselves have quantificational content.98

I think the universal reading is predominant for (39) for the same reason. The sentence

attains the universal meaning by the coordination between the bare nominal ‘donkeys of the same

age’ and the compound predicate ‘If Smith owns ...i, he places themi in the same stable.’99 The

nominal is neutral between the universal and existential readings, but the universal reading is favored

by the predicate. This makes the sentence interchangeable with manifestly universal sentences, (40a)

and (40b): in (39), the bare plural, which fills the gap marked by ‘...i’ to be the antecedent of ‘themi’,

must have the widest scope, as the nominal led by any does in (40a); and the predicate favors the

universal reading, while the nominal has no resistence to it.

Now, we can give the same account of the universality of (36), which involves a singular

indefinite: ‘a donkey’. Like their bare plural cousins, singular indefinites are by themselves neutral

with regard to the universal and existential readings, but predicates combining with them (often with

extra-linguistic factors) might favor one reading over the other. It is for this reason that (38a) and

(38b), for example, attain their existential and universal meanings, respectively. Similarly, (36) is

100 Compare them with ‘x+1 = 2’ and ‘Here comes she who must be obeyed!’

101 Frege (1892/1960) suggests this view in his discussion of donkey anaphora. While givingas an example ‘If a number is less than 1 and greater than 0, its square is less than 1 and greater than0’, he says: “In conditional clauses, also, there may usually be recognized to occur an indefiniteindicator, having a similar correlate in the dependent clause. . . . It is by means of this veryindefiniteness that the sentence [= the above-mentioned] acquires the generality expected of a law.It is this which is responsible for the fact that the antecedent clause alone has no complete thoughtas its sense and in combination with the consequent clause expresses one and only one thought,whose parts are no longer thoughts” (ibid., 71f; my italics).

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interchangeable with manifestly universal generalizations, because the predicate combining with ‘a

donkey’ in the sentence (i.e., ‘If Smith owns ...i, he beats iti’) strongly favors the universal reading.

On this analysis, the indefinite article a does not amount to a quantifier, existential or

universal. It does not signify universality in (38b), nor does it signify existence in (38a). And

nominals led by the article (i.e., singular indefinites), like bare plurals, are by themselves neutral or

‘indefinite’ between the universal and existential readings for sentences featuring the nominals, just

as free variables in mathematical equations (e.g., ‘x+y = y+x’ and ‘x+1 = 2’) are by themselves

neutral between the universal and existential readings for the equations. It is this neutrality, I think,

that makes room for the quantificational flexibility of sentences featuring indefinite nominals. The

equation ‘x+y = y+x’ can express the law of commutativity, even though the variables do not signify

universality, and ‘He who hesitates is lost’ makes a universal statement, even though the pronoun

he by itself has no quantificational content.100 Similarly, on my view, (36) attains universality and

(38) existentiality, even though ‘a donkey’ and ‘a whale’ have no quantificational content.101

This account applies to indefinite nominals led by numerals (e.g., ‘three donkeys’) as well.

Sentences featuring such nominals also have universal/existential flexibility. While ‘Three donkeys

are drawing a cart’ is interchangeable with the existential ‘There are three donkeys that are drawing

a cart’, ‘Three non-linear points determine a plane’ is interchangeable with ‘Any three non-linear

points determine a plane.’ Similarly, ‘If three points are non-linear, they determine a plane’ is

102 Note that while ‘One donkey is drawing a cart’ is existential, ‘One donkey cannot drawa cart alone’ is universal. (Consider also ‘One man’s enemy is another’s best friend’, ‘Two handsare better than one’ and ‘One hand is not as good as two.’)

103 The article a (or an) stems from the old English numeral n ‘one’; see, e.g., Jespersen(1954, 407), Hewson (1972, Chapter 1) and Sommerer (2011, 291). So Jespersen says the articleis “historically a weakened form of one” (1964, 174). Indefinite articles in many other languages,too, derive from numerals for one.

69

interchangeable with ‘If any three points are non-linear, they determine a plane.’ This

quantificational flexibility, on my account, is not due to the ambiguous quantificational meaning

inherent in the numeral three, but stems in part from the absence of any such meaning in the

numeral. Likewise with nominals led by other numerals, including one.102 In this respect, numerals

draw parallels with the indefinite article a.

Instead of comparing numerals to the indefinite article, however, it would be useful to

compare the article to numerals, for it derives from the Old English word for one.103 Like its numeral

cousin, one, on my account, the article is not equipped with any quantificational meaning. This does

not mean that sentences featuring nominals led by the article can have no quantificational reading,

any more than the lack of quantificational meaning in the numeral means that sentences featuring

nominals led by it can have no quantificational reading. The existential or universal meaning, we

have seen, can be attained by sentences with no quantificational components: (41a), (41b), ‘He who

hesitates is lost’ etc.

Finally, let me discuss indefinite nominals led by some. It is useful to discuss them together

with the indefinite pronouns something, somebody and someone. Such nominals might seem to carry

existential meaning. Unlike (38b) and (41b) (i.e., ‘A whale is a mammal’ and ‘Donkeys are

mammals’), ‘Something is a mammal’ and ‘Some donkeys are mammals’ cannot be taken to be

universal. But this does not mean that some-indefinites by themselves exclude the universal

meaning. Like other indefinites, they can figure as the antecedents of donkey anaphora:

104 Consider also ‘We’ll always ask [about it] if there is something we didn’t understand’(New York Times, Nov. 9, 2013), ‘If some arrows are green, they will hit the target’ (Harman 1970,295), and ‘Every farmer who owns some donkeys of the same age places them in the same stable.’

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(42) a. If someone comes in this room, he will trip the switch. (Evans 1977, 500)

b. If Smith owns some donkeys of the same age, he places them in the same

stable.

c. If some twin primes are greater than 217, their sum must be divisible by 6.104

These sentences are interchangeable with universal sentences resulting from replacing some or

someone with any or anyone. So I do not think some signifies existence. Although indefinite

nominals led by it give rise to a fairly strong presumption in favor of the existential reading, the

presumption is not inexorable. It can be overridden in contexts that strongly favor the universal

reading, such as compound predicates figuring in donkey sentences. This, I think, makes the

universal reading predominant for the donkey sentences, and is responsible for their

interchangeability with manifestly universal sentences.

Given the above treatment of some, it is necessary to revise the analyses of the determiner

and its cousins presented above (§5.2). We can amend the analyses by qualifying Def. 4(a):

Def. 4(a)*. (some()) df xs[(xs) (xs)], where and are plural predicates and

the predicate (as used in the context) does not override the existential

presumption for the sentence.

Replacing 4(a) with 4(a)* is sufficient, for the other clauses of Def. 4 appeal to (some()), and both

the pluralist GQT and its Russellian cousin appeal to the plural language analyses.

105 E.g., those satisfying Mostowski’s invariance condition.

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8. Concluding Remarks

Symbolic or regimented languages developed in tandem with modern logic can be considered

refinements of fragments of natural languages, and the standard quantifiers in the languages (‘’ and

‘’) refinements of some natural language expressions. Accordingly, there are a range of

expressions commonly taken to amount to those quantifiers: (a) every, any, each, all, everything,

anything etc. (universal); and (b) a, some, something, there is/are, exist etc. (existential). Many of

these are, or can be used, as determiners, which combine with common noun phrases to yield

nominals led by them. Determiners also include many other expressions often considered quantifiers

in a broad sense: the, most, numerals (e.g., one, two) etc.

GQT gives a semantic account of natural language determiners. The theory analyzes them

as functors that take noun phrases (or predicates) to yield second-order predicates, and takes the

resulting second-order predicates to be ‘generalized quantifiers’, quantifiers in a general sense. And

it assigns semantic functions to such predicates as well as to determiners: the former signify second-

order properties, and the latter signify third-order functions. In this respect, GQT deviates from

Russell (1905), according to whom denoting phrases (which include definite and indefinite

descriptions) are ‘incomplete symbols’. Russell supports this view by giving elementary language

analyses of determiners leading those phrases: every, a, the singular the etc. But taking the

determiners to amount to dyadic second-order predicates yields the same view, and this latter

analysis can accommodate the most that figures in Most boys are funny (the Rescher quantifier).

This yields the alternative to GQT that I call the Russellian GQT (or RGQT). On this theory,

determiners are dyadic second-order predicates and some of them105 are quantifiers in a liberal sense,

polyadic (or Lindström) quantifiers.

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We have discussed three constructions that give rise to problems for both GQT and RGQT:

(a) plural determiners, (b) Bach-Peters sentences, and (c) donkey anaphora. We can deal with Bach-

Peters sentences, we have seen, by making minor modifications of the theories. We can take ‘The

pilot who shot at it hit the Mig that chased him’, for example, to involve the double the construction

(‘the . . . the . . .’), which cannot in general be separated into two usual definite descriptions, and we

can analyze the pair of occurrences of the in the construction as either (a) a single triadic quantifier

that takes three two-place predicates (RGQT) or (b) a single binary functor that takes two two-place

predicates to yield a one-place predicate for relations (GQT). Resolving the other problems, I think,

requires making radical changes in the framework underlying most contemporary theories of

quantifiers, including GQT and RGQT.

Both theories give clearly incorrect accounts of plural determiners because they assume the

traditional view of plurals as devices for abbreviating singulars. But we can formulate improvements

of GQT and RGQT (the pluralist GQT and RGQT), as we have seen, by combining the GQT or

RGQT approach to determiners with analyses of basic plurals given in recent works that take an

approach to plurals (the pluralist approach) that departs radically from the traditional one.

Donkey sentences (e.g., ‘If Smith owns a donkey, he beats it’) feature indefinite nominals

that must be taken to have the widest scope to anchor pronouns that take them as antecedents: ‘a

donkey’, ‘donkeys (of the same age)’, ‘two donkeys (of the same age)’, ‘something’, ‘someone’,

‘some donkeys (of the same age)’ etc. But this gives rise to a serious difficulty; (a) the sentences are

interchangeable with universal sentences, but (b) indefinite nominals are usually taken to have

existential meaning. This view, (b), dates back to Russell (1905), whose theory of denoting includes

an account of indefinite descriptions that takes the article a to amount to the existential quantifier.

Some might modify the Russellian account by taking indefinite nominals to be ambiguous:

sometimes they have existential meaning, and sometimes universal meaning. On this

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view,‘someone’ must sometimes mean everyone; and the bare plural ‘donkeys’ must have a

component that signifies existentiality or universality, albeit ambiguously. I propose to make a

radical departure from the longstanding view of indefinite nominals as carrying quanificational

meaning. In the view that I propose, indefinite nominals have no quantificational meaning. The

nominals themselves are not equipped with the existential or universal meaning (ambiguously or

not), just as the variables x and y, the pronouns he and you, and the modified pronoun he who

hesitates, for example, have no such meaning. Still, sentences featuring indefinites (e.g., ‘You must

yield to donkeys’) can attain the universal or existential meaning in the same way that ‘x+y = y+x’

and ‘x+1 = 2’ attain their universal and existential meanings, respectively, and ‘He who hesitates is

lost’ attains the universal meaning. This, I think, explains why donkey sentences are interchangeable

with universal sentences.

On this account, English determiners are not semantically homogeneous, as they are assumed

in GQT. Some of them (e.g., every, any) have quantificational meaning, and the nominals led by

them might be taken to signify second-order properties. But some determiners (e.g., a, one, two)

have no quantificational meaning, and the nominals led by them cannot be taken to signify second-

order properties. One would need to add quantificational meaning, explicitly or implicitly. One way

to do so is to add a determiner with quantificational meaning (e.g., any), as in ‘Any two hands are

better than one.’ But this is not the only way. In ‘Two hands are {always, usually, often, sometimes}

better than one’, for example, adverbs are used for quantification, and some sentences (e.g., ‘Two

hands are better than one’) attain quantificational meaning with the aid of linguistic and extra-

linguistic contexts. So indefinite nominals (e.g., a hand, hands, two hands, some hands) cannot be

taken to signify second-order predicates. I think that they signify first-order properties, for they are

essentially nominalizations of first-order predicates: is a hand, are hands, are two hands, are some

106 They include (one-place) plural predicates, which signify plural properties.

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hands etc.106

Considering the diversity of determiners, it is not surprising that they are not semantically

homogeneous. GQT takes the so-called generalized quantifiers (or at least those with determiners)

to consist of two components: (a) a determiner and (b) a common noun or predicate phrase. But

consider, e.g., all the three students. This phrase has three determiners (all, the and three), and they

are all considered determiners in traditional English grammar. So Quirk et al. (1985, 253–265), for

example, divide English determiners into three kinds: predeterminers (e.g., all); central determiners

(articles); and postdeterminers (e.g., numerals). In GQT, however, all, the and three cannot all be

taken to figure as determiners in the phrase; if three, for example, is a determiner, then three

students, on GQT, must be a ‘generalized quantifier’ (i.e., a second-order predicate) and cannot

figure as an argument for another determiner, the. The theory shelves this problem by taking the full

phrase to have just one determiner: the compound all the three. In doing so, GQT adopts a notion

of determiner quite different from the traditional syntactic notion. It is doubtful that the GQT notion

can be justified via studies of English syntax. More importantly, one cannot use the theory at all to

analyze the meaning of the compound all the three in terms of those of its components. Once one

undertakes the task of analyzing the semantic contributions of the components, it should be clear that

the components cannot be semantically homogeneous.

On my account, the three determiners all, the and three have different kinds of semantic

function. This account can explain how the determiners can be successively added to bare noun

phrases (e.g., students). First, the numeral three amounts to a predicate, and it combines with

students to yield a noun phrase, three students, that is tantamount to a compound first-order

predicate; next, the determiner the can combine with the noun (or predicate) phrase to yield the

definite description; finally, the predeterminer all (which is essentially short for all of on my

107 Then those who think, as I do, that a predicate amounting to ‘H’ is logically more basicthan include might argue that such a predicate underlies include, but this is not necessary to meetBen-Yami’s objection. Those who consider the counterpart of include a basic predicate can define‘H’ in terms of it, for ‘H’ is logically equivalent to the singular reduct of ‘’ (viz., ‘x,ys (x, ys)’).For the notion of singular reduct, see my (2002, 25–28; 2005, 485; 2006, 243).

75

analysis) can take the description to yield all the three students. Moreover, it is straightforward, as

we have seen, to use these analyses of the three determiners to define a compound determiner

amounting to all the three.

Appendix: is one of and include

Ben-Yami (2009, 219–223) objects that plural languages deviate from natural languages by

introducing the predicate ‘H’ or ‘’. He argues that one cannot properly explain the meaning of

putative natural language counterparts of the latter (among and be some of) without using is one of,

but that the English is one of cannot be considered a predicate. While his discussion raises some

interesting issues about proper analyses of the English phrase (and its analogues in other natural

languages) and legitimate processes for introducing refinements of natural language expressions into

regimented languages, it is not necessary to resolve these issues to justify introduction of ‘H’ or ‘’

as a predicate amounting to a natural language expression. As noted in §4, the English include

amounts to the converse of ‘’, and one cannot deny that this is a legitimate predicate. So one can

introduce its counterpart (or the converse thereof) into plural languages.107 Moreover, the predicate

has logical significance. Consider, e.g., (43a) and (43b):

(43) a. The boys who lifted Bob include the boys in the club, and these boys are

funny.

b. Some of the boys who lifted Bob are funny.

108 Some might consider using the some of in (43b) to analyze include (or its converse). ButBen-Yami’s argument is directed against this very procedure. Note that we can use ‘’ or itsconverse to analyze some of (see §5.2 and §6).

109‘@’ is the term-connective amounting to and. In languages with ‘H’ or ‘’, we can defineit (see note 61). In Ben-Yami’s languages, we can introduce it directly as a counterpart of a naturallanguage expression.

76

These are logically equivalent, but one cannot explain this without using a language that has an

analogue of include (or other expressions one can use to analyze it), and as a logical expression.108

It is notable that while arguing against introducing ‘H’ or ‘’, Ben-Yami uses the plural

identity predicate ‘’ (presumably as a basic logical expression) in formulating [P3] (see §4), which

he calls “Leibniz Law” (ibid., 229). Note that we can use this predicate to define both ‘H’ and ‘’:

(a) xHysdf [x@ys] ys. (Or H df x,ys ([x@ys], ys).)

(b) xs ys df [xs@ys] ys. (Or df xs,ys ([xs@ys], ys).)109

This means that ‘H’, ‘’ and ‘’ are interdefinable. While I think it is more natural to use one of

the first two to define the third, Ben-Yami might argue that the only correct way to proceed is to

regard ‘’ as basic and analyze the others in terms of it, because ‘’ is the only one among the three

that amounts directly to a natural language predicate. But this is not correct, as we have seen. The

predicate include in (43a) is as natural as the are (or be) in ‘Russell and Whitehead are the authors

of Principica Mathematica.’ I do not think one can take the plural ‘’ to be unanalyzable. And

although the definitions given above, (a) and (b), give logical equivalents of ‘H’ or ‘’, they are not

as natural as the usual definitions of ‘’ in terms of ‘H’ or ‘’ (e.g., Def. 1(a)).

77

Acknowledgments

This paper is a sequel to my (2012b). The research for the two papers was supported in part by

SSHRC Standard Research Grants, which are hereby gratefully acknowledged. I presented the

ancestors of this paper (including the prequel) at the 6th International Conference of Cognitive

Science and Formal Grammar 10, and at Seoul National University and University of Padua. I would

like to thank the audiences of the presentations and Joseph Almog, Ranpal Dosanjh, Adam Harmer,

Kevin Kuhl, Chungmin Lee, Adam Murray, Seungho Nam, Jonathan Payton, Richard Zuber and two

anonymous referees for Oxford University Press for useful comments. Special thanks are due to

Hanoch Ben-Yami, Thomas McKay and Yiannis Moschovakis for comments and discussions. I

wish to dedicate this paper, together with its prequel, to David Kaplan in appreciation of his

pioneering work on plurals.

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Department of Philosophy, University of Toronto, Toronto, ON M5R 2M8, Canada

Department of Philosophy, Kyung Hee University, Dongdaemun-gu, Seoul, 130-701, South Korea

b.yi@utoronto.ca