## 1. Introduction


 This article concerns Discourse Representation Theory narrowly
defined as work in the tradition descending from Kamp (1981). The same
term is sometimes used more broadly, occasionally embracing Heim's
(1982) work and the developments initiated by Groenendijk and Stokhof
(1989, 1991). However, the more common common label for the extended
DRT family is "dynamic semantics", and that is the term we
will use, too.


Two features that set DRT apart from other varieties of dynamic
semantics is that it is representational and non-compositional. In the
1980s, the founding years of dynamic semantics, these features made
DRT a controversial theory, though by now those controversies have
abated. DRT's main innovation, beyond the Montagovian paradigm which
was then considered orthodox, is that it introduced a level of mental
representations, called "discourse representation
structures" (DRSs). The basic idea is rather straightforward. It
is that a hearer builds up a mental representation of the discourse as
it unfolds, and that every incoming sentence prompts additions to that
representation. This picture has always been commonplace in the
psychology of language. DRT's principal tenet is that it should be the
starting point for semantic theory, too.


A theory of the DRT family consists of the following
ingredients:


* A formal definition of the representation language, consisting
 of:
	+ a recursive definition of the set of all well-formed DRSs, and
	+ a model-theoretic semantics for the members of this set;
* a construction procedure, which specifies how to extend a given
 DRS when a sentence comes in.


Technically, this is very similar to earlier work in formal
semantics, with two exceptions: the interpretation process always
takes the previous discourse into account, and the level of semantic
representations is claimed to be essential. What has worried some
semanticists is not so much the fact that DRSs are mental
representations, but that an additional level is needed, and used in
ways that render the theory non-compositional; we will return to this
issue in Section 7.


## 2. Donkey pronouns


The relationship between a pronoun and its antecedent is one that
has received a great deal of attention in linguistics and philosophy.
We say that a pronoun is anaphoric, as opposed to deictic, for
example, if it depends for its interpretation on an antecedent
expression elsewhere in the sentence or the discourse. In some cases
the nature of this dependency seems straightforward:



[1]
Pedro beats his donkey.

(Here and henceforth we use underlining to highlight the anaphoric
links we are currently interested in.) Since "Pedro" is a
referential expression, it makes sense, in this case, to say that
"his" is a referential term, too, which derives its
reference from its antecedent. This construal is plausible enough,
but it doesn't apply across the board:



[2]
No farmer beats his donkey.

The term "no farmer" is not a referential expression,
so "his" cannot be coreferential with it. Rather, it would
seem that, in this case, the relationship between the pronoun and its
antecedent is one of **binding**, in the logical sense of
the word: "no farmer" and "his" are
interpreted as a quantifier and a variable, respectively, and the
former binds the latter.


It is fairly obvious that the bound-variable construal of pronouns
is subject to syntactic constraints. Most importantly, this type of
interpretation requires that the pronoun
be **c-commanded** by its antecedent, where c-command is
defined as follows:



> 
> *A* c-commands *B* iff *B* is, or is contained
> in, a sister constituent of *A*'s.
> 


The constituent structure of sentence [2] is [S
[NP No farmer][VP beat [NP his
donkey]]]. Here the pronoun is contained in the sister of its
antecedent, so the c-command constraint is met. In [3] "no
farmer" does not c-command "his", and so we predict,
correctly, that the pronoun cannot be bound.



[3]
His donkey likes no farmer.

Since the syntactic structure of [1] is identical to that of [2],
we might wonder if the pronoun in [1] may not be bound by its
antecedent, just as in [2]. If this were possible, there would be two
ways of construing the pronoun in [1]: it could function either as a
referential term or as a bound variable. There is evidence that this is
indeed the case:



[4a]
Pedro beats his donkey, and Juan does, too.
[4b]
Every farmer beats his donkey, and Juan does, too.

[4a] is ambiguous in a way [4b] is not. Its second conjunct can be
construed as saying either that Juan beats his own donkey or that Juan
beats Pedro's donkey. The second conjunct of [4b], by contrast, can
only be interpreted as saying that Juan beats his own donkey. If the
possessive pronoun in the first conjunct of [4a] can be either
referential or a bound variable, the contrast is readily
explained.


In large part, the motivation for developing dynamic theories of
interpretation, beginning with DRT, was the realization that the
dichotomy between referential and bound-variable (occurrences of)
pronouns is less natural than one might think--less natural in
the sense that some pronouns don't fit comfortably in either
category.



[5a]
Pedro owns a donkey.
[5b]
It is grey.

What is the relationship between "it" in the second
sentence and its antecedent expression in the first? On the one hand,
it cannot be coreference. If the pronoun were coreferential with its
antecedent, the indefinite "a donkey" would have to be a
referential term, which seems unlikely, e.g., because the negation of
[5a] says not that there is a donkey that Pedro fails to own, but
rather that he doesn't own any donkey, and furthermore, if [5a] is
negated, the anaphoric link between the pronoun and its antecedent is
severed:



[6a]
Pedro doesn't own a donkey. 
[6b]
\*It is grey. 

(The asterisk indicates that the sentence is infelicitous if
"a donkey" is to be interpreted as the antecedent of
"it".) Observations like these suggest rather strongly
that indefinites are quantifiers rather than referential
terms. However, if we construe "a donkey" as an
existential quantifier, how does it manage to bind the pronoun across
a sentence boundary?


The problem with [5] is related to the fact that the pronoun and
its indefinite antecedent occur in different sentences. The following
examples show, however, that similar problems arise within
sentences:



[7a]
 If Pedro owns a donkey, he beats it.
[7b]
 Every farmer who
owns a donkey beats
it. 

These are the infamous "donkey sentences" that were
already discussed by scholastic philosophers, and in modern times were
reintroduced into philosophy by Geach (1962). In these cases it is
obvious that the pronouns don't refer, so they can't be coreferential
with their antecedents, either. Nor are the pronouns bound by their
antecedents, for they aren't c-commanded by them. The constituent
structure of [7b] is roughly as follows:



> 
> 
> [S [NP Every [N farmer who owns
> [NP a donkey]]] [VP beats it]]


Whatever the syntactical details, "a donkey" is too
deeply embedded for it to c-command "it". The same goes
for [7a]. Hence, the neuter pronouns in [7a,b] cannot be construed as
bound variables.


Another way of showing that donkey pronouns are problematic is by
considering how we might render these sentences in predicate
logic. The most obvious interpretation of [7a] is that Pedro beats
every donkey he owns, and [7b] is naturally interpreted as claiming
that every farmer beats every donkey he owns. (So, somewhat
surprisingly, it appears that the indefinites in these sentences have
universal force.) These readings may be captured as follows:



[9a]

[?]*x*[[donkey(*x*) & own(Pedro,*x*)]
- beat(Pedro,*x*)]
[9b]

[?]*x*[?]*y*[[farmer(*x*) &
donkey(*y*) & own(*x*,*y*)] -
beat(*x*,*y*)]

So the problem is not that predicate logic fails to capture the
meanings of [7a] and [7b]. The problem is, rather, how to derive [9a]
and [9b] from [7a] and [7b] in a principled way. In order to derive
these logical forms, we have to assume not only that an indefinite
expression buried in a subordinate position ends up having wide scope,
but also that it acquires universal force in the process. A theory
based on such assumptions might capture the facts, but it would be
clearly ad hoc, and would generate a host of false predictions, as
well.


The upshot of the foregoing observations is that, apparently,
indefinites are neither quantifiers nor referential terms, and this
problem entrains another one, for as long as it is unclear what
indefinites mean, it will also remain obscure how they can serve as
antecedents to pronouns. Later on we will see that very similar
problems arise in seemingly disparate domains, like the interpretation
of tense and presuppositions.


## 3. Basic DRT


This section introduces DRT in fairly informal terms. We show how a
hearer builds up a mental model of the ongoing discourse, dealing not
only with simple sentences, but also with conditionals,
quantification, and anaphora, and we discuss DRT's treatment of
indefinites, and how they are set apart from quantifying
expressions.


### 3.1 The key ideas


A discourse representation structure (DRS) is a mental
representation built up by the hearer as the discourse unfolds. A DRS
consists of two parts: a universe of so-called "discourse
referents", which represent the objects under discussion, and a
set of DRS-conditions which encode the information that has
accumulated on these discourse referents. The following DRS represents
the information that there are two individuals, one of which is a
farmer, the other a donkey, and that the former chased the latter:



[1]
 **[x,
y: farmer(x), donkey(y), chased(x,y)]**

The universe of this DRS contains two discourse
referents, **x**
and **y**, and its condition set is
{**farmer(x)**,
**donkey(y)**,
**chased(x,y)**}.


A DRS like the one in [1] can be given a straightforward
model-theoretic interpretation. In DRT this is done by means
of **embedding functions**, which are partial functions
from discourse referents to individuals in a given model *M*;
or put otherwise, an embedding function is a partial variable
assignment. An embedding function *f* **verifies**
[1] in *M* iff the domain of *f* includes at
least **x**
and **y**, and according to *M*
it is the case that *f*(**x**)
is a farmer, *f*(**y**) is a
donkey, and *f*(**x**)
chased *f*(**y**).


Meanwhile it will have become clear that the DRS in [1] is designed
to reflect the intuitive meaning of:



[2]
 A farmer chased a
donkey. 

Indeed, it is claimed that, in the absence of any information about
the context in which this sentence is uttered, [1] is the semantic
representation of [2]. So the indefinite expressions "a
farmer" and "a donkey" prompt the introduction of
two new discourse referents, **x**
and **y**, and contribute the
information that **x** is a farmer
and **y** a donkey, while the verb
contributes the information that the former chased the latter.


If a discourse begins with an utterance of [2], the DRS in [1] is
constructed, and this DRS forms the background against which the next
utterance is interpreted. Suppose now that [2] is followed by a token
of [3a]:



[3a]
 He caught
it. 
[3b]
 **[v, w:
caught(v,w)]**

[3b] is the DRS that reflects the semantic content of [3a] before
the pronouns are resolved. In this DRS, the anaphoric pronouns
"he" and "it" in [3a] are represented by the
discourse referents **v**
and **w**, respectively. These
discourse referents are underlined to indicate that they require an
antecedent. Since [3a] is uttered in the context of [1], the next step
in the interpretation of this sentence is to merge the DRS in [3b]
with that in [1]. The result of this merging operation is [4a]:



[4a]
 **[x,
 y, v, w: farmer(x), donkey(y), chased(x,y),
caught(v,w)]**

Since [3a] is immediately preceded by [2], the antecedent of
"he" is probably (though not necessarily) "a
farmer", while "it" is anaphorically dependent on
"a donkey". At the DRS level, this is represented by equating
the discourse referents **v**
and **w**
with **x**
and **y**, respectively. These
operations yield [4b], which is equivalent to [4c]. Either DRS is
verified in a model *M* iff *M* features a farmer who
chased and caught a donkey.




[4b]
 **[x, y, v, w:
v = x, w = y,
  

farmer(x), donkey(y), chased(x,y), caught(v,w)]**
[4c]
 **[x,
y: farmer(x), donkey(y), chased(x,y),
caught(x,y)]**

Thus far, we have only considered DRSs with simple conditions, but
in order to account for negated and conditional sentences, say,
complex conditions are required.



[5a]
 Pedro doesn't have
a donkey. 
[5b]
 **[1
x: Pedro(x), ![2 y: donkey(y), owns(x,y)]
]**

[5b] is the sentence DRS corresponding to [5a]. This DRS contains a
condition that consists of a DRS prefixed by a negation sign. For ease
of reference we will sometimes adorn DRSs with numerical labels, as we
have done in [5b], and use these in names like
"[5b1]", "[5b2]", and so
on. In general, labeling of DRSs will be top-down and left-to-right,
so the main (or principal) DRS will always be number one.


A function *f* verifies [5b1] in a
model *M* iff *f*
maps **x** onto an individual
in *M* which "is a Pedro", i.e., which is called
"Pedro", and *f* cannot be extended to a
function *g* which verifies [5b2]. That is to say,
no such *g* should map **y**
onto a donkey that Pedro owns.


[5b2] contains a token of the discourse
referent **x** which is introduced
externally, in the DRS in which [5b2] is embedded, i.e.,
[5b1]. Apart from that, [5b2] also introduces a
discourse referent of its own,
i.e., **y**, which is associated with
the indefinite NP "a donkey", and whose scope is delimited
by [5b2]. Consequently, it doesn't make sense to refer
to **y** outside of [5b2].
In DRT, this is taken to explain why the "lifespan" of the
individual introduced by the indefinite NP in [5a] is delimited by the
scope of the negation operator. If [5a] were followed by [6a], for
example, the pronoun could not be linked to the indefinite:



[6a]
It is
grey. 
[6b]
**[z:
grey(z)]**
[6c]
**[1
x, z: Pedro(x), ![2 y:donkey(y), owns(x,y)],
grey(z)]**

If we merge [5b] and [6b], which is the sentence DRS corresponding
with [6a], we obtain [6c]. In this representation, the discourse
referent **z** does not have access
to **y**,
because **y** is introduced in a DRS
that is not accessible to the DRS in
which **z** is introduced, and
therefore it is not possible to
bind **z**
to **y**. In other words, if [6a] is
preceded by [5a], the pronoun cannot be anaphorically linked to the
indefinite. This prediction appears to be correct.


Accessibility is in the first instance a relation between DRSs;
derivatively, it is also a relation between discourse
referents. [6c1] is accessible to [6c2], but not
the other way round, and therefore the discourse referents introduced
in [6c1], i.e., **x**
and **z**, are accessible from
[6c2], but conversely, if we are in [6c1] we
have no access to [6c2] and its discourse referents,
i.e., **y**. Thus in [6c] anaphora is
not possible because **y** is not
accessible to **z**. In [4a], by
contrast, anaphora is possible,
because **x**
and **y** are accessible
to **v**
and **w** (the accessibility relation
being reflexive). The notion of accessibility is crucial to DRT's
account of anaphora, and it is important to note that it is not
stipulated, but is entailed by the semantics of the DRS
language. [6c1] is accessible to [6c2] because
every embedding function that must be considered for [6c2]
is an extension of an embedding function for [6c1], and it
is for this reason that every discourse referent in [6c1]
is also defined in [6c2]. The converse, however, does not
hold.


Like negated sentences, conditionals give rise to complex
DRS-conditions, too. (7) gives an example:



[7a]
 If Pedro owns a
donkey, he beats it. 
[7b]
 **[1
x: Pedro(x),   
[2 y: donkey(y), owns(x,y)] =
[3 v, w: beats(v,w)] ]**

[7b] is the sentence DRS which corresponds to [7a], and assuming
for convenience that this sentence is uttered in an empty context, it
is also the initial DRS of the discourse. The complex condition in
this structure is interpreted as follows: if *f* is to verify
[7b1] in the current model,
then *f*(**x**) must be an
individual called "Pedro", and every extension
of *f* which verifies [7b2] must itself be extendable
to a function that verifies [7b3]. It follows from this that
[7b1] is accessible to [7b2], which in its turn is
accessible to [7b3], and
therefore **v** may be linked up
to **x** (accessibility being a
transitive relation) and **w**
to **y**. The result is [7c], which
is equivalent to [7d], both DRSs saying that Pedro beats every donkey
he owns.





[7c]
 **[1
x, v: v = x, Pedro(x),   
 [2 y, w: w = y, donkey(y),
owns(x,y)] = [3 : beats(v,w)] ]**
[7d]
 **[1
x: Pedro(x),   
[2 y: donkey(y), owns(x,y)] =
[3 : beats(x,y)] ]**

### 3.2 Quantifiers vs. indefinites


The interpretation of quantified donkey sentences is very similar
to what we have just seen:



[8a]
 Every farmer who
owns a donkey beats it. 
[8b]
 **[1
: [2 x, y: farmer(x), donkey(y),
owns(x,y)]([?]x)[3 v, w: beats(v,w)]
]**

There are two ways of spelling out the interpretation of
so-called **duplex conditions** of the
form *K*(**[?]***x*)*K'*. On its
weak interpretation, [8a] means that every farmer who owns a donkey
beats at least one of the donkeys he owns; on its strong
interpretation the sentence says that every farmer beats every donkey
he owns. While the strong reading is the most natural choice for [8a],
other donkey sentences prefer a weak reading:



[9]
 Every farmer who
owns a tractor uses it to drive to church on Sundays. 

The most likely interpretation for [9] to have is that every
tractor-owning farmer uses one of their tractors to drive to church.


The distinction between weak and strong interpretations is somewhat
mysterious, because it doesn't seem to be an ordinary ambiguity. For
example, donkey sentences with existential quantifiers don't seem to
license strong readings:



[10]
 Some farmers who
own a donkey beat it. 

This can only be construed as saying that some donkey-owning
farmers beat at least one of their donkeys. More generally, the
universally quantified donkey sentences seem to be the only ones to
license both interpretations. It is not entirely clear why this should
be the case (see Kanazawa 1994, Geurts 2002).


Whether weak or strong, the interpretation of a condition of the
form *K(Qx)K'*,
where *Q* may be any quantifier,
makes *K* accessible
to *K'*, and in this respect
(which is the fulcrum of the DRT analysis) conditionals and quantified
sentences are the same. Consequently, the discourse
referents **x**
and **y** in [8b2] are
accessible to **v**
and **w** in [8b3], and the
latter may be equated to the former. The resulting representation is
[8c], which is equivalent to [8d].



[8c]
**[1
: [2 x, y, v, w: v = x, w = y, farmer(x), donkey(y),
owns(x,y)]([?]x)[3 : beats(v,w)] ]**
[8d]
**[1
: [2 x, y: farmer(x), donkey(y),
owns(x,y)]([?]x)[3 : beats(x,y)] ]**

The DRT analysis of quantified expressions like "all"
or "most" is fairly standard. A quantifier binds a
variable and delivers the truth conditions one should
expect. Indefinites are different. An indefinite like "a
donkey" is treated not as a quantifier but simply as device for
introducing a discourse referent and one or more conditions; on the
DRT account, indefinites have no quantifying force of their own. What
quantifying force they seem to have is not theirs, but derives from
the environment in which they occur. If the semantic material
associated with "a donkey" is introduced in the main DRS,
as in [4c], the quantifying effect will be existential, owing to the
fact that this DRS is verified in a model *M* iff **there
is** a way of verifying it in *M*. If the semantic
material associated with "a donkey" is introduced in the
antecedent of a conditional, as in [7d], the quantifying effect will
be universal, owing to the fact that a
condition of the form *K***=***K'* is
verified in *M* iff **every** way of
verifying *K* can be extended to a
way of verifying *K'*. This
view on indefinites lies at the heart of DRT.


DRT was one of the first semantic theories to go beyond the
sentence boundary, and take into account how the interpretation of an
expression may depend on the preceding discourse. Looking back at the
examples discussed in the foregoing, we see that, if the DRT approach
is on the right track, sentence boundaries are not as important as the
Fregean (or Chomskyan) conception of language would have it. In particular, there is
essentially no difference between the DRT analysis of cross-sentential
anaphora, as exemplified by the mini-discourse [2]-[3a], and
sentence-internal anaphora, as in [7a] or [9a]. In either case, the
pronoun simply serves to pick up an accessible discourse
referent. This raises the question of how DRT's new-fangled notion of
anaphora relates to the dichotomy between referential and
bound-variable pronouns. Curiously, this issue doesn't seem to have
received much attention yet.


### 3.3 One- vs. two-level versions of DRT


In the foregoing we assumed that the contribution of a sentence to
the discourse, as represented by a DRS, was obtained in two steps. In
the first one, a sentence DRS was constructed in a compositional way;
this part of the construction process we took for granted, but its
implementation wouldn't be too difficult. In the second step, the
sentence DRS was merged with the DRS representing the prior discourse,
and anaphoric references were resolved. This two-stage procedure has
become the industrial standard, but the original version of DRT (Kamp
1981, Kamp and Reyle 1993) was monostratal in the sense that one set
of rules took care of both tasks at once. In the meantime, Kamp and
his co-workers have adopted the two-step method, too (e.g., Kamp et
al. 2011).


In one-level versions of DRT, a single set of rules is used to
obtain the semantic contribution of a sentence. To take a simple
example, if the sentence is "It is grey", the first rule
to apply will say that the semantic correlate of the subject is an
argument to the predicate expressed by "is grey". Then the
two main parts of the sentence are analysed further, and the pronoun
"it" triggers a rule which at once deals with the
pronoun's lexical content and its context dependence; i.e., it says
that we must select a new discourse referent, link it to a discourse
referent made available by the preceding discourse, and update the DRS
so as to record these changes. In two-level versions of DRT there is
no such rule. Its duties are divided between two separate
mechanisms. In the first stage, a separate DRS is constructed for the
sentence, in which pronouns prompt the introduction of new discourse
referents that are marked as being anaphoric (we use underlining for
this purpose). In the second stage, there is a general mechanism for
dealing with discourse referents thus marked.


The distinction between one- and two-level versions of DRT is of
interest for at least two reasons. First, once we go beyond basic DRT,
the two-stage system may actually be more economical overall. For
example, if DRT is extended for dealing with presuppositions (see
Section 5), the one-pass approach becomes unwieldy. Secondly, and
perhaps more importantly, two-stage theories may be viewed as
implementing a distinction between semantics and pragmatics which has
some prima facie plausibility: in the first stage a representation is
computed which is projected from the sentence's lexico-grammatical
structure, while in the second stage, context-dependent aspects of
meaning are dealt with. This division of labor is reminiscent of the
traditional dichotomy between compositional semantics and
non-compositional pragmatics, but it should be noted that the two do
not coincide. Usually, the "semantical" part of a
two-stage version of DRT is not compositional in the Fregean sense,
because its output is essentially incomplete; it may contain anaphoric
"gaps", for example. Therefore, even if a sentence DRS
admits of a truth-conditional interpretation, it will typically fall
short of capturing everything that is conventionally thought to belong
to truth-conditional content.


## 4. The DRS language: syntax, semantics, accessibility


In this section we give a more precise description of the DRS
language than we have done so far. We define the syntax of the
language, present a model-theoretic interpretation, and discuss the
notion of accessibility, which figures so prominently in DRT's account
of anaphora.


### 4.1 Syntax


DRSs are set-theoretic objects built from **discourse
referents** and **DRS-conditions**. A
DRS-condition is either atomic, in which case it consists of a
predicate and a suitable number of discourse referents, or complex, in
which case it embeds one or two DRSs. Hence, the definition of the DRS
language is by simultaneous recursion:



**DRSs and DRS-conditions**


* A DRS *K* is a pair
<*UK*,
 *ConK*>,
 where
 *U**K*
 is a set of discourse referents,
and *Con**K* is a
set of DRS-conditions.
* If *P* is an n-place predicate, and
*x1*, ...,
*xn*
are discourse referents, then
*P(x1, ..., xn)*
 is a DRS-condition.
* If *x* and
*y*
 are discourse referents, then
*x***=***y* is a
DRS-condition.
* If *K* and
*K'* are DRSs, then
**!***K*, 
*K***=***K'*, and
*K***[?]***K'* are
DRS-conditions.
* If *K* and
*K'* are DRSs and
*x* 
 is a discourse referent, then
*K*(**[?]***x*)*K'*
is a DRS-condition.



Note that there are no conditions corresponding to conjoined
sentences. Such sentences are dealt with, rather, by merging the DRSs
associated with their parts, where the merge of two DRSs
*K* and
*K'*
is defined as their pointwise union:



**DRS-merge**


* *K*[?]*K'* =  
 <*U**K* [?]
 *U**K'*,
 *Con**K* [?]
 *Con**K'*>



The merge operation is also used for combining a sentence DRS with
the DRS representing the preceding discourse, so the idea is that
there is no principled distinction between (clausal) conjunction and
sentence concatenation (see Muskens 1996 for discussion).


Officially, DRSs are set-theoretical objects, but in this article
we use a linear notation. Many sources, including Kamp's own work,
employ a graphical "box notation" which is sometimes
clearer, but less parsimonious and more difficult to type. To
illustrate the various schemes, here are three ways of representing
the content of a conditional donkey sentence:




|  |  |  |
| --- | --- | --- |
| Official DRS: |  | **<{}, <{x, y}, {farmer(x), donkey(y), owns(x,y)}> = <{}, {beats(x,y)}>} >** |
| Linear notation: |  | **[: [x, y: farmer(x), donkey(y), owns(x,y)] = [: beats(x,y)]]** |
| Box notation: |  | 

|  |
| --- |
|  |
| 

|  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- |
| 

|  |
| --- |
| **x y** |
| **farmer(x)**
**donkey(y)**
**owns(x,y)** |

 | **=** | 

|  |
| --- |
|  |
| **beats(x,y)** |

 |

 |

 |


The choice between these formats is one of convenience, and does
not have any theoretical significance.


### 4.2 Accessibility


Accessibility is a relation between DRSs that is transitive and
reflexive, i.e., it is a preorder. More in particular, it is the
smallest preorder for which the following holds, for all
DRSs *K*, *K'*,
and *K''*:
if *Con**K*
contains a condition of the form ...


* **!***K'*,
 then *K* is accessible
 to *K'*
* *K'***[?]***K''*,
 then *K* is accessible
 to *K'*
 and *K''*
* *K'***=***K''*,
 then *K* is accessible
 to *K'*
 and *K'* is accessible
 to *K''*
* *K'*(**[?]***x*)*K''*,
 then *K* is accessible
 to *K'*
 and *K'* is accessible
 to *K''*


To illustrate, in the following schematic representations, every
DRS is accessible to all and only those DRSs whose number doesn't
precede its own (so every DRS is accessible to itself):


* **[1 ... [2
 ... ]([?]x)[3 ... ![4
 ... ] ] ]**
* **[1 ... [2
 ... ] = [3 ... ![4 ...
 ] ] ]**


The **accessible domain** of a
DRS *K*, *A**K*,
is the set of discourse referents that occur in
some *K'* that is accessible
to *K*,
i.e., *A**K* =
{*x*:  *K'* is accessible
to *K*
and *x*
[?] *U**K'*}.
The main constraint which DRT imposes on the interpretation of
anaphora is this:



**Accessibility constraint**



A pronoun is represented by a discourse
referent *x* which must be equated
to some discourse referent *y*
[?] *A**K*,
where *K* is the DRS in
which *x* is introduced.



It is important to note that neither the notion of accessibility
nor the accessibility constraint need be stipulated. For, as we will
presently see, both follow from the way the DRS language is
interpreted (cf. the notion of variable binding in standard predicate
logic). Thus, we are not at liberty to modify the accessibility
relation, should we wish to do so, unless we simultaneously revise the
truth conditions associated with the DRS language. The accessibility
constraint is a semantic one.


As discussed in Section 2, the main syntactic constraint on
bound-variable pronouns is that they be c-commanded by their
antecedents. The domain of the accessibility constraint overlaps with
that of the c-command constraint, but it is wider. For one thing,
sentence boundaries don't affect accessibility. For another, within
the confines of a sentence, the semantic correlate of an
expression *E1* may be accessible to that
of *E2*, even if *E1* doesn't
c-command *E2*. To illustrate this point, consider
the conditional donkey sentence in [1a] and its sentence DRS in
[1b]:



[1a]
[S
[S If [S [NP
a farmer] [VP owns
 a donkey]] [S
he beats
 it] ]
[1b]
**[1
: [2 x, y: farmer(x), donkey(y), owns(x,y)] =
[3 u**, **v: beats(u,v)****] ]**

Syntactically speaking, neither "a farmer" nor "a
donkey" can bind (in the sense of Chomsky 1986) an expression
beyond the smallest clause in which they occur. In particular, neither
can bind any of the pronouns in the main clause of [1a]. But the
discourse referents associated with these indefinites,
i.e., **x**
and **y**, are in the accessible
domain of [1b3], and therefore the anaphoric discourse
referents in this embedded DRS can be linked
to **x**
and **y**. Put otherwise, they are
bindable at a semantic but not at a syntactic level.



### 4.3 Semantics


The truth-conditional semantics of the DRS language is given by
defining when an embedding function **verifies** a DRS in
a given model. An embedding function is a partial mapping from
discourse referents to individuals. Given two embedding
functions *f* and *g* and a
DRS *K*, we say that *g*
extends *f* with respect
to *K*,
or *f*[*K*]*g* for
short, iff *Dom*(*g*) = *Dom*(*f*)
[?] *U**K*, and
for all *x*
in *Dom*(*f*): *f*(*x*)
= *g*(*x*). Viewing
functions as set-theoretic objects, this can be formulated more
succinctly as follows:




> *f*[*K*]*g*
> iff *f* [?] *g* and *Dom*(*g*) = *Dom*(*f*)
> [?]
>  *U**K*


In the following we define what it takes for an embedding function
to verify a DRS or DRS-condition in a given model. As usual, a
model *M* is a pair *<D,I>*, where *D* is a
set of individuals, and *I* is an interpretation function that
assigns sets of individuals to one-place predicates, sets of pairs to
two-place predicates, and so on. To enhance the legibility of our
definition somewhat we omit the qualifier "in M"
throughout:



**Verifying embeddings**


* *f* verifies a DRS *K*
iff *f* verifies all conditions
in *Con**K*.
* *f* verifies *P(x1,
..., xn*) iff
<*f*(*x1*),
..., *f*(*xn*)>
[?] *I*(*P*).
* *f* verifies *x**=**y*
iff *f*(*x*)
= *f*(*y*).
* *f* verifies **!***K* iff
there is no *g* such
that *f*[*K*]*g*
and *g* verifies *K*.
* *f* verifies *K***[?]***K'* iff there is a *g* such
that *f*[*K*]*g*
and *g* verifies *K*
or *K'*.
* *f* verifies *K***=***K'* iff, for
all *f*[*K*]*g* such
that *g* verifies *K*, there
is an *h* such
that *g*[*K'*]*h*
and *h*
verifies *K'*.
* *f*
verifies *K*(**[?]***x*)*K'* iff,
for all individuals *d* [?] *DM* and for
all *f*[*K*]*g* such
that *g*(*x*) = *d*
and *g* verifies *K*,
every *h* such
that *g*[*K'*]*h*
verifies *K'*.



The last clause gives a strong interpretation to universal duplex
conditions (cf. Section 3.2); the weak interpretation is obtainable as
follows:


* *f*
verifies *K*(**[?]***x*)*K'* iff,
for all individuals *d* [?] *DM* for which
there is a *g* such
that *f*[*K*]*g*, *g*(*x*)
= *d*, and *g*
verifies *K*, there is an *h*
such
that *g*[*K'*]*h*
and *h*
verifies *K'*.


A DRS is true in a given model iff we can find a verifying
embedding for it, as follows:



**Truth**


A DRS *K* is true in a
model *M* iff there is an embedding function *f* such
that *Dom*(*f*) =
 *U**K*
 and *f* verifies
*K* in *M*.



To see how accessibility follows from these definitions, the
critical junctures are the points at which extensions of embedding
functions are called for. For example, consider a schematic DRS with a
conditional in it:



[2]
 **[x:
A(x), [y: B(x,y)] = [z: C(x,y,z)] ]**

This DRS is true iff we can find an embedding
function *f*, *Dom*(*f*) =
{**x**}, which verifies all conditions
in [2], which is to say
that *f*(**x**) must be
an **A**, and *f* verifies the
complex condition **[y: B(x,y)] = [z:
C(x,y,z)]**. The latter requirement is met iff every
extension *g* of *f*, *Dom*(*g*)
= {**x,y**}, such that *g*
verifies **[y: B(x,y)]** can be
extended to a function *h*, *Dom*(*h*)
= {**x,y,z**}, such that *h*
verifies **[z: C(x,y,z)]**. Hence, no
matter how we
choose *g*, *g*(**x**)
must be the same individual
as *f*(**x**), and for
any *g*, no matter how we
choose *h*, *h*(**x**)
and *h*(**y**) must be the same
as *g*(**x**)
and *g*(**y**),
respectively. This is why **x** is in
the accessible domain associated with the antecedent of the
conditional, and why **y** is in the
accessible domain associated with the consequent.


## 5. Beyond the basics


Having focussed thus far on the interpretation of anaphoric
pronouns, the purpose of this section is to illustrate how the ideas
lying at the heart of DRT have been applied to other phenomena. We
will restrict ourselves to three topics: tense, presupposition, and
propositional attitudes. In the first two cases, we will encounter
interpretative dependencies reminiscent of the dependency relation
between an anaphor and its antecedent, and we will describe, if only
in outline, how basic DRT has been extended so as to capture the
similarities. DRT's representationalism comes into its own in its
treatment of attitudes and attitude ascriptions, which is the third
topic on our list. In section 6 we introduce SDRT, a significant extension of the DRT framework designed to model discourse structure more accurately.


### 5.1 Tense


In the early 1970s, it was pointed out that there are systematic
parallels between tenses and pronouns (Partee 1973), and DRT was
partly born from the conviction that a theory of interpretation should
account for these parallels. Partee (1984) herself worked out some of
these ideas in the DRT framework. In the following we will briefly
discuss some of Partee's observations, and outline a DRT
treatment.



[1a]
Pedro owns a
donkey. He beats it. 
[1b]
Yesterday, Pedro
tried to kiss Juanita. She slapped him. 

Intuitively, the indefinite description "a donkey" in
the opening sentence of [1a] serves to introduce an entity that is
subsequently picked up by the pronoun "it" in the second
sentence, and the DRT account of indefinites and pronouns seeks to
formalize this intuitive picture. [1b] shows that a very similar
phenomenon occurs in the temporal domain, for in this case it is
natural to construe the past tense of the second sentence as being
anaphorically dependent on the content of the first: presumably,
Juanita slapped Pedro right after he tried to kiss her. In order to
capture this idea, DRT adopts an event-based semantics, treating
events as semantic values of a designated class of discourse
referents. Sentences are now construed like indefinite descriptions in
that they, too, serve to introduce discourse referents. For example,
the first sentence in [1b] introduces a discourse referent of the
event type that represents Pedro's attempting to kiss Juanita, and the
tense morpheme in "slapped" is construed as referring
back, very much like an anaphoric pronoun, to that event: by default,
the time of Juanita's slapping Pedro will be taken to be immediately
after the time of him trying to kiss her.


Our second example involves what one might call "donkey
tense":



[2a]
Every farmer who
owns a donkey beats it. 
[2b]
Whenever Pedro
tried to kiss Juanita, she slapped him.

In an event-based semantics, it is natural to construe [2b] as
quantifying over events, just like [2a] quantifies over
individuals. In conjunction with the DRT treatment of quantification,
this allows us to interpret the tense in "slapped" as
referring back to the event introduced in the subordinate clause,
along the following lines (where **[?][?]** stands for
the "right after"
relation, **e''** is the variable
introduced by the tense in "slapped",
and **e''** indicates
that **e''** requires an
antecedent):



[3a]
**[x,
y: Pedro(x), Juanita(y),**  

**[e: try-to-kiss(e,x,y)]([?]e)[e', e'' : e'' [?][?] e', slap(e',y,x)] ]**
[3b]
**[x,
y: Pedro(x), Juanita(y),**  

**[e: try-to-kiss(e,x,y)]([?]e)[e': e [?][?] e', slap(e',y,x)] ]**

In [3a] the tense introduces an anaphoric element, which can be
bound in the restrictor of the quantifier, just as an ordinary donkey
pronoun might be bound. The result, after a minor simplification, is
[3b], which says that every event e of Pedro trying to kiss Juanita is
immediately followed by an event e' of her slapping him. Note
that, although this DRS captures the most natural interpretation of
the past tense in "slapped", it leaves the past tense in
"tried" unaccounted for. But this is as it should be, for
if tense is anaphoric, then that tense, too, should be linked to a
salient time point, which seems to be right: in the absence of a
context that furnishes such a time, [2b] is simply infelicitous; it is
like saying "He is handsome" when it is not clear whom the
pronoun is intended to refer to.


### 5.2 Presupposition


Presuppositions are chunks of information associated with
particular lexical items or syntactic constructions. There are many
such items and constructions, and the following is just a small
selection: 



**Factive verbs**



[4a]
Juan knows that
Pedro beats his donkey.
[4b]
Pedro beats his
donkey.

**It-clefts**



[5a]
It's in the stable
that Pedro beats his donkey.
[5b]
Pedro beats his
donkey.

**Definites**



[6a]
Pedro beats his
donkey.
[6b]
Pedro has a
donkey.


Someone who utters any of the [a] sentences commits himself to the
truth of the respective [b] sentence. Of course, this does nothing to
distinguish presuppositions from ordinary entailments, but the
difference becomes apparent when we embed presuppositional expressions
or constructions in non-entailing environments, as in:



[7a]
It isn't in the
stable that Pedro beats his donkey.
[7b]
Maybe it's in the
stable that Pedro beats his donkey.

Here [5a] is embedded in the scope of a negation and a modal
operator, respectively, and it appears that these sentences commit a
speaker to the truth of [5b] just as much as [5a] does. This behavior
sets presuppositions apart from ordinary entailments.


Generally speaking, presuppositions tend to escape
from **any** embedded position in the sense that, if a
sentence *S* contains a presupposition-inducing
expression *P*, an utterance of *S* will usually imply
that *P* holds. This is only generally speaking, because this
rule, though correct in the majority of cases, does not hold without
exceptions:



[8a]
It may be that
Pedro is a mean farmer, and that he beats his donkey.
[8b]
It may be that
Pedro has a donkey, and that he beats his donkey.
[9a]
If Pedro is a mean
farmer, he beats his donkey.
[9b]
If Pedro has a
donkey, he beats his donkey.

While the [a] sentences would normally commit a speaker to the
claim that Pedro has a donkey, the same does not hold for the [b]
sentences. It appears, therefore, that presuppositions are typically
though not invariably inherited by the sentences in which they
occur. This is the so-called "projection problem" for
presuppositions.


Considering that anaphora and presupposition were widely discussed
from the late 1960s onwards, it took a remarkably long time before it
was discovered that, in some respects at least, the two phenomena are
very similar (van der Sandt 1992, Kripke 2009). To see how, consider
the following pairs:



[10a]
Pedro owns a
donkey. Juan knows {it / that he owns a donkey}.
[10b]
If Pedro owns a
donkey, Juan knows {it / that he owns a donkey}.
[10c]
Every farmer who
owns a donkey knows {it / that he owns a donkey}.

It is true that the variants with the
"that"-complements are slightly odd, but actually the fact
that they are proves the point we want to make; for it is clear that,
in these examples, the pronoun "it" and the clause
"that he owns a donkey" perform the same duty, and if the
latter sounds somewhat off it is presumably because the former does
the job just as well.


While the factive verb "know" triggers the
presupposition that its clausal complement is true, this
presupposition is neutralized in [10b] and [10c]. If presuppositions
behave similarly to anaphoric pronouns, it is clear why that should be
so: in both cases, the presupposition is **bound**
sentence-internally, just like the neuter pronoun is bound.


Presupposed information is information that is presented as given,
and according to the so-called "binding theory" of
presupposition this means that presuppositions want to have discourse
referents to bind to (van der Sandt 1992, Geurts 1999). However,
whereas anaphoric pronouns are rarely interpretable in the absence of
a suitable antecedent, the same does not hold for all
presupposition-inducing expressions. For instance, a speaker may
felicitously assert that he met "Pedro's sister" even if
he knows full well that his audience wasn't aware that Pedro has a
sister. In such cases, Karttunen (1974) and Stalnaker (1974)
suggested, presuppositions are
generally **accommodated**, which is to say that the
hearer accepts the information as given, and revises his
representation of the context accordingly. Accommodation, thus
understood, is a form of exploitation in Grice's sense: the purpose of
presuppositional expressions is to signal that this or that
information is given, and if some information is new but not
particularly interesting or controversial (like the fact that somebody
has a sister) the speaker may choose to "get it out of the
way" by presuppositional means.


The binding theory incorporates the notion of accommodation as
follows. Presuppositions, according to the theory, introduce
information that prefers to be linked to discourse referents that are
already available in the DRS, and in this respect they are like
pronouns. However, if a suitable discourse referent is not available,
a new one will be accommodated, and the presupposition is linked to
that. Generally speaking, accommodation is not an option in the
interpretation of pronouns, and a possible reason for this is that a
pronoun's descriptive content is relatively poor. Being told that
"she" is wonderful is not particularly helpful if it isn't
clear who the pronoun is meant to refer to. By contrast, if the
speaker refers to "Pedro's sister" there is more to go on,
and accommodation becomes feasible. Hence, the binding theory views
pronouns as a special class of presuppositional expressions. While all
presupposition triggers prefer to be bound, pronouns almost always
must be bound. According to the binding theory, this is because
pronouns are descriptively attenuated, and therefore cannot be
construed by way of accommodation.


### 5.3 Attitudes and attitude ascriptions


Propositional attitudes (mental states like belief, desire, or
imagination) and attitude ascriptions (sentences like "John
thinks/hopes that Mary likes him") have been a rich source of
philosophical puzzles. DRT's representationalism offers fresh
solutions to some of them.


A central issue in the treatment of attitudes is that of logical
omniscience: humans are cognitively limited agents, and are not aware
of all the consequences of their beliefs and desires. On this basis,
Kamp (1990) argues that the objects of thought should not be seen as
purely model-theoretic entities, for example a set of belief worlds,
as is common in logics of belief from Hintikka (1962) onwards. Rather,
the logic of belief and other attitudes must involve structured mental
representations. The problem of logical omniscience is then resolved
in terms of the failure of human agents to perform logically complete
computations over these representations. Modeling the objects of
attitudes in this way, as representational entities, allows us to make
the fine distinctions necessary to distinguish between believing that
it is raining and believing that if it is not raining then 37 is not
prime. However, traditional versions of such a sententialist or
Language-of-Thought approach to omniscience typically run into the
problem of intentionality: to be useful to us, our attitudes must
somehow be about the external world we live in, so we still have to
explain how these syntactic representations connect with reality. In a
Hintikka-style possible worlds semantics the link between world and
attitude is hardwired into the semantics: believing that John is happy
means that the agent takes the actual world to be an element of the
set of worlds where John is happy. The DRT approach combines these two
strategies: use the classic, model-theoretic semantics of the DRS
language to cash out intentionality in terms of possible worlds, but
hold on to the representational level to model resource-bounded
computations in practical reasoning.


 A special type of intentionality is that of singular or *de
re* attitudes: beliefs and desires that directly involve objects
the attitude holder is acquainted with. To capture such attitudes,
Kamp (1990) introduces anchors. An internal anchor is a descriptive
mode of presentation, in the form of a DRS with a distinguished
discourse referent, while an external anchor maps that discourse
 referent to an actual individual. A full-fledged entity representation comprises an internal and an external anchor, and is a construct closely related to the Mental Files of Recanati (2012) and other neo-Fregean approaches to reference.


A complete mental state
representation then consists of (i) a set of internal anchors and
attitude descriptions, each consisting of a DRS paired with a mode
indicator (e.g., **ANCH**, 
**BEL**, 
**DES**) and possibly sharing discourse
referents, and (ii) an external anchor mapping the internally anchored
discourse referents to objects of acquaintance. By way of
illustration, the mental state representation below describes
my *de re* belief that this cup *c* I'm holding now
contains coffee, along with the desire that it is still warm:



[1a]
internal mental state:  
 **{  <ANCH x, [x: cup(x), hold(i,x)]>,   
   <BEL, [y: contains(x,y), coffee(y)]>,   
   <DES, [: warm(y)]> }**
[1b]

 external anchor: {<**x**,*c*>}
Compartmentalized mental state representations like this have been
turned into semantic theories of attitude ascriptions and used to
solve a variety of familiar puzzles, such as those involving Kripke's
(1979) Pierre who believes Londres is pretty but London is not (Asher
1986, Kamp 2015), Geach's (1967) Hob who believes a witch killed his
cow while Nob believes she, the same witch, blighted his mare (Asher
1987), specific indefinites (Kamp and Bende-Farkas 2019), presupposition projection out of
(non-doxastic) attitude ascriptions (Maier 2015), and referring to fictional entities (Maier 2017).


Not all DRT treatments of attitude reports rely on a
representational theory of the underlying mental states
themselves. For example, Geurts (1999) and Maier (2010) develop
analyses of attitude ascriptions in versions of DRT that use only a
standard Hintikka-style possible worlds semantics for attitudes. In
the same vein, Maier (2009) and Hunter (2013) explore extensions of
DRT to deal with directly referential expressions like indexicals and
proper names without resorting to anchors or mental state
representations.


## 6. Discourse relations in SDRT


We've seen in sections 3 and 4 above how anaphora resolution
is constrained by accessibility. But there are many other factors that play a role in
the resolution process, from local, grammatical factors like subject
antecedents being generally more prominent than object antecedents in pronoun
resolution, to more global, discourse structural factors. Focusing on the latter,
Hobbs (1979) argues that the interpreter's desire to create a maximally coherent
interpretation will often play a decisive role in choosing between multiple
accessible antecedents for a pronoun, as he illustrates with the following contrast:



[1a]

 John can open Bill's safe. He knows the combination. 
[1b]
 John can open Bill's safe. He'll have to change the combination soon.
 

In [1a] the pronoun binds to John, because that delivers the most coherent
interpretation of the two-sentence discourse as a whole. In [1b], by contrast, the
pronoun is bound to Bill. This difference corresponds to a difference in the kind of
coherence relations exemplified by the two discourses: [1a] exemplifies a coherence
relation known as Elaboration, while [1b] is an instance of Result. That discourse
relations play an important role in discourse interpretation is shown by examples like
 [2] (again from Hobbs 1979).



[2]

 John took a train from Paris to Istanbul. He likes spinach. 
 

 Reading [2] you might wonder if Turkish spinach is really that much better than
what you can get in France, which shows to what lengths we go to accommodate a
suitable coherence relation, in this case Explanation, before we give up our quest for
 a coherent interpretation.


 Asher and Lascarides (2003) take the idea that pronoun resolution and the
inference of coherence relations between discourse units are closely related processes
as the starting point for their Segmented Discourse Representation Theory (SDRT),
combining DRT with insights from discourse coherence theories like Hobbs's. SDRT
assumes a set of discourse relations (Narration, Result, Contrast, Explanation,
Elaboration, Correction, to name a few) that we may infer to connect the
propositions expressed by the individual sentences (or clauses), in order to create a
maximally coherent interpretation of the discourse as a whole. Different discourse
relations may impose different constraints on anaphora resolution over and
above DRT's accessibility. Take the example in [3] (Asher and Lascarides
 2003).



[3a]

John had a great evening last night. 
 
[3b]

He had a great meal. 
 
[3c]

He ate salmon.
 
[3d]

[3d] He devoured lots of cheese.
 
[3e]

He won a dancing competition.
 
[3f]

??It was a beautiful pink.
 

In standard DRT the salmon introduced in [3c] would provide a suitable and
accessible antecedent for the pronoun in [3f], but in fact this resolution appears to be
unavailable. SDRT explains this observation in terms of a discourse structural
constraint known as the Right Frontier Constraint. This constraint is best illustrated
by drawing a discourse structure graph like the one below. A horizontal arrow in this
graph indicates a coordinating discourse relation (where neither relatum dominates
the other), in this case Narration. A vertical arrow indicates a subordinating
discourse relation (where a nucleus dominates a satellite), in this case Elaboration.





|  |
| --- |
| [3a] |
|             |Elaboration |
| 

|  |  |  |
| --- | --- | --- |
| [3b] | - | [3e] |
|             |Elaboration | Narration |  |
| 

|  |  |  |
| --- | --- | --- |
| [3c] | - | [3d] |
|  | Narration |  |

 |  |  |

 |


SDRT's version of the Right Frontier Constraint says that the pronoun in [3f] can
only find an antecedent introduced in the node that was just introduced ([3e]), or one
 that dominates it ([3a]).


 More formally, an SDRS consists of such a discourse structure graph, where each
vertex corresponds to a discourse relation and each node to a DRS. A common SDRS
notation extends the familiar box notation of DRT (see section 4) with labels for
 elementary and complex discourse units:




|  |
| --- |
| **p1 p6** |
| **p1:[x: John(x), great-time(x)]** |
| 

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| **p6:** | 

|  |
| --- |
| **p2 p5 p7** |
| **p2:[: great-meal(x)]** |
| **p5:[: won-dance(x)]** |
| **Narration(p2,p5)** |
| 

|  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- |
| **p7:** | 

|  |
| --- |
| **p3 p4** |
| **p3:[: ate-salmon(x)]** |
| **p4:[: devoured-cheese(x)]** |
| **Narration(p3,p4)** |

 |

 |
| **Elaboration(p2,p7)** |

 |

 |
| **Elaboration(p1,p6)** |


Anaphora and presupposition resolution rely on a modified notion of
accessibility that incorporates the Right Frontier Constraint. The
model-theoretic interpretation of an SDRS is built around a set of
special interpretation rules for the various discourse
relations. Narration for instance comes with a semantic interpretation
rule that states that the both relata hold and moreover the main
events they introduce stand in a relation of temporal
progression. Some notational
shortcuts: *K**p* is the DRS
associated with label *p* (i.e. the *K* such
that *p* : *K* occurs somewhere in the
SDRS); *e**p* is the unique event
discourse referent that occurs
in *U*(*K**p*) (for readability
we left out event referents from the example SDRS above, but for the purpose of modelling narrative temporal progression in SDRT we should treat every verb phrase as introducing an event discourse referent).



> 
> *f* verifies **Narration(p1,p2)** iff there is an embedding function *g* such that *f*[*K***p****1**]*g*
> and *g* verifies *K***p****1**, and there is an embedding *h* such that *g*[*K***p****2**]*h* and *h* verifies *K***p****2** and moreover *h*(*e***p****1**) *[?][?]**h*(*e***p****2**)
> 


Note that conditions of the form *p**n* : *K* are not interpreted directly as semantic conditions on their own, but instead serve to associate DRS contents with
 proposition labels.


## 7. Representationalism and compositionality


DRT is a representational theory of interpretation. In the DRT
framework, sentences have meanings, if at all, only in a derivative
sense: a sentence prompts the hearer to modify his DRS, and it is DRSs
that have a truth-conditional interpretation. Thus, a sentence causes
a DRS *Ki* to be
transformed into a
DRS *Ki+1* (usually,
though not invariably, the process will consist in adding information
to *Ki*), and at a remove
there is an associated transition at the semantical level,
from *Ki*'s
interpretation
to *Ki+1*'s. Hence, if
we could cut out the representational middleman, we would have a
theory that defines sentence meaning in dynamic terms, as a transition
from one semantic object to another. Thus arises the question of
whether or not the representational level can be dispensed with. Kamp
(1981) claimed that representations were essential to
his **ur**-version of DRT, which was a subset of our
basic DRT, but in the meantime he was proved wrong by a slew of
dynamic theories that recast classical DRT in purely semantical terms:
Barwise (1987), Rooth (1987), Groenendijk and Stokhof (1989, 1991),
Muskens (1996), and many more. However, Kamp's claim still holds for
various extensions of classical DRT, like the theories of
presupposition and attitude ascription discussed in the foregoing.


These theories are allegedly "non-representational",
but what exactly does that mean? Consider, for example, Groenendijk
and Stokhof's (1991) Dynamic Predicate Logic (DPL), which provides a
dynamic semantics for the language of classical predicate logic, in
which "Pedro owns a donkey and beats it" may be rendered
as follows:



[1]

[?]*x*[donkey(*x*) & own(Pedro,*x*)]
& beat(Pedro,*x*)

DPL's semantics ensures that, unlike in classical logic, the last
occurrence of *x* is bound by the existential quantifier. The
way this works is that DPL analyzes sentence meanings in terms of
variable assignments, which are manipulated by the logical constants
in such as way that [1] is true in a model *M* iff *M*'s
domain features an individual which, when assigned to the
variable *x*, makes each of the atomic sentences in [1] come
out true in *M*. Formally, this is implemented by stipulating
that the meaning of a sentence is a relation between variable
assignments.


A variable assignment is a mapping from variables to individuals,
and variables are linguistic expressions, so it can hardly be claimed
that assignments are non-representational entities. Therefore, it is a
bit puzzling that, in contrast to DRT, DPL is widely seen as a
non-representational variety of dynamic semantics. True, DRT's
representationalism goes quite a bit further than DPL's, and the two
theories are obviously very different in this respect, but both
theories import representational elements into semantics, and this
holds for dynamic theories of semantics generally (though see Dekker
2012 for exceptions).


Note, incidentally, that DRT's representationalism is not
particularly troubling from a psychological point of view. For if we
adopt a cognitivist perspective, and view DRT as a (somewhat abstract)
theory of the psychology of interpretation, then its
representationalism wouldn't be particularly controversial, since for
most psychologists, the controversy is not over whether there exist
mental representations, but over what they are like.


 One way in which DRT's representationalism manifests itself as
being more radical than other theories' is that it is a
non-compositional theory as well. These two features are intertwined.
Consider, for instance, the way pronouns are interpreted in basic DRT,
by first setting up a discourse referent, which is subsequently linked
to another discourse referent. This is a non-deterministic process,
but even if it were not, it is clear that the anaphoric link is not
part of the meaning of the pronoun. A standard statement of
compositionality would say that the meaning of a compound expression
must be a function of the meaning of their components and their mode
of combination. But if some of the components, like pronouns, do not
introduce into the DRS any object that can naturally be described as
the meaning of that object, then it is clear that we do not have a
compositional system.


How bad is it that we lose compositionality? It is generally agreed
that some of the standard arguments for compositionality are not
particularly compelling. In particular, we don't need compositionality
to explain why people can understand an indefinite number of
sentences, and we don't need it for explaining how languages can be
learned. To explain such facts, it suffices that there is some
procedure for establishing the meanings of sentences, but it is not
strictly necessary for this procedure to be compositional.
Compositionality is often viewed as a methodological principle
(Groenendijk and Stokhof 1989, 1991; Janssen 1997; Dever 1999), since
any model of interpretation can be made compositional if we are
sufficiently relaxed about the nature of lexical meanings and/or the
syntactic structures over which the compositional theory is
defined. In the case of DRT, the standard approaches to making the
system compositional involve dispensing with Kamp's construction
algorithm, and replacing it with a system in which DRSs can be
composed either through suitable merge operations (Zeevat 1989b), or,
in dynamic semantic systems like those of Groenendijk and Stokhof
(1989) and Muskens (1996), through use of function application in a
lambda calculus. But while most such systems capture meanings for
sentences that are closely related to those of original DRT, they do
not capture every aspect of DRT. In particular, whereas anaphora
resolution is an integral part of DRT, in compositional theories anaphora resolution is
assumed to be performed by a separate component of the theory.


Representationalism and compositionality are issues that have
produced great disagreement amongst semanticists and philosophers of
language, though less so nowadays than they did in the 1980s. The
methodological view of compositionality has reduced what appeared once
to be a substantive argument to a matter of taste, and as regards DRT
it has been shown that researchers who care about compositionality are
free to adopt a variant of the original system with many of its
features, but none of its non-compositionality. But
representationalism may be a more serious issue, for if it is true, as
we have suggested, that dynamic theories generally rely on a notion of
meaning that contains at least some form of representation, then that
notion could do with a bit more philosophical attention than it has
received so far. 


## 7. Further reading


* **Introductions:**
Kamp and Reyle (1993), Kamp (1995), Blackburn and Bos (2005), van
Eijck (2006), van Eijck and Kamp (1997), Kamp et al. (2011).
* **Donkey anaphora:**
Heim (1990), Evans (1980), Neale (1990), Kanazawa (1994), Geurts (2002), Krifka (1996).
* **Presupposition:**
van der Sandt (1992), Geurts (1999), Beaver (2002), Beaver and Zeevat (2007), Kamp (2001), Kamp and Rossdeutscher (1994), Krahmer (1998), Blackburn and Bos (2005).
* **Tense and aspect:**
Kamp and Reyle (1993), Kamp et al. (2011), Partee (1984).
* **Quantification and plurality:**
Kamp and Reyle (1993).
* **Attitude reports:**
Asher (1986, 1989), Zeevat (1996), Kamp (1990), Geurts (1999), Maier (2010).
* **Discourse structure and SDRT:**
Asher (1993), Asher and Lascarides (2003), van Leusen (2007).
* **Inference systems for DRT:**
Kamp and Reyle (1996), Saurer (1993).