Dependent Indefinites in Donkey Sentences

Abstract

This chapter revisits the problems raised by donkey-sentences without resorting to an analysis in terms of scope. We show the limits of the quantificational approach on the one hand and of DRT analyses on the other hand. These accounts are problematic because they treat indefinite DPs on a par with quantified expressions. Pursuing the line of analysis developed in Chap. 6, we show that by implementing dependency relations in functional terms and by treating both indefinite DPs and pronouns occurring in donkey-sentences as functional terms, we can solve the proportion problem, we can account for the whole range of available readings of donkey-sentences (symmetric/asymmetric, weak/strong) and we can predict when and why an indefinite DP can or cannot serve as the antecedent of an anaphoric pronoun.

Appendix

Here we will present two pre-DRT proposals that attempted to solve the problems raised by donkey sentences. The first is due to Egli (1979), who elaborated an algorithm that translates a fragment of English (including DPs headed by every, indefinite DPs, proper names and pronouns) into formulas of predicate logic. Egli’s proposal presents two drawbacks: it predicts some readings that do not exist and it postulates rules of translation for indefinite DPs that are context-dependent. It turns out that this semantics of indefinite DP’s is not compositional (see Sect. 8.1). The second proposal, due to Evans (1980) relies on the idea that pronouns in donkey sentences are not bound pronouns but rather ‘E(vans)-type pronouns’, which are disguised definite descriptions, since they are equivalent to definite descriptions that can be recovered from the context (see Sect. 8.6.1). Cooper (1979) proposed a formal analysis of pronouns, which can be used to account for donkey sentences. The major problem with Cooper’s proposal is that it predicts too many interpretations due to the fact that the interpretation of the pronoun does not depend on the syntactic environment of the pronoun but on pragmatic factors (see Sect. 8.6.2).

8.1.1 Egli’s (1979) Solution

Egli (1979) argues that in predicate logic, the formulas in (1a) and (1b) are equivalent, as long as there is no free occurrence of x in ϕ.

(1)	a.	(∃x Φ)  →  ϕ
	b.	∀x (Φ  →  ϕ)

An existential quantifier in the antecedent of a conditional clause is equivalent to a universal quantifier taking scope over the conditional. This equivalence can explain the universal interpretation of indefinite DPs in donkey sentences if there is no anaphoric pronoun in the consequent. However, since the equivalence in (1) only holds for cases where there is no free occurrence of x in ϕ, it cannot account for cases where there is a pronoun in the consequent, which acts as a free variable.

Egli suggests that the logical equivalence in (1) holds in natural language, where the restriction concerning the variables in ϕ does not apply. According to him, it is always possible to replace the existential quantifier in the antecedent of a conditional with a universal quantifier having wide scope over the conditional, regardless of whether or not the consequent contains unbound occurrences of the free variable. His hypothesis is supported by other cases of binding of pronouns that occur outside the scope of their antecedent.

(2)	a.	Somebody left, and he went home.
	b.	Somebody left. He went home.
	c.	Somebody left and went home.

In other words, on Egli’s proposal, there is a mismatch between scope in logic and scope in natural language.

In order to account for the interpretation of indefinite DPs in conditionals, Egli posits the existence of a specific rule, without, however, taking a clear position on the precise nature of this rule. Apparently, it is not a grammatical rule but a rule that applies to an intermediate level, where the meaning of the sentence is computed. For example, in analyzing (3), Egli first builds the intermediate representation (3′a) in which the quantified DP is translated as a universal quantifier that scopes over the implication. He then applies a conversion rule, which turns the existential ∃ corresponding to the indefinite DP a donkey into the universal ∀ resulting in the representation in (3′b).

(3)		Every farmer who owns a donkey beats it.
(3′)	a.	∀x [(man(x) Ù x owns a donkey )  →  (x beats it )]
	b.	∀x∀y [(man(x) Ù donkey(y) Ù x owns y)  →  (x beats y )]

Egli’s solution has several advantages but also a number of shortcomings. One advantage is that it captures the interpretation of examples such as (4a), which cannot be paraphrased as in (4b):

(4)	a.	One of my friends who owns a donkey beats it.
	b.	For each donkey, if one of my friends owns it, then, he beats it.
(4′)	a.	∃x (friend-of-mine(x) Ù x owns a donkey Ù x beats it)
	b.	∃x ∃y (friend-of-mine(x) Ù own(x,y) Ù donkey(y) Ù beat(x,y))

The fact that (4a) cannot be paraphrased as (4b) is due to the nature of the DP on which the relative depends. The DP one of my friends (unlike every farmer in (3) above) is not a universally quantified DP; rather, it is an indefinite DP that is associated with existential quantification. The sentence in (4a) has the intermediate representation in (4′a). Notice that the conversion rule that transforms the existential ∃ corresponding to the indefinite DP a donkey into the universal ∀ does not apply in this case, because there is no logical implication in (4′a) (contrary to what happens in (3′a)). The representation in (4′b) is obtained by replacing a donkey in (4′a) with a variable bound by an existential quantifier. To sum up, when the DP on which the relative clause depends is not universally quantified, we are not in a donkey context anymore.

However, Egli’s solution raises at least two problems. As we have already said, it is not clear what the nature of the rule that converts ∃ to ∀ is. It is also unclear why the constraint on the absence of free variables disappears in natural language.

Note moreover that in addition to the equivalence in (1), the equivalence in (5) also holds in predicate logic. According to this equivalence, a universal quantifier can scope out of the antecedent of a conditional once it is assigned an existential value, provided there are no free occurrences of x in ϕ:

(5)	a.	(∀x Φ)  →  ϕ
	b.	∃x (Φ  →  ϕ)

The question is why the logical equivalence in (5) does not hold for natural language sentences (possibly with the difference observed for (1) concerning the variables in ϕ). Egli observes this difference between predicate calculus and natural language LFs but does not provide an explanation. He points out that binding is not possible in examples such as (6) but does not consider sentences such as (7a), which would come out equivalent to (7b), if the rule in (5) applied. And yet, (7a) and (7b) have clearly different meanings:

(6)		*If every farmer works, he becomes rich.
(7)	a.	If every farmer vaccinates his donkey, the disease will not spread.
	b.	There is a farmer such that, if he vaccinates his donkey, the disease will not spread.

By changing the notion of scope, Egli treats connectives like Ù and  →  as dynamic connectives but this provides only a partial and somewhat ad hoc solution to the problem of donkey sentences. Egli’s solution is not entirely satisfactory because he analyzes indefinite DPs as quantified expressions and it is precisely this assumption that needs to be changed.

8.1.2 E-type Analyses

Another way of accounting for donkey sentences is to analyze the pronouns that occur in these contexts not as bound pronouns but as disguised definite descriptions.

8.1.2.1 E-type Pronouns: Evans (1980)

Evans (1980) puts forth a different classification of pronouns. The originality of this approach consists in introducing a novel class of pronouns, which he calls E-type pronouns, where E is the first letter of his name. E-type pronouns are those pronouns whose antecedent is a quantified expression and which are not in the scope of the quantifier that the antecedent introduces. To put it differently, Evans makes a distinction between bound pronouns, which act as variables bound by a quantifier, and E-type pronouns. The pronoun his in (8) for example is a bound pronoun and them in (9) is an E-type pronoun.

(8)	Every man loves his mother.
(9)	John owns some sheep and Harry vaccinates them.

There are two tests that distinguish E-type pronouns from bound pronouns. On the one hand, unlike what happens in the case of bound pronouns, the antecedent of an E-type pronoun cannot be replaced with an expression such as no N:

(10)	a.	Every farmer who owns a donkey beats it.
	b.	*Every farmer who owns no donkey beats it.
(11)	a.	Every man loves his mother.
	b.	No man loves his mother.

On the other hand, E-type pronouns, but not bound pronouns, have interrogative counterparts:

(11)	c.	??Who does every man love? Every man loves his mother.
(12)		John owns some sheep and what does Harry vaccinate? John owns some sheep and Harry vaccinates them.

E-type pronouns are reminiscent of Russell’s view of pronouns as abbreviations of complex nominal expressions, which are used instead of full DPs for stylistic reasons. It is however difficult to determine what exactly these pronouns replace. One possibility would be to assume that they replace their antecedent, which would suffice to copy in order to obtain a paraphrase. It is however rarely the case that we obtain a good paraphrase by replacing the pronoun with its antecedent:

(13)	a.	John is tall. He is handsome.
	b.	John is tall. John is handsome.
(14)	a.	A man walks in the garden. He smokes.
	b.	A man walks in the garden. A man smokes.
	c.	A man walks in the garden. This man smokes.
	d.	A man walks in the garden. The man who walks in the garden smokes.

While (13a) and b are equivalent, this is not the case for (14a) and b. We can paraphrase (14a) either by replacing the pronoun with a demonstrative DP, as in (14c), or by reconstructing a definite description, as in (14d).

The problem observed in (14) also appears with E-type pronouns. Substituting the pronoun with its antecedent changes the meaning of the sentence, as shown in (15) and (16).

(15)	a.	Every farmer who owns a donkey beats it.
	b.	Every farmer who owns a donkey beats a donkey.
	c.	Every farmer who owns a donkey beats (this donkey/the donkey he owns).
(16)	a.	John owns some sheep and Harry vaccinates them.
	b.	John owns some sheep and Harry vaccinates some sheep.
	c.	John owns some sheep and Harry vaccinates (these sheep/the sheep that John owns).

In these examples, the E-type pronouns were replaced with definite descriptions. This step is intuitively correct but raises several problems. On the one hand, in order to make the procedure compositional, one needs to explain how the definite description is built, a non-trivial task considering that the reference of the noun is always restricted by the relative clause. This is particularly problematic for those sentences where the antecedent of the E-type pronoun is a quantified expression such as someone, as in (17):

(17)	a.	If someone lives in Paris, he does not live in London.
	b.	If someone lives in Paris, this someone does not live in London.
	c.	If someone lives in Paris, the person who lives in Paris does not live in London.

On the other hand, the reconstructed definite description carries a uniqueness presupposition, which seems counter-intuitive. Consider example (15) again. In (15c), the donkey he owns is a definite description that presupposes that the person in question owns only one donkey. Yet, it seems obvious that the sentence in (15a) does not presuppose that there is no farmer who owns more than one donkey. It also seems difficult to assume that the sentence only refers to those farmers who own a single donkey and does not say anything about the other farmers. Many scholars have tried to understand what exactly (15a) says about farmers who own more than one donkey. Is the sentence in (15a) appropriate or inappropriate in a context where a farmer owns several donkeys and beats only one? This brings us back to the proportion problem, discussed in Sect. 8.1.3 and Sect. 8.3 above.

Let us return to example (17a) and its version in (17b), where the E-type pronoun was replaced with a definite description. Heim (1982) showed that an E-type pronoun analysis induces the presupposition that there is only one person who lives in Paris, which is untenable. One solution proposed by Davidson, Parsons and others is to say that the uniqueness presupposition is associated with situations/events, rather than with donkeys or inhabitants of Paris. We may then consider a minimal situation where there is only one person living in Paris.

There still remains one problematic case, namely the case of symmetric situations pointed out by van Eijck and Kamp (1997) and illustrated in (18):

(18)	a.	If a man lives with another man, he shares the desk with him.
	b.	If a cardinal meets another cardinal, he blesses him.

These examples are problematic for the minimal situation approach, since the minimal situation in which the sentence is true contains two individuals and therefore the uniqueness presupposition does not hold.

It seems nevertheless quite clear that the pronouns that are used in donkey ­sentences are E-type pronouns. In the next section, we review the formal implementation of this kind of analysis, due to Cooper (1979).

8.1.2.2 A Formal Analysis with Lambda Operators

Cooper (1979) proposed that pronouns should be analyzed as incomplete definite descriptions. This can be formally represented by means of a lambda operator, which abstracts over a free variable denoting an n-ary relation. The value of this n-ary relation, just like the value of the relation holding between the variables v₁,…, v_n−1, is determined by the context:

(19)	λK ∃x [∀y [R(v1, v2, …, vn−1, y) ↔ x  =  y] Ù K(x)]

This implementation has the advantage of giving a uniform account of pronouns: it accounts for E-type pronouns as well as ordinary pronouns. To allow a pronoun to pick up the same referent as its antecedent, it suffices to empty the content of the R-relation by reducing it to, say, an identity relation. On this view, the representation in (19) is reduced to (20):

(20)

λ K K(x)

This analysis can also account for the so-called laziness pronouns, as in (21):

(21)	John gave his paycheck to his mistress. Everyone else kept it for himself.

The pronoun it in the second sentence corresponds to the definite description his own paycheck and can be formally represented by means of a n-ary relation:

(22)	λK ∃x [∀y [paycheck(v1, y) ↔ x  =  y] Ù K(x)]

This kind of account also captures the interpretation of E-type pronouns.

Its major drawback is that anaphora resolution is relegated to pragmatics, since it is the job of pragmatics to determine how the n-ary R relation is instantiated. Moreover, according to this analysis, all pronouns are analyzed in the same way, regardless of whether their antecedent is a proper name, a definite, an indefinite or a quantificational DP. It is important to note that Cooper’s solution is based exclusively on the analysis of pronouns; the representation of the indefinite DP in sentences such as (15a) is taken to be irrelevant. The element that acquires a universal interpretation (see ∀ in the formula in (19)) is not the indefinite DP but the definite description that stands for the pronoun. When the reconstructed definite description is in the singular, the N W carries a uniqueness presupposition and when it is in the plural, the N Ws reads as all of the N W.