## 1. Unproblematic Anaphora



The simplest sorts of anaphoric pronouns are those that "pick
up" a reference from a previous referring expression whether in
the same sentence or another. Consider for example:



(3)
 John left. He said he was ill.
(11)
 John left his wallet on the table.


on the readings of these sentences on which "he" and
"his" co-refer with "John". In such cases, the
pronouns are anaphoric, and the expression "John" is
called *the antecedent* of the anaphoric expression. The
semantics of such anaphoric pronouns is very simple: the referent of
the anaphoric pronoun is the referent of its antecedent.



As indicated above, there are also anaphoric pronouns with quantifier
(rather than referring expression) antecedents. Examples include
 (2)
 above and:



(12)
 Every
male skier loves his mother.


Again, on the readings of these sentences on which "he"
and "his" "look back" to their antecedents for
interpretation rather than being assigned independent reference (e.g.,
by pointing to Chris when uttering "his" in (12)). It is
widely held that in such cases, the pronouns function semantically as
variables bound by their quantifier antecedents. Thus, their semantic
function is just like that of bound variables of first order logic.
 [3]
 The insight that some pronouns with quantifier antecedents function
like bound variables in first order logic goes back at least to Quine
(1960, see chapter IV section 28). Historically, these cases have been
of more interest to linguists than philosophers. For example, one
thing that needs to be explained is that in a sentence like
"Sarah likes her", "Sarah" and
"her" cannot co-refer (in order to get a co-reference
reading, we must say "Sarah likes herself"), though we can
get the co-reference reading (or not) in a sentence like "Sarah
likes her sister". Accounting for these types of patterns of
sentence-internal anaphora is the central concern of the area of
linguistics called *binding theory* (see May 1980; Higginbotham
and May 1981; Chomsky 1981; Reinhardt 1983a,b; Buring 2005).
Though these insights are all important, if examples like
 (3),
 (11), and
 (12)
 were the only kinds of pronominal anaphora, they currently would not
be of much interest to so many philosophers and linguists. 


## 2. Problematic Anaphora



Significant interest in anaphoric pronouns grew out of the realization
that there are anaphoric pronouns that cannot be understood as having
their references fixed by their antecedents (as in
 (3)
 and
 (11)
 above) nor as being variables bound by their quantifier antecedents
(as in
 (2)
 and
 (12)
 above). The three sorts of examples of this discussed here have
figured prominently in the literature on anaphora.



First, there is *discourse anaphora*: cases in which an
anaphoric pronoun has an antecedent in another sentence, where that
antecedent at least appears to be a
 quantifier.[4]
 Examples include:



(13)
 An
anthropologist discovered the skeleton called "Lucy". He
named the skeleton after a Beatle's song.


(14)
 Few
professors came to the party. They had a good time.


There are at least two reasons for thinking that the pronouns in (13)
and (14) are not variables bound by their quantifier antecedents. Both
reasons are discussed by Evans (1977). The first is that such a
treatment clearly yields the wrong truth conditions for examples like
(14). If "they" is a bound variable in (14), the two
sentences should be equivalent to



(14a)
 Few
professors : *x* (*x* came to the party and *x* had a
good time)


(Or, more colloquially, "Few professors are such that they both
came to the party and had a good time.") This is clearly
incorrect, since the sentences of (14) entail that few professors
attended the party (i.e., the first sentence entails this), whereas
(14a) could be true if many professors
 attended.[5]



The second reason for thinking pronouns in cases of discourse anaphora
aren't bound variables is that it seems committed to the claim
that the following anomalous sentences aren't anomalous:



\*15.
 John
bought no sheep and Harry vaccinated them.
\*16.
 Every professor came to the party. He had a great time.


If the (apparent) quantifiers in
 (13)
 and
 (14)
 can bind variables in sentences after those in which they occur, why
can't the quantifiers in (15) and (16)? If this could happen,
(15) and (16) should be fine and should together be equivalent to,
respectively:



(15a)
 No
sheep were both bought by John and vaccinated by Harry.
(16a)
 Every professor came to the party and had a great time.


But they are not. Thus, pronouns in discourse anaphora are not
variables (syntactically) bound by their quantifier
 antecedents.[6]



Furthermore, there are cases of (plural) anaphora in which the
denotation of "they" is derived in a more indirect way
from the antecedent, as in (17), an example of complement
anaphora:



(17)
 Few
students came to the party. They were busy studying.


Unlike in (14), where "they" picks out the professors who
came to the party, in (17), "they" picks out the students
who *didn't* come to the party. (For more about
complement anaphora, see Nouwen (2003a), for complement anaphora in the
psycholinguistics literature, see Sanford & Moxey (1993).)



Nor are these like example
 (3),
 in which the pronoun simply refers to the same thing as the
antecedent. Indefinite descriptions like "an
anthropologist" are commonly thought to be quantifiers, and
expressions like "few professors" are certainly
quantifiers, not referring expressions. On most theories of indefinite
descriptions, the first sentence of
 (13)
 is true just in case there is at least one anthropologist who
discovered the skeleton called "Lucy"; its truth does not
depend on any particular anthropologist. Thus there is no referential
antecedent in the first sentence for the pronoun in the second
sentence to be co-referential with.



A further problem with thinking of these pronouns as referential can
be seen by considering a slightly more complex example:



(18)
 A man
broke into Sarah's apartment. Scott believes he came in the
window.


The crucial point is that the second sentence has a reading on which
it attributes to Scott a general belief instead of a belief about a
particular person. This reading would be true, for example, if Scott
believed that some man broke into Sarah's apartment by coming in
the window, but had no idea about who might have broken in. This is
(*prima facie*) evidence that the pronoun in the second
sentence is not a referring expression, because if it were, the second
sentence of (18) would only have a reading on which it attributes to
Scott a belief about the particular person the pronoun refers to. But
this is
 incorrect.[7]
 (On the other hand, for a treatment of such *de dicto*
readings while maintaining that the pronoun is a referring expression,
see Elbourne 2005: 99-106.)



Thus, with pronouns in discourse anaphora, we have examples of
pronouns that can neither be understood as picking up their referents
from their antecedents nor as being variables bound by their
antecedents. Discourse anaphora provides further interesting examples
to philosophers and linguists when the antecedents are under the scope
of quantifiers, modals, or negation. For example, the pronouns in
(19), (20), and (22) are infelicitous, but those in (21) and (23) are
felicitous. This is something further that an account of discourse
anaphora needs to
 explain.[8]



(19)
 Bryan
didn't buy a bottle of wine. #It is a pinot noir.
(20)
 Bryan might buy a bottle of wine. #It is a pinot noir.
(21)
 Bryan might buy a bottle of wine. It would be a pinot noir. (This
an example of *modal subordination*. For an introduction to
modal subordination see Roberts (1987, 1989, 1996).)
(22)
 Bryan bought a bottle of wine at every store in the
neighborhood. #It was a pinot noir.
(23)
 Bryan bought a bottle of wine at every store in the
neighborhood. It was always a pinot noir. (This is an example
of *quantificational subordination*. For more on
quantificational subordination see Brasoveanu 2006.)


A second sort of anaphoric pronoun that cannot be understood as a
referring expression or as a bound variable is in fact a special case
of discourse anaphora. However, it deserves separate mention because
it has generated so much interest. Consider the following discourse,
which we shall call a *Geach Discourse*, adapted from the
analogous conjunction in Geach 1967:



(24)
 Hob
thinks a witch blighted Bob's mare. Nob wonders whether she
killed Cob's sow.


There is a reading of this discourse on which both sentences in it are
true even if there are no witches, so that "a witch" in
the first sentence must take narrow scope with respect to "Hob
thinks". But then the scope of "a witch" cannot
extend to the second sentence to bind the pronoun "she",
since the "scope" of "Hob thinks"
doesn't extend to the second sentence. Hence on the reading in
question, which we shall call the *Geach Reading*, the pronoun
"she" is not a bound variable. Further, since there are no
witches, and "she" is anaphoric on "a witch",
"she" in the second sentence must in some sense being used
to "talk about" a non-existent witch. Thus, it apparently
cannot be a referring term either, since its alleged referent
doesn't exist. So here again we have an anaphoric pronoun that
cannot be understood as a referring expression nor as a bound
variable. Examples of this sort are sometimes referred to
(misleadingly, in our view) as instances of *intentional
identity*.



The third sort of case in which an anaphoric pronoun cannot be
understood as a referring expression nor as a bound variable is that
of "donkey
 anaphora".[9]
 Here there are two varieties, which are called *conditional*
and *relative clause* donkey sentences, respectively:



(25)
 If Sarah
owns a donkey, she beats it.
(26)
 Every woman who owns a donkey beats it.


On the readings we are concerned with, neither (25) nor (26) is
talking about any particular donkey, and so the pronoun
"it" cannot be a term referring to a particular donkey.
Further, in the case of (25), all independent evidence available
suggests that a quantifier can't take wide scope over a
conditional and bind variables in its consequent (\*"If John owns
every donkeyi, he beats iti"). This
suggests that the (apparent) quantifier "a donkey" in (25)
cannot bind the pronoun in the consequent. In addition, even if
"a donkey" could magically do this in (25), assuming it is
an existential quantifier, we still wouldn't get the intuitive
truth conditions of (25), which require that Sarah beats
*every* donkey she owns. Similarly, the independent evidence
available suggests that quantifiers can't scope out of relative
clauses (\*"A man who owns every donkeyi beats
iti"), and so again the pronoun in (26) is not within
the scope of its quantifier antecedent and so is not bound by it.
Thus, the pronouns in both conditional and relative clause donkey
sentences can be neither understood as referring expressions nor as
bound variables.



So now we have three cases of anaphoric pronouns that cannot be
understood as referring expressions nor as bound variables: 1)
pronouns in discourse anaphora; 2) pronouns in Geach discourses and 3)
pronouns in (conditional and relative clause) donkey sentences.
Let's call these cases of *problematic anaphora*. The
recent interest in anaphora is largely an interest in finding a
semantic theory for problematic anaphora. In the next section, we
outline the main theories that have arisen to fill this void.


## 3. Recent Theories of Problematic Anaphora



Before discussing recent theories of problematic anaphora, a few
caveats are in order. First, our discussion will not be exhaustive. We
cover what we take to be the best known and most promising theories.
Second, because each theory is a formal, sophisticated semantic
theory, to describe a single theory in detail would itself be a paper
length project. Thus, we try instead to give a simple, informal sketch
of the main features of each theory. The notes and references point
the interested reader to places where he/she can get more detail.
Third, we shall confine ourselves to briefly describing how each
theory handles simple discourse anaphora of the sort exhibited by
 (13)
 above, in which a pronoun in one sentence is anaphoric on an
indefinite noun phrase in a previous sentence, and donkey anaphora
like
 (25)
 and
 (26).
 Readers interested in more details on how these theories deal with
embedding under quantifiers, negation, or modals should consult the
readings cited both in the previous section and in the sections below.
Readers interested in Geach discourses or "intentional
identity" should begin by consulting Asher (1987), Edelberg
(1986), Geach (1967), Kamp (1990), King (1994), Braun (2012), and the
works mentioned therein.



The first two of the theories discussed below, discourse
representation theory and dynamic semantics, represent departures from
traditional semantics, departures which were largely motivated at the
beginning by the problematic anaphora cases. The second two theories
discussed below, d-type theories and the CDQ theory, represent ways in
which the problematic anaphora data is dealt with within a traditional
semantic framework.


### 3.1 Discourse Representation Theory



In the early 1980s, Irene Heim (1982) and Hans Kamp (1981)
independently formulated very similar semantic theories that were in
part designed to handle problematic anaphora, particularly donkey and
discourse anaphora. The theories developed by Heim and Kamp have come
to be known as Discourse Representation Theory or DRT (see entry on
 Discourse Representation Theory).[10]
 We shall not attempt to describe differences between the formulations
of Heim and Kamp. Indeed, in our exposition we shall combine elements
of the two theories. Readers interested in the differences between the
two accounts should consult Heim (1982) and Kamp (1981) directly.



We believe it is fair to say that it was the development of DRT that
made the semantics of anaphora a central issue in philosophy of
language. One reason for this was the following bold statement by Kamp
(1981):



> 
> 
> A theory of this form differs fundamentally from those familiar from
> the truth-theoretical and model-theoretical literature, and thus a
> substantial argument will be wanted that such a radical departure from
> existing frameworks is really necessary. The particular analysis
> carried out in the main part of this paper should be seen as a first
> attempt to provide such an argument. The analysis deals with only a
> small number of linguistic problems, but careful reflection upon just
> those problems already reveals, I suggest, that a major revision of
> semantic theory is called for. (Kamp 1981: 278) 
> 
> 
> 



The problems that Kamp goes on to address are the treatment of donkey
anaphora and simple discourse anaphora. Hence Kamp appears to be
saying that these problems cannot be handled within more traditional
frameworks and thus that a DRT approach is necessary. Obviously, the
claim that the semantics of anaphora requires a radical revision in
semantic theory got the attention of philosophers of language. Thus,
the study of problematic anaphora blossomed during the 1980s and
1990s.



The first way in which DRT departs from more traditional approaches is
that it claims that indefinite noun phrases such as "an
anthropologist" or "a donkey" are essentially
predicates with free variables rather than existential quantifiers.
Thus, the above indefinites might as well look as follows at the level
of "logical form":




anthropologist(*x*)
  

donkey(*x*)




In effect, an indefinite introduces a "novel" variable,
(i.e., in DRT's terminology, establishes a *discourse
referent*) and a pronoun anaphoric on an indefinite is interpreted
as the same variable as was introduced by its indefinite antecedent.
Hence a simple discourse such as:



(27)
 A man
loves Annie. He is rich.


in effect can be represented
 as[11]



(27a)
 man(*x*)
  

*x* loves Annie
  

*x* is rich


In addition to this, DRT builds in to the assignment of truth
conditions default existential quantification over free variables.
Thus, (27a) is true iff there is some assignment to the variable
"*x*" that is in the extension of "man",
"loves Annie" and "rich", that is, iff
something is a man who loves Annie and is rich. Thus, that indefinites
appear to have the force of existential quantifiers in cases like (27)
is not because they are existential quantifiers but because of the
default existential quantification of free variables.



Let us turn now to the DRT treatment of donkey anaphora. First, note
that both relative clause and conditional donkey anaphora appear to
have a sort of "universal force": the truth of
 (25)
 and
 (26)
 above, repeated here, require that Sarah beats every donkey she owns
and that every donkey owning woman beats every donkey she
 owns.[12]



(25)
 If Sarah owns a
donkey, she beats it.
(26)
 Every woman who owns a donkey beats it.


Thus, the indefinite here mysteriously has universal force and
expresses something about every donkey owned by someone. Recall that
according to DRT, an indefinite is effectively a one-place predicate
with a free variable. The central idea of DRT in the case of both
conditional and relative clause donkey sentences is that the universal
force of the indefinite results from the variable in it being bound by
an operator with genuine universal force. In the case of (25), the
"conditional operator" has universal force, since it in
effect says that in every case (assignment to free variables) in which
the antecedent is true, the consequent is true. So (25) claims that
every assignment to "*x*" that makes "Sarah
owns *x*" and "*x* is a donkey" true, also
makes "Sarah beats *x*" true. So (25) is true iff
Sarah beats every donkey she owns. In (26), by contrast, the universal
quantifier (determiner) "every" not only binds the
variables associated with the predicative material "woman who
owns a donkey" (for example, presumably there is such a variable
in the subject argument place of "beats"), but it also
binds the variable introduced by the predicate-with-free-variable
"a donkey". So it is as though (26) has the "logical
 form".[13]



(26a)
 Every
*x*,*y* (woman(*x*) & donkey(*y*) &
*x* owns *y*) (*x* beats *y*)


Note that this account requires allowing quantificational determiners
("every") to bind multiple variables. This, again, is a
departure from more classical
 approaches.[14]



Now since the DRT approach claims that indefinites get their apparent
quantificational force from other elements that bind the variables in
them, it predicts that when different determiners are involved in
relative clause donkey sentences, as in



(28)
 Most
women who own a donkey beat it.


the indefinite should appear to have the quantificational force of the
new determiner ("Most"). So (16) should be true if most
pairs of women and donkeys they own are such that the women beat the
donkeys. Similar remarks apply to conditionals containing
"non-universal" quantifiers such as "usually",
as in



(29)
 Usually,
if a woman owns a donkey, she beats it.


This should be true if most pairs of women and donkeys they own are
such that the women beat the donkeys. But this prediction,
particularly in the case of (28), seems clearly false. If there are
exactly ten donkey owning women and one woman owns ten donkeys and
beats them all, while the nine other women own a donkey each and
don't beat them, (28) intuitively seems false: most donkey
owning women fail to beat the donkeys they own. However, the DRT
account as formulated claims (28) is true in this situation. This
difficulty is one of the main criticisms of classical DRT in the
literature and is often called *the proportion problem* (Heim
(1990) claims that Nirit Kadmon so dubbed it). The criticism is
damaging, because it appears to refute what was claimed to be a
central insight of DRT: that the apparent quantificational force of
indefinites comes from other elements that bind the variables in
them.



A second difficulty with classical DRT as formulated here involves
cases such as
 (18)
 above, repeated here:



(18)
 A man broke into
Sarah's apartment. Scott believes he came in the
window.


As mentioned above, (18) has a reading on which the second sentence of
the discourse attributes a general belief to Scott (something like the
belief that a man who broke into Sarah's apartment came in
through the window). But as formulated, DRT doesn't get this
reading. For the default existential quantification of free variables
acts in effect like a wide scope existential quantifier over the
entire discourse. Thus, it is as if (18) were as follows:




([?]*x*)(*x* is a man & *x* broke into
Sarah's apartment & Scott believes *x* came in the
window).




But this attributes a belief about a specific person to Scott. Hence
it can't capture the reading mentioned. Similarly, consider the
following sentences:



(30)
 Every
women who has a secret admirer thinks he is stalking her.
(31)
 If a woman has a secret admirer, usually she thinks he is
stalking her


These sentences also appear to have readings on which they attribute
general or *de dicto* beliefs to the women in question. That
is, they have readings on which they attribute to the women in
question *general* beliefs to the effect that they are being
stalked by secret admirers. This is why these sentences can be true
even though the women in question don't know who their secret
admirers are, and so have no beliefs about *particular* persons
stalking them. For reasons exactly similar to those given for the case
of the analogous reading of the second sentence of
 (18),
 these readings can't be captured by DRT as formulated here. We
shall see below that dynamic approaches have exactly similar problems.
Asher (1987) and Kamp (1990) attempt to remedy this problem (among
others). For further elaboration of the DRT framework, see also Kamp
and Reyle (1993) and van Eijck and Kamp (1997). For expansions of the
DRT framework, e.g., SDRT (segmented discourse representation theory),
see Asher and Lascarides (2003), for compositional versions of DRT,
see Muskens (1996) and Brasoveanu (2006, 2007, 2008).


### 3.2 Dynamic Semantic Approaches



As its name suggests, Discourse Representation Theory was designed to
capture the way in which certain features of a discourse, particularly
inter-sentential relations such as inter-sentential anaphora, affect
the interpretation of sentences in the discourse. At the same time,
Discourse Representation Theory as originally formulated in Kamp
(1981) failed to be compositional, at least in the sense of that term
familiar from Montagovian
 approaches.[15]



The initial motivation for a dynamic semantic (see entry on
 dynamic semantics)
 approach to discourse and donkey anaphora was on the one hand to
preserve the "dynamic" elements of DRT, that is the view
that what an expression means is given by the way in which the
addition of the expression to a discourse changes the information
available to a hearer of the discourse, ("meaning as potential
for changing the state of information"). On the other hand,
dynamic semantic approaches wanted to adhere to compositionality. This
is made very clear in the introduction of the classic statement of the
dynamic semantic approach to discourse and donkey anaphora, namely
Groenendijk and Stokhof (1991). We shall here discuss their treatment
of discourse and donkey anaphora, *Dynamic Predicate Logic*
(henceforth DPL), and gesture at other treatments in the dynamic
semantic tradition. First generation dynamic semantic theories of anaphora like DPL give an account of singular cross-sentential anaphora as well as donkey sentences. Subsequent work in dynamic semantics has adapted and added to these tools to account for things like plural anaphora, quantificational subordination, and other issues. (See for example van den Berg (1996), Krifka (1996a), Nouwen (2003b), Brasoveanu (2007, 2010), and Keshet (2018).) Of the theories discussed in this entry, this is
the most difficult to explain informally. We shall keep the discussion
as informal as possible, and urge interested readers to consult the
works cited directly for more detail.



To begin with, let's look at how simple discourse anaphora is
handled on DPL. So consider again:



(27)

A man loves Annie. He is rich.


Now in DPL, indefinites such as "a man" are treated as
existential quantifiers. Further, DPL treats consecutive sentences in
discourses as being conjoined. So we can think of (27) as follows:



(27b)
([?]*x*)(man *x* & *x* loves Annie) &
*x* is rich


Here we have rendered the anaphoric pronoun "He" as the
variable "*x*", the same variable that is the
variable of its quantifier antecedent. This represents the anaphoric
connection. The important point to notice is that the anaphoric
pronoun/variable in (27b) is not within the syntactic scope of its
quantifier antecedent. This corresponds to the fact that in DPL, the
syntactic scopes of quantifiers are confined to the sentences in which
they occur, as current syntactic theory tells us they should be.



The key to understanding the DPL account of discourse anaphora lies in
understanding its semantics of the existential quantifier and
conjunction. Let's begin with existential quantification. The
basic idea here it that when we interpret an existential quantifier,
the output of that interpretation may affect the interpretation of
subsequent expressions. In standard predicate logic, the
interpretation of an existential quantifier is a set of assignment
functions. In DPL, it is a set of *ordered pairs* of input and
output assignment functions. The output assignment functions act as
the input to subsequent formulas. Take for example the simple formula
"\((\exists x)(\mbox{man } x)\)". In DPL, such a formula
takes all the input functions *g*, and for each one outputs all
the possible assignment functions *h* such that they differ from
*g* at most in that they assign *x* to an object in the
interpretation of "man" (i.e., they assign *x* to a
man). More generally, accounting for possibly more complex formulas in
the scope of the existential (including another quantifier), a pair of
sequences \(\langle g, h\rangle\) is in the interpretation of an
existential formula "\((\exists x)\Phi\)" just in case
there is an assignment function *k* differing from *g* at
most on *x* such that \(\langle k,h\rangle\); is in the
interpretation of "\(\Phi\)".[16]
 So note how interpreting the existential quantifier results in
shifting from the input function *g* to *k*, where *k*
is now the input to "\(\Phi\)". This makes the existential
quantifier *internally dynamic*, capable of affecting the
interpretation of expressions within its syntactic scope. Further, the
fact that the output assignment functions of interpreting the whole
existentially quantified sentence are allowed to be different from the
input to the interpretation means that the processing of the
existentially quantified formula may affect the interpretation of
expressions *after* the existentially quantified formula, and
hence *outside* the scope of the existential quantifier. This
is to say that the existential quantifier is *externally
dynamic*, capable of affecting the interpretation of expressions
*outside* its syntactic scope. As we will see, an expression
can be internally dynamic and externally static (as well as internally
and externally static). At any rate, putting things very roughly, the
idea here is that once the existential quantifier "resets"
the value of "*x*" so that it satisfies the formula
the quantifier embeds, that value stays reset (unless there is a
subsequent existential quantifier attached to the same variable) and
can affect the interpretation of subsequent formulas.



Turning now to conjunction, the idea here is similar. Again, the
fundamental idea is that the interpretation of the left conjunct can
affect the interpretation of the right conjunct. A bit more formally,
a pair of assignment functions \(\langle g,h\rangle\); is in the
interpretation of a conjunction just in case there is an assignment
function *k* such that \(\langle g,k\rangle\); is in the
interpretation of the left conjunct and \(\langle k,h\rangle\); the
right
 conjunct.[17]
 So note how interpreting the left conjunct changes the input sequence
for the interpretation of the right conjunct. Again, this means that
conjunction is internally dynamic, possibly affecting the
interpretation of expressions in its scope. And again, that the output
of interpreting a conjunction, here *h*, can differ from the
input, here *g*, means that a conjunction is capable of affecting
things outside of it and hence outside of the scope of that
conjunction sign. Again, this is to say that conjunction is externally
dynamic.



To go through an example, we also need to understand the semantics of
an atomic formula like "man *x*". Atomic formulas act
like a test on the input assignments, allowing those assignments that
satisfy the condition to pass through and act as input to subsequent
assignments and rejecting the rest. However, atomic formulas do not
*change* assignment functions. More specifically, an input
output pair \(\langle g,h\rangle\) is in the interpretation of an
atomic formula "\(Rt\_1 \ldots t\_n\)" just in case \(h=g\)
and *h* assigns "\(t\_1\)"...
"\(t\_n\)" to something in the interpretation of
*R*.



Putting these elements all together, we are now in a position to see
how DPL accounts for (27b). Informally speaking, the first sentence of
(27b) takes all the input assignment functions *g* and outputs
all those assignments *h* in that they differ at most from
*g* in assigning "*x*" to a man who loves Annie.
Thus the output of the first sentence is all (and only) the possible
assignments of "*x*" (given the model) to men who
love Annie. The output of the first sentence is the input to the
second. The second sentence tests these assignments, allowing only
those through which also assign "*x*" to something in
the interpretation of "rich". The output of the second
sentence includes all and only the assignments of
"*x*" to rich men who love Annie. As long as the
model includes at least one rich man who loves Annie, there will be at
least one such output assignment. Hence the discourse is true iff
there is at least one rich man who loves Annie. Since conjunction is
externally dynamic, we can keep adding sentences with anaphoric
pronouns to similar effect. Thus in a discourse such as



(27c)
 A man
loves Annie. He is rich. He is famous.


the discourse is all true iff some rich famous man loves Annie.



Thus the dynamic semantic treatment yields a truth conditional
equivalence between 

\[
(\exists x)(\Phi) \mathbin{\&} \Psi 
\]

 and 

\[
(\exists x)(\Phi \mathbin{\&} \Psi) 
\]

 even when
"\(\Psi\)" contains free occurrences of
"*x*". Hence it is often said that in DPL quantifiers
*semantically bind* free variables outside their syntactic
scope. This doesn't encounter the same problems as the syntactic
binding treatment rejected in
 section 2,
 since it allows quantifiers to provide values for subsequent
anaphoric pronouns without actually binding them in a way that falsely
predicts, e.g., that "Exactly one man loves Annie. He is
rich" is truth-conditionally equivalent to
\(\mbox{"}(\exists !x)(\mbox{man } x \mathbin{\&} x \mbox{
loves Annie} \mathbin{\&} x \mbox{ is rich})\mbox{"}\).



Because the treatment of donkey anaphora is a bit more complicated
technically, and because some of the main ideas of DPL are now on the
table, we will be more suggestive here. Again, we urge the interested
reader to consult Groenendijk and Stokhof (1991) directly.



First, consider conditional donkey anaphora:



(32)
 If a
farmer owns a donkey, he beats it.


(32) gets regimented in DPL as follows:



(32a)

\((\exists x)(\mbox{farmer } x \mathbin{\&} \exists y(\mbox{donkey
} y \mathbin{\&} \mbox{own }x,y)) \rightarrow \mbox{beat }
x,y\)


There are three crucial points to the DPL treatment here. (1) The
existential quantifier is externally dynamic and hence may affect the
interpretation of variables outside its scope, and in particular
"*x*" in the consequent of (32a). (2)
"\(\rightarrow\)" is internally dynamic and allows the
interpretation of its antecedent to affect the interpretation of its
consequent (just as is conjunction). 1 and 2 together mean that the
quantifier in the antecedent of (32)/(32a) can semantically bind the
variable in the consequent, even though it is not in the syntactic
scope of the quantifier. But without doing anything further, we would
be left with (32a) having the truth conditions of



(32b)
\(
(\exists x)(\mbox{farmer } x \mathbin{\&} \exists y(\mbox{donkey }
y \mathbin{\&} \mbox{own } x,y)) \rightarrow \mbox{beat }
x,y)\)


where "\(\rightarrow\)" is the standard material
conditional. This doesn't give the intuitive truth conditions of
(32) on the reading that concerns us, i.e., that every donkey-owning
farmer beats every donkey he owns. The third and final element we need
to get the truth conditions to come out right is to say that a pair of
assignments \(\langle h,h\rangle\) is in the interpretation of a
conditional iff for all *k* such that \(\langle h,k\rangle\)
satisfies the antecedent, there is a *j* such that \(\langle
k,j\rangle\) satisfies the
 consequent.[18]
 This says, roughly, that for any output assignment *k* of a pair
of assignments satisfying the antecedent of the conditional
(assignment of a donkey-owning farmer to *x* and a donkey owned
by *x* to *y* in the case of (32)/(32a)), *k* is the
input of a pair \(\langle k,j\rangle\) that satisfies the consequent,
for some *j*. Since the consequent of the conditional in
(32)/(32a) is an atomic formula, *j*=*k*. So the account
claims that any output of a pair of sequences that satisfies the
existentially quantified antecedent also satisfies "*x*
beats *y*", and so also assigns to "*x*"
and "*y*" to something that stands in the beating
relation. That is, the truth of (32)/(32a) requires every
donkey-owning farmer to beat every donkey he owns.



Turning now to our relative clause donkey sentence, (repeated
here)



(26)

Every woman who owns a donkey beats it.


we shall be even more schematic. This gets regimented in DPL as
follows, (giving the predicate letters here the obvious meanings):



(26a)
\((\forall x)((Wx
\mathbin{\&} (\exists y)(Dy \mathbin{\&} Oxy)) \rightarrow
Bxy)\)


Note in particular that the "*y*" in
"*Bxy*" is not in the scope of the existential
quantifier. Now given a quite straightforward treatment of the
universal quantifier, on which it allows dynamic effects in its
 scope,[19]
 in all essentials, the example works like (32)/(32a). For stripping
the universal quantifier away, we have:



(26a')
\( ((Wx
\mathbin{\&} (\exists y)(Dy \mathbin{\&} Oxy))\rightarrow
Bxy\)


And overlooking the free variables left by stripping away the
universal quantifier (which anyway were in its scope and bound by it),
we simply have another conditional with an existential quantifier in
its antecedent and a formula in the consequent containing an
occurrence of the variable of that existential quantifier. So the
treatment goes essentially as it did for (32)/(32a) itself, with the
externally dynamic existential quantifier, internally dynamic
conditional, and universal quantification over assignments in the
semantics of the conditional all working their magic so that
(26)'s truth requires every donkey owning woman to beat every
donkey she owns.



Though DPL cannot handle relative clause donkey sentences such as:



(33)
 Most
women who own a donkey beat it.


since it is working within the framework of a first order logic
without generalized quantifiers, this is only a limitation of this
particular formulation and not of dynamic approaches generally. Others
have formulated systems of dynamic semantics with generalized
quantifiers that are capable of dealing with examples like
 (33).[20]



On the other hand, DPL and dynamic approaches generally do face a
problem. Put crudely, DPL (and dynamic approaches generally) solve the
problems of discourse and donkey anaphora by formulating semantics for
quantifiers that allows quantifiers to semantically bind variables
that aren't in their syntactic scopes. In this they
(self-consciously) resemble DRT. But then they face a problem similar
to one faced by DRT and mentioned above. Consider again our discourse
 (18), repeated here



(18)

A man broke into Sarah's apartment. Scott believes
he came in the window.


As mentioned above, (18) has a reading on which the second sentence of
the discourse attributes a general belief to Scott (something like the
belief that a man who broke into Sarah's apartment came in
through the window). On a dynamic approach to (18), the quantifier in
the first sentence semantically binds the variable in the second
sentence. But then this semantically amounts to quantification into
the verb of attitude, and so will not result in a reading of the
second sentence on which it attributes a general belief to Scott.
Hence, dynamic approaches need to invoke some other mechanism to get
the reading of the second sentence in
 question.[21]



Similarly, again consider the following sentences discussed in
connection with DRT above :



(30)
 Every woman who
has a secret admirer thinks he is stalking her.
(31)
 If a woman has a
secret admirer, usually she thinks he is stalking her.


As mentioned there, these sentences also appear to have readings on
which they attribute general or *de dicto* beliefs to the women
in question. That is, they have readings on which they attribute to
the women in question *general* beliefs to the effect that they
are being stalked by secret admirers. This is why these sentences can
be true even though the women in question don't know who their
secret admirers are, and so have no beliefs about *particular*
persons stalking them. For reasons exactly similar to those given for
the case of the analogous reading of the second sentence of (18),
these readings can't be captured by DPL or dynamic approaches
generally.



Paul Elbourne (2005) reviews three other serious problems for dynamic
approaches. The first is the problem of disjunctive antecedents
(discussed in detail in Stone 1992). The problem is with sentences
like the following:



(34)
 If Mary
hasn't seen John lately, or Ann misses Bill, she calls him.
(Elbourne 2005: 19)


This is problematic because the pronouns cannot co-refer with the
antecedent (since it is disjunctive), but the antecedent also does not
introduce any suitable variables or quantifiers to provide values for
the pronouns, since they are definites (names) and not indefinites or
other quantificational expressions.



The other two problems involve anaphoric pronouns that don't
have a proper linguistic antecedent. The following example, originally
from Jacobson (2000), is one in which there is no linguistic
antecedent for the pronoun "it", though the pronoun has a
covarying interpretation with the quantifier "most faculty
members":



(35)
 *A
new faculty member picks up her first paycheck from her mailbox.
Waving it in the air, she says to a colleague:* Do most faculty
members deposit it in the Credit Union? (Elbourne 2005:
20)


The second kind of example involves what is commonly called a
*paycheck pronoun*:



(36)
 John
gave his paycheck to his mistress. Everybody else put it in the bank.
(Elbourne 2005: 21)


The problematic (salient) reading is the one is which everybody else
put *their own* paychecks in the bank. The first sentence only
introduces *John's* paycheck, hence "it"
doesn't have a proper antecedent. Standard dynamic accounts
don't have the machinery for dealing with pronouns without the
proper linguistic antecedent, since it is the antecedent's
effect on the context that accounts for the anaphoric pronoun's
 interpretation (but see Keshet (2018) for a recent dynamic semantics of anaphora that includes an account of paycheck pronouns).[22]


### 3.3 Descriptive Approaches



There have been many accounts of the semantics of anaphora according
to which anaphoric pronouns in some sense function like definite
descriptions. Though there are important differences between such
theories, examples of theories of this sort include Evans (1977),
Parsons (1978, Other Internet Resources), Davies (1981), Neale (1990),
Heim (1990) and Elbourne (2005). Theories of this sort are often
called *E-Type* or
*D-type* approaches. Though it is beyond the scope of this
article to compare all the different accounts, some of the main
differences in types of accounts are as follows. E-type accounts are
sometimes distinguished as ones in which a definite description fixes
the referent of an anaphoric pronoun, whereas D-type accounts are
ones in which the pronoun itself has the semantics of a definite
description. However, it is common to describe the latter as 'E-type' as well. These are further sub-divided into those that treat
pronouns as definite descriptions merely at the level of semantics and
those that also treat them as descriptions at the level of syntax.
Finally, accounts differ in how the descriptive material is recovered.
Some accounts hold that the descriptive material has to be recovered
linguistically, from prior discourse. Others hold that it can be any
contextually salient description.



We will discuss two of the best known versions of the view, Neale
(1990) and Elbourne (2005). We should add that though Neale developed
the view in question in greater detail, Davies (1981) had earlier
defended essentially the same view in all crucial
 respects.[23]
 Thus, the view we go on to describe should probably be called *the
Davies-Neale view*. But since we shall focus on Neale's
presentation of the view, we shall talk of Neale's view.



On Neale's view, in all instances of problematic anaphora,
anaphoric pronouns "go proxy for" definite descriptions
understood as quantifiers along roughly Russellian lines. First,
consider discourse anaphora. Neale's view is that in a discourse
such as:



(37)
 John
bought a donkey. Harry vaccinated it.


the pronoun "it" "goes proxy for" the definite
description "the donkey John bought." Hence the second
sentence of such a discourse is equivalent to the sentence
"Harry vaccinated the donkey John bought" with the
description understood in the standard Russellian fashion. Within a
generalized quantifier type framework, where "the" is
treated as a determiner that, like other determiners, combines with a
set term to form a quantified NP, the evaluation clause for sentences
containing a singular description (with wide scope) can be given as
follows



(38)

"[The \(x:Fx](Gx)\)" is true iff \(|\mathbf{F}|=1\) and
\(|\mathbf{F-G}|=0\)


So the second sentence of (37) is true iff Harry vaccinated the unique
donkey John bought. Thus far, then, the view is that pronouns
anaphoric on singular indefinites are interpreted as Russellian
definite descriptions.



There is, however, a further complication in Neale's theory that
is invoked in the explanation of donkey anaphora. In particular, Neale
introduces what he calls a "numberless description": a
description that, unlike semantically singular descriptions, puts no
cardinality constraint on the denotation of the set term that combines
with the determiner to form the quantified NP (other than that it must
be nonempty--note above how in the singular case \(|\mathbf{F}|\)
is constrained to equal one.) Following Neale, let "whe"
be the determiner (corresponding to "the") used to form
"numberless descriptions." Then the evaluation clause for
sentences containing numberless descriptions, analogous to (38) above,
would be



(39)

"[whe \(x: Fx](Gx)\)" is true iff \(|\mathbf{F}| \ge 1\)
and \(|\mathbf{F-G}|=0\)


Thus numberless descriptions are in effect universal quantifiers.



In addition to going proxy for Russellian singular descriptions in the
way we have seen, Neale claims that anaphoric pronouns sometimes go
proxy for numberless descriptions. In particular, Neale holds that
pronouns anaphoric on singular existential quantifiers (but outside of
their scope) can be interpreted *either* as standard Russellian
descriptions *or* as numberless descriptions. Now if the
pronouns in our conditional and relative clause donkey sentences
(repeated here)



(25)

If Sarah owns a donkey, she beats it.
(26)
Every woman who owns a donkey beats it.


are interpreted as a numberless description, (25) asserts that if
Sarah owns a donkey, she beats all the donkeys she owns and (26)
asserts that every donkey owning woman beats every donkey she owns.
Thus, Neale's account of donkey anaphora requires the pronouns
here to be interpreted as numberless descriptions.



Having seen how Neale's theory handles discourse anaphora and
donkey anaphora, we turn to difficulties with the account. An obvious
question concerning an account like this that allows pronouns
anaphoric on singular existential quantifiers to go proxy for both
Russellian *and* numberless descriptions is: what determines
whether such a pronoun is going proxy for a Russellian, as opposed to
a numberless, description? This question is pressing for Neale's
account, since there will be a substantial difference in the truth
conditions of a pronoun-containing sentence depending on whether the
pronoun receives a numberless or Russellian interpretation. In his
most explicit statement about the matter (1990: 237) Neale makes clear
that it is primarily whether the utterer had a particular individual
in mind in uttering the indefinite description that determines whether
a pronoun anaphoric on it receives a Russellian or a numberless
 interpretation.[24]



If this is correct, then discourses of the form



(40)
 A(n)
*F* is *G*. He/she/it is *H*.


generally ought to display both readings (in the suitable contexts),
depending on whether the utterer of the discourse had a particular
individual in mind in uttering "A(n) *F*". So the
second sentences of discourses of the form of (40) ought to have
readings on which they mean the unique *F* that is *G* is
*H* (Russellian) *and* on which they mean every *F*
that is *G* is *H* (numberless). But this does not seem to
be the case. In particular, such discourses do not have readings
corresponding to the numberless interpretation of the pronoun.
Consider the discourse anaphora analogue of the donkey conditional
(25):



(25a)
 Sarah
owns a donkey. She beats it.


It seems clear that this discourse has no reading on which the second
sentence means that Sarah beats every donkey she owns, even if we
imagine that the utterer of the discourse had no particular donkey in
mind when she uttered the first sentence. Suppose, for example, that
the Homeland Security and Donkey Care Bureau comes to town and wants
information about local donkey ownership and beating. The speaker
tells them that she really don't know how many donkeys anybody
owns, and has never seen or had any other contact with particular
local donkeys. But she tells them that she has received some
information from reliable sources and it has been deemed
"credible". Asked what she has heard, she responds:
"Sarah owns a donkey and she beats it".



Even though she has no particular donkey in mind in uttering these
sentences, we simply don't get a numberless reading here. If
Sarah beats some donkey she owns, the speaker has spoken truly even if
she owns others she fails to beat. Or again, suppose we are debating
whether anybody has an eight track tape player anymore, and one of us
says "I'll bet the following is true: some guy with a
'68 Camaro owns an eight track player and he still uses
it." Again, there is no numberless reading for the pronoun in
the second sentence, even though the speaker clearly has no particular
eight-track player in mind. If some '68 Camaro driving guy owns
and uses an eight-track player, the sentence is true even if he owns
other eight track players that aren't
 used.[25]



So it appears that Neale's account has no explanation of why the
pronouns in discourses like (25a) never have numberless readings.
Neale's account has similar problems with sentences like:



(41)
 Some
woman who owns a donkey beats it.


Here again, Neale's theory predicts that this sentence has a
reading on which its truth requires that some woman beats every donkey
she owns. And again, even if we imagine the sentence being uttered
without any particular woman or donkey in mind, we don't get
this reading of the sentence predicted by Neale's theory, (say
we are discussing women's tendencies towards animals they own,
and one of us utters (41) simply thinking it is statistically likely
to be true). So Neale's account has no explanation as to why the
second sentence of discourse (25a) and sentence (41) lack the relevant
readings assigned to those sentences by his theory.[26]



Elbourne (2005) proposes a different D-type theory. The main thesis of
the book is that pronouns of all types, proper names, and definite
descriptions have a unified syntax and semantics: they are all of type
\(\langle s, e\rangle\) (functions from situations to individuals) and
are syntactically comprised of a definite determiner that takes two
arguments: an index and an NP (noun phrase). 



His theory of unbound pronouns is *NP-deletion theory.*
NP-deletion is when an NP at the level of syntax is (felicitously)
unpronounced at the surface level as in:



(42)
 I like
Bill's wine, but Max's is even better.


On Elbourne's theory, pronouns undergo *obligatory*
NP-deletion. That is, a sentence like
 (25),
 at the level of syntax is actually (25b):



(25)
 If Sarah owns a
donkey, she beats it.
(25b)
 If Sarah owns a donkey, she beats [it donkey].


Following Elbourne, we'll suppress mention of the index, since
it doesn't do any work here (anaphoric pronouns have a null
index on his view, though he does provide arguments for why this null
index is present--see Elbourne (2005: 118-126)). Pronouns
have the same semantics as the determiner "the"
(abstracting away from the phi-features of pronouns) so that (25b) has
the same semantics as (25c):



(25c)
 If
Sarah owns a donkey, she beats the donkey.


This is motivated by the insight that pronouns pattern with
*minimal* definite descriptions rather than long ones, as other
D-type theories have it. For example, a theory like Neale's
predicts that (43a) has the semantics of (43b):



(43a)
 A man
murdered Smith, but John does not believe that he murdered
Smith.
(43b)
 A man murdered Smith, but John does not believe that the man who
murdered Smith murdered Smith.


But this is problematic, because (43a) clearly has a salient, true
reading, whereas (43b) does not. However, Elbourne observes that (43c)
has the same salient true reading as (43a):


* (43c) A man
murdered Smith, but John does not believe that the man murdered
Smith.



Treating pronouns as determiners is motivated more generally by Postal
(1966)'s arguments
that they at least sometimes have the same semantics as determiners
based on examples such as:



(44)
 You
troops will embark but the other troops will remain.


Finally, positing an NP-deletion theory provides an elegant solution
to the problem of the formal link, which is that any good account of
anaphora has to explain why (45a) is felicitous but (46b) is not:



(45a)
 Every
man who has a wife is sitting next to her.
(46b)
 #Every
married man is sitting next to her. (Heim 1982, 1990)


Since NP-deletion generally requires a linguistic antecedent, if
pronouns are determiners that have undergone NP-deletion, this
explains the contrast. 



As mentioned above, pronouns have the semantics of determiners.
Specifically, Elbourne proposes a situation semantics
 (situations in natural language semantics),
 where the semantics of pronouns (and determiners) are treated in a
Fregean way, as follows:

 

\[
[[\mbox{it}]]^g = \lambda\_{\langle\langle s,e\rangle\langle s,t\rangle\rangle}. \lambda s: \exists!x(\lambda s'. x)(s) 
= 1. \iota x\ f(\lambda s'. x)(s)
= 1
\]


That is to say, pronouns come with presuppositions that there is a
unique individual in each situation that satisfies the predicate, and
(when that presupposition is satisfied), they refer to that individual
(in that situation). Conditional donkey anaphora works as follows. A
sentence like (47) has the structure at LF of (48):



(47)
 If a man
owns a donkey, he beats it.
(48)
 [[always
[if [[a man] [\(\lambda\)6 [[a donkey][ \(\lambda\)2 [t6 owns
t2]]]]]]][[he man] beats [it donkey]]] (Elbourne 2005:
 52)[27]


The semantics of "always" is crucial to getting the truth
conditions right:




\([[\mbox{always}]]^g = \lambda p\_{\langle s,t\rangle}. \lambda
q\_{\langle s,t\rangle}. \lambda s\). for every minimal situation
\(s'\) such that \(s' \le s\) and \(p(s') = 1\), there is a situation
\(s''\) such that \(s'' \le s\) and \(s''\) is a minimal situation
such that \(s' \le s''\) and \(q(s'') = 1\) 




Basically, "always" takes two sentences, and returns true
iff every minimal situation that satisfies the first sentence has a
minimal extension that satisfies the second sentence. A minimal
situation is one that contains one or more *thin
particulars*--an individual abstracted away from its
properties--and one or more properties or relations that the thin
particular(s) instantiate(s). An extension of a minimal situation
includes all those same particulars and properties and relations, with
(possibly) some more particulars, properties, or relations in
addition. On Elbourne's account "if" doesn't
play any role in contributing to the truth conditions of a donkey
conditional; it is semantically vacuous. It is the (sometimes silent)
adverb of quantification along with the semantics of the antecedent
and consequent that do the work in yielding the truth conditions. The
following are the truth conditions for (47) on Elbourne's
account. (We won't go through the derivation for relative clause
donkey anaphora, but it works similarly. See Elbourne 2005: 53.)



> 
> 
> \(\lambda s\_1\). For every minimal situation \(s\_4\) such that
> 
> \(s\_4 \le s\_1\) and there is
> an individual *y* and a situation \(s\_7\) such that \(s\_7\) is a
> minimal situation such that \(s\_7\le s\_4\) and *y* is a man in
> \(s\_7\), such that there is a situation \(s\_9\) such that \(s\_9\le
> s\_4\) and \(s\_9\) is a minimal situation such that
> 
>  \(s\_7\le s\_9\) and
> there is an individual *x* and a situation \(s\_2\) such that
> \(s\_2\) is a minimal situation such that \(s\_2\le s\_9\) and *x*
> is a donkey in \(s\_2\), such that there is a situation \(s\_3\) such
> that \(s\_3\le s\_9\) and \(s\_3\) is a minimal situation such that \(s\_2
> \le s\_3\) and *y* owns *x* in \(s\_3\),
> 
>  there is a situation \(s\_5\) such that
> 
>  \(s\_5\le s\_1\) and \(s\_5\) is a minimal situation
> such that \(s\_4\le s\_5\) and \(\iota x\ x\) is a man in \(s\_5\) beats
>  in \(s\_5\) \(\iota x\ x\) is a donkey in \(s\_5\). (2005: 52)
>  
> 
> 



Thus on Elbourne's account (47) is true iff each donkey-owning
man beats every donkey he owns.



Elbourne surprisingly never goes through a derivation of a case of
cross-sentential anaphora as in:



(49)
 A woman
walked in. She sat down.


His view is that this has the same semantics as (49a):



(49a)
 A
woman walked in. The woman sat down.


But he doesn't provide an account of how this works, neither in
description nor detailed derivation. One worry about this is that in
the conditional and relative clause donkey sentences, the fact that
the pronoun(s) co-varies with its antecedent is accounted for by
"always" and "every". These expressions
quantify over situations in a way that guarantees that the man who
beats the donkey is the same man who owns that same donkey,
effectively acting as (semantic) binding. But there is no such
mechanism in place to guarantee that "the woman" in (the
situation in) the second sentence of (49) in any sense co-varies with
the woman (in the situation) in the first. It is not clear how
Elbourne's view can accomplish this without employing either a
dynamic notion of binding or a more traditional d-type account with
complete descriptions.



However, Elbourne's account doesn't encounter the same
problem as Neale's in predicting numberless readings where there
are none, as he doesn't posit a numberless interpretation of the
pronoun at all. This highlights a difference between the way that
Neale's theory approaches the truth conditions of conditional
donkey sentences and the way DRT, dynamic semantics, the CDQ theory
(discussed in the next section) and other D-type theories like
Elbourne's and Heim (1990)'s do (Heim 1990 also employs a
situation semantics to account for conditional donkey anaphora). In
these other theories, the requirement that all the donkey-owning men
beat all the donkeys they own for (47) to be true arises due to the
interaction of the semantic of indefinites, the semantics of anaphoric
pronouns and the semantics of
 conditionals.[28]
 Indeed, it is the latter that is primarily implicated in (47)'s
truth requiring that *all* donkeys owned by donkey-owning men
be beaten (since the theories posit some sort of universal
quantification in the semantics of conditionals). By contrast, on
Neale's view, the requirement that the men beat all the donkeys
they own for (47) to be true (on one of its readings) essentially
falls out of the semantics of the anaphoric pronoun alone, since on
one of its readings, it expresses universal quantification over
donkeys Sarah owns (the numberless description reading).



One problem that D-type theories face is that pronouns come with a uniqueness requirement (either as a presupposition or part of the asserted content). But it has been observed that pronouns anaphoric on indefinites do not have a uniqueness requirement; it seems that the truth conditions for such sentences are existential. This observation is nicely captures by Discourse Representation Theory, File Change Semantics, dynamic semantics, and the Context Dependent Quantifier approach (discussed in the next section). For example, consider (49) again. This seems true even if many women walked into the contextually salient place at the contextually salient time, and regardless of whether they sat down or not, so long as at least one woman walked in and sat down. Mandelkern & Rothschild (2020) call this phenomenon "definiteness filtering" and Lewis (forthcoming) calls it "the problem of non-uniqueness". Both of the cited works argue that there is evidence of uniqueness requirements on definites more generally, and the non-uniqueness is specific to the kind of constructions in (49). Mandelkern & Rothschild tentatively propose a D-type theory that employs situation semantics, while Lewis (forthcoming) argues for a D-type theory that takes definite descriptions to be ambiguous between presupposing worldly uniqueness and discourse uniqueness.[29]




Another problem that all D-type theories must address is the problem of
*indistinguishable participants.* Since on D-type theories,
pronouns in one way or another have the semantics of definite
descriptions, they come with uniqueness presuppositions. When it comes
to conditional donkey anaphora, the way to meet these uniqueness
presuppositions is by employing minimal situations--the definite
description then picks out the unique object satisfying the
description in the minimal situation. (With the exception of
Neale's theory: since he employs numberless pronouns, the
pronouns have no uniqueness presupposition.) Thus uniqueness is
satisfied within the situation, even if it can't be satisfied in
the larger world. But in examples of indistinguishable participants,
uniqueness cannot even be satisfied within a minimal situation.
Consider the typical example, (50):



(50)
 If a
bishop meets a bishop, he blesses him.


In the minimal situation that satisfies the antecedent, there are
*two* bishops that have the same property (*meeting another
bishop*). Thus descriptions like "the bishop" or
"the bishop who meets a bishop" do not denote uniquely.
DRT and dynamic theories don't have any problem at all with
these examples because pronouns are treated as dynamically bound
variables, not definite descriptions, and each instance of "a
bishop" in the antecedent of (50) is associated with a different
variable, which prescribe the anaphoric links to the two pronouns in
the consequent. For D-type solutions to this puzzle see Heim(1990),
Ludlow (1994), Elbourne (2005, 2016), Lewis (forthcoming).



Finally, it should be mentioned that there are some who propose
*mixed* approaches, wherein some pronouns are treated as
dynamically bound variables whereas others are treated as D-type (see
Chierchia 1995; Kurafuji 1998, 1999). On these views, some pronouns
are ambiguous between the two readings, while others have only the
dynamically bound variable or D-type reading. This type of theory
avoids many of the problems raised for each type of account, since
they use dynamic semantics for the examples most amenable to a dynamic
explanation and d-type pronouns for example most amenable to that sort
of explanation. However, the theory comes at a considerable
theoretical cost in that it predicts a systematic ambiguity in
pronouns that is (arguably) not explicitly marked in any language.
(See Elbourne 2005 for further criticisms of these views.)


### 3.4 The Context Dependent Quantifier Approach



The Context Dependent Quantifier, or CDQ, account of anaphora was
suggested in Wilson (1984) and subsequently developed in King (1987,
1991,
 1994).[30]
 The idea underlying the application of CDQ to discourse anaphora is
that these expressions look like quantifier-like expressions of
generality, where the precise nature of the generality they express is
determined by features of the linguistic context in which they occur.
On the CDQ account, anaphoric pronouns with quantifier antecedents in
discourse anaphora are contextually sensitive devices of
quantification. That is, these anaphoric pronouns express
*quantifications*; and *which* quantifications they
express is partly a function of the *linguistic environments*
in which they are embedded. Consider the following discourses:



(51)
 A man
from Sweden climbed Mt. Everest alone. He used no oxygen.
(52)
 Most students passed the exam. They didn't get scores below
70%.


Looking at (51), suppose that in fact at least one Swede has soloed
Mt. Everest without oxygen. Then it would seem that the sentences of
(51) are true. If this is correct, then it appears that the second
sentence of (51) expresses a (existentially) *general* claim.
CDQ claims that the pronoun "He" in the second sentence is
itself a (existential) quantifier, and this explains why the second
sentence expresses a general claim: the generality is a result of the
presence of this quantifier in the sentence. Similar remarks apply to
(52), (except that "They" expresses a universal
quantifier). Further, consider the following example, which is similar
to our example (18) above:



(53)
 A man
killed Alan last night. Michelle believes he used a knife to kill
him.


The second sentence of this discourse appears to have two different
readings. On one reading, it asserts that concerning the man who
killed Alan last night, Michelle believes of *that very man*
that he used a knife. This would be the case if, for example, Michelle
knew the man who killed Alan, believed that he killed Alan and based
on his well-known fondness of knives, believed he used this sort of
weapon. But the second sentence has another reading on which it
ascribes to Michelle the general belief to the effect that a man
killed Alan with a knife last night. On this reading the sentence
would be true if e.g., on the basis of conversations with personnel at
the hospital and having no particular person in mind, Michelle
believed that a man fatally stabbed Alan last night.



Again, CDQ claims these facts are to be explained by holding that the
pronoun in the second sentence is a quantifier. For then we should
expect that, like other quantifiers, it could take wide or narrow
scope relative to "Michelle believes". On the wide scope
reading of the pronoun/quantifier, the second sentence attributes to
Michelle a belief regarding a particular person. On the narrow scope
reading, it attributes to Michelle a general belief.



Occurrences of "ordinary quantifiers", such as
"every man" have what we might call a *force*, in
this case universal; what we might call a *restriction*, in
this case the set of men; and *scope* relative to other
occurrences of quantifiers, verbs of propositional attitude, and so
on. CDQ claims that the anaphoric pronouns in question also have
*forces* (universal, existential, etc.), *restrictions*
("domains over which they quantify") and *scopes*
relative to each other, verbs of propositional attitude, etc. However,
unlike "ordinary" quantifiers, these anaphoric pronouns
qua quantifiers have their forces, restrictions and relative scopes
determined by features of their linguistic environments. King (1994)
lays out how the forces, restrictions and relative scopes of these
anaphoric pronouns are determined, and we shall not describe those
details here.



As to donkey anaphora, without going through the details, let me just
say that CDQ assigns to a relative clause donkey sentence such as
 (26)
 above (repeated here)



(26)
 Every woman who
owns a donkey beats it.


truth conditions according to which (26) is true iff every woman who
owns a donkey beats some donkey she owns (see King 2004 for details
and discussion). Some think that the truth of (26) requires every
woman who owns a donkey to beat *every* donkey she owns, and as
we saw, DRT, DPL, and the D-type theories discussed assign these truth
conditions to (26) (though this is not an essential feature of DRT or
dynamic semantic approaches more generally). The truth conditions CDQ
assigns to (26) correspond to what is often called the *weak
reading* of donkey sentences (Chierchia (1995) calls this the
\(\exists\)-reading) and the truth conditions the accounts discussed thus far
correspond to the *strong reading* of donkey sentences
(Chierchia calls this the \(\forall\)-reading). There actually has been a debate
in the literature as to which truth conditions sentences like (26)
have. There are sentences that are exactly like (26) except for the
descriptive material in them that clearly seem to have (only) the weak
reading. An example is:



(54)
 Every
person who had a credit card paid his bill with it.


It seems clear that the truth of this sentence does not require every
person with a credit card to pay his bill with each credit card he
has. We will discuss these matters further in
 section 4,
 but for now let us simply note that CDQ differs from the other
accounts discussed on what truth conditions should be assigned to
sentences like (26) and (54) and that it is simply unclear which truth
conditions are the correct ones.



As for conditional donkey anaphora, the CDQ account is rather
complicated and we are only able to provide the briefest outline of
the account here (interested readers should consult King 2004). As we
saw above, a conditional donkey sentence such as
 (25)




(25)
 If Sarah owns a
donkey, she beats it.


is true iff Sarah beats every donkey she owns. Thus, "a
donkey" somehow seems to have ended up with universal force. The
CDQ account holds that this illusion of universal force for the
indefinite is really the result of the interaction of the semantics of
the conditional, the indefinite, understood as an existential
quantifier, and the CDQ "she", understood as a
context dependent quantifier with existential force ranging over
donkeys Sarah owns. Roughly, the account goes as follows. The
antecedent of (25) is equivalent to



(25a)
 \((\exists x)(x
\mbox{ is a donkey} \mathbin{\&} \mbox{Sarah owns }
x)\)


Given the CDQ "it" and its context in (25), the consequent
of (25) is equivalent to



(25b)
 \((\exists x)(x
\mbox{ is a donkey} \mathbin{\&} \mbox{Sarah owns } x
\mathbin{\&} \mbox{Sarah beats } x)\)


The semantics of the conditional involves universal quantification
over minimal situations. In particular, a conditional claims that for
every minimal situation \(s\_1\) in which its antecedent is true, there
is a situation \(s\_2\) that \(s\_1\) is part of in which its consequent
is true. In the case of (25), a minimal situation in which the
antecedent is true consists of Sarah and a single donkey she owns. The
final element here is that the definiteness and/or anaphoricness of
the CDQ "it" in the consequent of (25) makes a difference
to its truth conditions. The definiteness and anaphoricness of
"it" in (25) induces a sort of "familiarity
 effect".[31]
 In particular, for any (minimal) \(s\_1\) in which the antecedent is
true, there must be an \(s\_2\) that \(s\_1\) is part of in which the
consequent (understood as expressing the claim that Sarah beats a
donkey she owns) is true. But *in addition*, because of the
"familiarity" condition induced by the anaphoric definite
"it", there must be a donkey in \(s\_2\) that is also in
\(s\_1\) and that makes the consequent true. In other words,
familiarity requires that a donkey that makes the CDQ-containing
consequent true in \(s\_2\) also be present in \(s\_1\). In this sense,
the donkey is "familiar", having been introduced by the
antecedent and the situation \(s\_1\) in which it is true. To see what
this means, consider a situation \(s\_1\) that is a minimal situation
in which the antecedent is true. \(s\_1\) consists of Sarah owning a
single donkey. If e.g., Sarah owns ten donkeys, there are ten such
minimal situations. For (25) to be true, each such \(s\_1\) must be
part of a situation \(s\_2\) such that \(s\_2\) is a situation in which
Sarah beats a donkey that she owns and that is in \(s\_1\). Now the
only way that every minimal \(s\_1\) in which Sarah owns a donkey can
be part of an \(s\_2\) in which Sarah beats a donkey she owns in
\(s\_1\) is if Sarah beats every donkey she owns. Thus, the CDQ account
claims that (25) is true iff Sarah beats every donkey she owns.



Turning now to difficulties with CDQ, a main difficulty is that it
isn't clear whether the use of the notion of
*familiarity* in the account of conditional donkey sentences
can be ultimately upheld. Recall that the idea was that because
"it" is a definite NP, and because definite NPs generally
are thought to involve some sort of familiarity, the pronoun in the
donkey conditional induces a sort of familiarity effect. There are
really two distinct problems here. One is that though the pronoun
"it" is "syntactically" definite in that the
pronoun "it" is thought to be a definite NP, according to
CDQ it is "semantically" indefinite in (25), since it
expresses an existential quantification (over donkeys owned by Sarah).
But then if "it" really is semantically indefinite in
(25), why should it induce familiarity effects at all? (Jason Stanley
(p.c.) raised this difficulty). One might reply that it is the fact
that "it" is "syntactically" definite that
triggers the familiarity effects CDQ posits. But familiarity probably
is not well enough understood to allow us to assess this response. A
second, and perhaps more pressing, difficulty is this. Generally,
familiarity has something to do with whether what an expression is
being used to talk about is familiar or salient to the audience being
addressed. This is vague, of course, but the idea is that if one says
"The dog is hungry" to an audience who isn't even
aware of any dog that is around or relevant to the conversation, the
remark is somewhat infelicitous. That is because the definite NP
"The dog" was used to talk about something not familiar to
the audience. Now the question is: is it plausible to claim that this
and related phenomena are related to the rather complex effect CDQ
claims is induced by the anaphoricness/definiteness of
"it" in donkey conditionals? In the latter case, CDQ
claims the effect of familiarity is to make the CDQ "it"
in the consequent of donkey conditionals quantify over donkeys in the
minimal situations introduced by the antecedents of the conditionals
(in the case of (25), situations consisting of Sarah and a single
donkey she owns). The CDQ can only quantify over
"familiar" donkeys--those introduced by the
antecedent. One may well wonder whether the effect CDQ posits here can
really be seen to be a manifestation of phenomena that have
traditionally been explained in terms of familiarity.


## 4. How Many Readings Do Donkey Sentences Have?



We have already mentioned that there is some disagreement regarding
the truth conditions of sentences like



(26)
 Every woman who
owns a donkey beats it.


Some think that the truth of (26) requires every woman who owns a
donkey to beat *every* donkey she owns. As mentioned above,
this is called the *strong reading* of a donkey sentence.
Others think that the truth of (26) requires that every donkey owning
woman beats *some* donkey she owns. As above, this is the
*weak reading* of (26). As we mentioned above, there are
sentences that are exactly like (26) except for the descriptive
material in them that clearly seem to have (only) the weak reading.
The example we gave was:



(54)
 Every person who
had a credit card paid his bill with it.


It seems clear that the truth of this sentence does not require every
person with a credit card to pay his bill with each credit card he
has, but merely with some credit card he has. Our comments to this
point have suggested that the debate with respect to (26)/(54) and
weak vs. strong readings is over which one of the two sets of truth
conditions (26)/(54) have. Each of the theories discussed in this
entry only assign *one* of the existential or universal truth
conditions to relative clause donkey sentences (though this may be
more of an artifact of the specific implementations of the theories
than the theories themselves).



But some think that (at least some) donkey sentences have
*both* weak and strong readings. Chierchia (1994) and Kanazawa
(1994b) are examples. In their favor, for a given determiner, one can
find pairs of relative clause donkey sentences fronted by that
determiner such that one has the existential truth conditions (on its
most natural interpretation) and the other has the universal truth
conditions (on its most natural reading). For example, consider the
following pairs:



(55)
Weak Readings:
 
a.
Every person who had a
credit card paid his bill with it.
b.
 Most women who have a dime will put it in the meter. 
c.
 No man with a teenage son lets him drive the car on the weekend. 

(56)
Strong Readings:
 
a.
 Every student who
borrowed a book from Peter eventually returned it. 
b.
 Most parents who have a teenage son allow him to go out on the
weekend. 
c.
 No man with an umbrella leaves it home on a day like
this. 


Elbourne, for example, explicitly addresses this issue, since his
semantics captures only the strong reading. He claims that while this
may present a problem for his semantics, it is not a problem for
NP-deletion theory more generally, since replacing the pronoun in a
typical weak reading example with a minimal definite description also
yields a weak reading, e.g.:



(57)
 Everyone
who has a dime put the dime in the meter. (Elbourne 2005:
84)


However, he doesn't present an account of how this reading might
be captured.



These examples and others suggest that whether a given relative clause
donkey sentence appears to favor the strong reading or weak reading is
influenced by a variety of factors, including the monotonicity
properties of the determiner on the wide scope quantifier, the lexical
semantics of the predicates occurring in the sentence, and general
background assumptions concerning the situations in which we are to
consider the truth or falsity of the sentences. However, it is very
hard to find significant generalizations regarding under what
conditions a given reading is
 favored.[32]
 Further, it is very hard to find sentences that clearly allow both a
strong and a weak reading.[33] This makes the view that the sentences
actually possess both readings as a matter of their semantics at least
somewhat suspect. If they really do possess both readings, why is it
so hard to find sentences that clearly allow both readings? And
finally, relative clause donkey sentences fronted by the determiner
"some" seem always to only have the existential truth
conditions:



(58)
 Some
women who own a donkey beat it.


Obviously, the facts here are quite complicated.



Brasoveanu (2006, 2007, 2008) points out that there are also
*mixed* readings of donkey sentences, such as:



(59)
 Every
person who buys a book on amazon.com and has a credit card uses it to
pay for it.


Sentence (59) intuitively is true iff for *every* book that a
credit-card owner buys on amazon.com, there is *some* credit
card or other that she uses to pay for the book. Hence the mixed
strong and weak reading. Brasoveanu accounts for these readings, along
with the weak/strong ambiguity in general, within his system of Plural
Compositional Discourse Representation Theory (PCDRT) by positing an
ambiguity at the level of indefinites. In his account, indefinites are
underspecified as to the presence of a maximization operator (which is
present in the strong indefinite but not the weak); the decision of
which indefinite to use in a specific case is a "context-driven
online process".



Theories that assign both sets of truth conditions to relative clause
donkey sentences generally do so by positing some sort of ambiguity,
either in the quantifiers, the pronouns, or the
 indefinites.[34]
 Though the matter isn't entirely clear, it seems plausible that
the theories discussed in this entry also may be able to assign both
the universal and the existential truth conditions to relative clause
donkey sentences by positing some sort of ambiguity. These matters
require further
 investigation.[35]


 A recent theory that does not rely on any ambiguity is presented in Champollion, Bumford & Henderson (2019). Champollion et al. argue for a trivalent semantics that allows for truth-value gaps for donkey sentences, along with a pragmatic theory that both fills in the gaps and predicts when hearers get a strong or weak interpretation. This explains the different readings of donkey sentences in terms of underspecification rather than ambiguity. The (dynamic) semantics delivers that a sentence like (26) is true iff every female donkey-owner beats every donkey she owns, false if at least one female donkey-owner does not beat any donkey she owns, and otherwise neither true nor false. Relying on Kriz (2016)'s framework for plural definites, speakers can use a sentence to address the question under discussion (QUD) that is true enough at a world w even if it lacks a truth value at w, so long as it is not false at any world equivalent to w. If the QUD is such that it requires that every female donkey-owner beat every donkey she owns, then worlds in which (say) some female donkey-owners who own multiple donkeys only beat one of their donkeys will be equivalent to worlds in which some female donkey owners do not beat any of their donkeys. However, if the QUD is such that it requires that every female donkey-owner beat some donkey she owns, such worlds will be equivalent for conversational purposes to worlds in which each female donkey-owner beats all the donkeys she owns, and thus speakers will judge the weak reading true in this scenario.


## 5. Anaphora in Sign Language



Another recent development in the study of anaphora comes from a
series of papers by Philippe Schlenker on anaphora in sign language,
particularly ASL (American Sign Language) and LSF (French Sign
Language) (2010, 2011, 2012a,b, 2013a,b, 2014, 2015). In sign
language, an antecedent is associated with a particular position in
signing space called a *locus*, and an anaphoric link to the
antecedent is obtained by pointing at that same locus. Unlike spoken
languages, which have a limited number of lexically encoded pronouns,
in sign language there seems to be no upper bound on how many loci can
simultaneously be used, aside from limitations of performance (since
signers have to remember what the loci are assigned to, and be able to
distinguish one from another). Schlenker (2015) gives an example of a
short discourse that involves 7 distinct loci. Given their unbounded
number and their overt connection to antecedents, Schlenker (as well
as others) posits that they are the overt realization of formal
indices, i.e., the variables that mark anaphoric connections in
theories like DRT and dynamic semantics. Despite their differences,
sign language and spoken language pronouns have enough in common that
a uniform theory is desirable, and Schlenker argues that sign language
provides evidence applicable to at least two debates in the anaphora
literature. First, there is good evidence that in sign language, there
is a single system of anaphora using loci for denoting individuals,
times, and worlds. This provides evidence in favor of those who think
that there are temporal and modal variables and pronouns in spoken
language. Second, Schlenker argues that the use of loci, which look
like an overt use of indices, very much resembles the dynamic semantic
account of pronominal anaphora since antecedents introduce variables
(loci) and anaphoric connections are made by repeating the same
variables (loci). Just as in spoken language, this occurs even when
the anaphoric pronouns are outside the syntactic scope of their
antecedents. Furthermore, according to Schlenker one of the most
decisive examples is the bishop example
 (50),
 repeated here:



(50)
 If a bishop meets
a bishop, he blesses him.


In sign language, the only way to obtain the intended reading of the
sentence is for the pronouns to index different antecedents. This
shows that a theory need not only get the truth conditions right (as
some D-type theories can), but must account for the formal link
between each antecedent and pronoun. For criticism of the thesis that
loci are overt variables see Kuhn (2016). 



The second major conclusion from Schlenker's research is that
sign language pronouns have an *iconic* element. For example,
the verb "ask" is signed near the chin/neck area of a
locus. If the locus denotes a person *standing* on a branch,
"ask" is signed at a fairly high spot in the locus. If the
locus denotes a person *hanging upside-down* from a branch,
"ask" is signed lower in the locus. This and many other
examples of iconicity in pronouns provides evidence that there needs
to be an account of iconicity integrated into the formal
semantics.