Discourse Representation Theory

First published Tue May 22, 2007; substantive revision Thu Jul 11, 2024

In the early 1980s, Discourse Representation Theory (DRT) wasintroduced by Hans Kamp as a theoretical framework for dealing withissues in the semantics and pragmatics of anaphora and tense (Kamp1981a); a very similar theory was developed independently by IreneHeim (1982). The distinctive features of DRT, to be discussed below,are that it is a mentalist and representationalist theory ofinterpretation, and that it is a theory of the interpretation not onlyof individual sentences but of discourse, as well. In these respectsDRT made a clear break with classical formal semantics, which duringthe 1970s had emanated from Montague’s pioneering work (Montague1974), but in other respects it continued the tradition, e.g., in itsuse of model-theoretical tools. In the meantime, DRT has come to serveas a framework for explaining a wide range of phenomena, and givenrise to numerous extensions, modifications, applications andimplementations. We will confine our attention to topics related tophilosophical debates, ignoring applications related more tolinguistics, logic, or AI.

1. Introduction

This article concerns Discourse Representation Theory narrowly definedas work in the tradition descending from Kamp (1981a). The same termis sometimes used more broadly, occasionally embracing Heim’s(1982) work and the developments initiated by Groenendijk and Stokhof(1990, 1991). However, the more common common label for the extendedDRT family is “dynamic semantics”, and that is the term wewill use, too.

Two features that set DRT apart from other varieties of dynamicsemantics is that it is representational and non-compositional. In the1980s, the founding years of dynamic semantics, these features madeDRT a controversial theory, though by now those controversies haveabated. DRT’s main innovation, beyond the Montagovian paradigmwhich was then considered orthodox, is that it introduced a level ofmental representations, called “discourse representationstructures” (DRSs). The basic idea is rather straightforward. Itis that a hearer builds up a mental representation of the discourse asit unfolds, and that every incoming sentence prompts additions to thatrepresentation. This picture has always been commonplace in thepsychology of language. DRT’s principal tenet is that it shouldbe 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 givenDRS when a sentence comes in.

Technically, this is very similar to earlier work in formal semantics,with two exceptions: the interpretation process always takes theprevious discourse into account, and the level of semanticrepresentations is claimed to be essential. What has worried somesemanticists is not so much the fact that DRSs are mentalrepresentations, but that an additional level is needed, and used inways that render the theory non-compositional; we will return to thisissue inSection 7.

2. Donkey pronouns

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

[1]: Pedro beatshis donkey.

(Here and henceforth we use underlining to highlight the anaphoriclinks we are currently interested in.) Since “Pedro” is areferential expression, it makes sense, in this case, to say that“his” is a referential term, too, which derives itsreference from its antecedent. This construal is plausible enough, butit doesn’t apply across the board:

[2]: No farmer beatshis donkey.

The term “no farmer” is not a referential expression, so“his” cannot be coreferential with it. Rather, it wouldseem that, in this case, the relationship between the pronoun and itsantecedent is one ofbinding, in the logical sense ofthe word: “no farmer” and “his” areinterpreted as a quantifier and a variable, respectively, and theformer binds the latter.

It is fairly obvious that the bound-variable construal of pronouns issubject to syntactic constraints. Most importantly, this type ofinterpretation requires that the pronoun bec-commanded by its antecedent, where c-command isdefined as follows:

A c-commandsB iffB is, or is containedin, a sister constituent ofA’s.

The constituent structure of sentence[2] is [_S [_NP No farmer][_VP beat[_NP his donkey]]]. Here the pronoun is contained in thesister of its antecedent, so the c-command constraint is met. In[3] “no farmer” does not c-command “his”, and sowe predict, correctly, that the pronoun cannot be bound.

[3]: His donkey likesno 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 itsantecedent, just as in [2]. If this were possible, there would be twoways of construing the pronoun in [1]: it could function either as areferential term or as a bound variable. There is evidence that thisis indeed the case:

[4a]: Pedro beatshis donkey, and Juan does, too.
[4b]: Every farmer beatshis donkey, and Juan does,too.

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

In large part, the motivation for developing dynamic theories ofinterpretation, beginning with DRT, was the realization that thedichotomy between referential and bound-variable (occurrences of)pronouns is less natural than one might think—less natural inthe sense that some pronouns don’t fit comfortably in eithercategory.

[5a]: Pedro ownsa donkey.
[5b]: It is grey.

What is the relationship between “it” in the secondsentence and its antecedent expression in the first? On the one hand,it cannot be coreference. If the pronoun were coreferential with itsantecedent, the indefinite “a donkey” would have to be areferential term, which seems unlikely, e.g., because the negation of[5a] says not that there is a donkey that Pedro fails to own, but ratherthat he doesn’t own any donkey, and furthermore, if [5a] isnegated, the anaphoric link between the pronoun and its antecedent issevered:

[6a]: Pedro doesn’t owna donkey.
[6b]: *It is grey.

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

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

[7a]: If Pedro ownsa donkey, he beatsit.
[7b]: Every farmer who ownsa donkey beatsit.

These are the infamous “donkey sentences” that werealready discussed by scholastic philosophers, and in modern times werereintroduced into philosophy by Geach (1962). In these cases it isobvious that the pronouns don’t refer, so they can’t becoreferential with their antecedents, either. Nor are the pronounsbound 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 deeplyembedded 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 byconsidering 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 everydonkey he owns. (So, somewhat surprisingly, it appears that theindefinites in these sentences have universal force.) These readingsmay be captured as follows:

[9a]: \(\forall x[[\textrm{donkey}(x) \amp\textrm{own}(\textrm{Pedro},x)] \rightarrow\textrm{beat}(\textrm{Pedro},x)]\)
[9b]: \(\forall x \forall y[[\textrm{farmer}(x) \amp \textrm{donkey}(y)\amp \textrm{own}(x,y)] \rightarrow \textrm{beat}(x,y)]\)

So the problem is not that predicate logic fails to capture themeanings 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 theselogical forms, we have to assume not only that an indefiniteexpression buried in a subordinate position ends up having wide scope,but also that it acquires universal force in the process. A theorybased on such assumptions might capture the facts, but it would beclearly ad hoc, and would generate a host of false predictions, aswell.

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

3. Basic DRT

This section introduces DRT in fairly informal terms. We show how ahearer builds up a mental model of the ongoing discourse, dealing notonly with simple sentences, but also with conditionals,quantification, and anaphora, and we discuss DRT’s treatment ofindefinites, and how they are set apart from quantifyingexpressions.

3.1 The key ideas

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

[1]: \(\cb{[\R{x},\R{y}:\textrm{farmer}(\R{x}),\textrm{donkey}(\R{y}),\textrm{chased}(\R{x},\R{y})]}\)

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

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

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

[2]: A farmer chased a donkey.

Indeed, it is claimed that, in the absence of any information aboutthe context in which this sentence is uttered,[1] is the semantic representation of[2]. So the indefinite expressions “a farmer” and “adonkey” prompt the introduction of two new discourse referents,\(\cb{\R{x}}\) and \(\cb{\R{y}}\), and contribute the information that\(\cb{\R{x}}\) is a farmer and \(\cb{\R{y}}\) a donkey, while the verbcontributes 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 thenext utterance is interpreted. Suppose now that [2] is followed by atoken of[3a]:

[3a]: He caught it.
[3b]: \(\cb{[\ulR{v}, \ulR{w}: \textrm{caught}(\R{v},\R{w})]}\)

[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 thediscourse referents \(\cb{\R{v}}\) and \(\cb{\R{w}},\) respectively.These discourse referents are underlined to indicate that they requirean antecedent. Since [3a] is uttered in the context of[1], the next step in the interpretation of this sentence is to merge theDRS in [3b] with that in [1]. The result of this merging operation is[4a]:

[4a]: \(\cb{[\R{x}, \R{y}, \ulR{v}, \ulR{w}: \textrm{farmer}(\R{x}),\textrm{donkey}(\R{y}), \textrm{chased}(\R{x},\R{y}),\textrm{caught}(\R{v},\R{w})]}\)

Since[3a] is immediately preceded by[2], the antecedent of “he” is probably (though notnecessarily) “a farmer”, while “it” isanaphorically dependent on “a donkey”. At the DRS level,this is represented by equating the discourse referents \(\cb{\R{v}}\)and \(\cb{\R{w}}\) with \(\cb{\R{x}}\) and \(\cb{\R{y}}\),respectively. These operations yield[4b], which is equivalent to[4c]. Either DRS is verified in a model \(M\) iff \(M\) features a farmerwho chased and caught a donkey.

[4b]: \(\cb{ [\R{x}, \R{y}, \R{v}, \R{w}: \R{v} = \R{x}, \R{w} = \R{y},\textrm{farmer}(x), \textrm{donkey}(\R{y}),\textrm{chased}(\R{x},\R{y}), \textrm{caught}(\R{v},\R{w})] }\)
[4c]: \(\cb{[\R{x}, \R{y}: \textrm{farmer}(\R{x}),\textrm{donkey}(\R{y}), \textrm{chased}(\R{x},\R{y}),\textrm{caught}(\R{x},\R{y})]}\)

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

[5a]: Pedro doesn’t have a donkey.
[5b]: \(\cb{[_{1} \R{x}: \textrm{Pedro}(\R{x}), \neg [_{2} \R{y}:\textrm{donkey}(\R{y}), \textrm{owns}(\R{x},\R{y})] ]}\)

[5b] is the sentence DRS corresponding to[5a]. This DRS contains a condition that consists of a DRS prefixed by anegation sign. For ease of reference we will sometimes adorn DRSs withnumerical labels, as we have done in [5b], and use these in names like“[5b₁]”, “[5b₂]”, and soon. 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 \(\cb{\R{x}}\) onto an individual in \(M\) which “is aPedro”, i.e., which is called “Pedro”, and \(f\)cannot be extended to a functiong which verifies[5b₂]. That is to say, no suchg should map\(\cb{\R{y}}\) onto a donkey that Pedro owns.

[5b₂] contains a token of the discourse referent\(\cb{\R{x}}\) 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., \(\cb{\R{y}}\), which is associated with the indefinite NP“a donkey”, and whose scope is delimited by[5b₂]. Consequently, it doesn’t make sense to referto \(\cb{\R{y}}\) outside of [5b₂]. In DRT, this is takento explain why the “lifespan” of the individual introducedby the indefinite NP in [5a] is delimited by the scope of the negationoperator. If[5a] were followed by[6a], for example, the pronoun could not be linked to the indefinite:

[6a]: It is grey.
[6b]: \(\cb{[\ulR{z}: \textrm{grey}(\R{z})]}\)
[6c]: \(\cb{[_{1} \R{x}, \ulR{z}: \textrm{Pedro}(\R{x}), \neg [_{2}\R{y}:\textrm{donkey}(\R{y}), \textrm{owns}(\R{x},\R{y})],\textrm{grey}(\R{z})]}\)

If we merge[5b] and[6b], which is the sentence DRS corresponding with[6a], we obtain[6c]. In this representation, the discourse referent \(\cb{\R{z}}\) doesnot have access to \(\cb{\R{y}}\), because \(\cb{\R{y}}\) isintroduced in a DRS that is not accessible to the DRS in which\(\cb{\R{z}}\) is introduced, and therefore it is not possible to bind\(\cb{\R{z}}\) to \(\cb{\R{y}}\). In other words, if [6a] is precededby[5a], the pronoun cannot be anaphorically linked to the indefinite. Thisprediction 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 otherway round, and therefore the discourse referents introduced in[6c₁], i.e., \(\cb{\R{x}}\) and \(\cb{\R{z}}\), areaccessible from [6c₂], but conversely, if we are in[6c₁] we have no access to [6c₂] and itsdiscourse referents, i.e., \(\cb{\R{y}}\). Thus in[6c] anaphora is not possible because \(\cb{\R{y}}\) is not accessible to\(\cb{\R{z}}\). In[4a], by contrast, anaphora is possible, because \(\cb{\R{x}}\) and\(\cb{\R{y}}\) are accessible to \(\cb{\R{v}}\) and \(\cb{\R{w}}\)(the accessibility relation being reflexive). The notion ofaccessibility is crucial to DRT’s account of anaphora, and it isimportant to note that it is not stipulated, but is entailed by thesemantics of the DRS language. [6c₁] is accessible to[6c₂] because every embedding function that must beconsidered for [6c₂] is an extension of an embeddingfunction for [6c₁], and it is for this reason that everydiscourse referent in [6c₁] is also defined in[6c₂]. The converse, however, does not hold.

Like negated sentences, conditionals give rise to complexDRS-conditions, too.[7] gives an example:

[7a]: If Pedro owns a donkey, he beats it.
[7b]: \(\cb{[_{1} \R{x}: \textrm{Pedro}(\R{x}), [_{2} \R{y}:\textrm{donkey}(\R{y}), \textrm{owns}(\R{x},\R{y})] \Rightarrow [_3\ulR{v}, \ulR{w}: \textrm{beats}(\R{v},\R{w})] ]}\)

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

[7c]: \(\cb{[_1 \R{x}, \R{v}: \R{v} = \R{x}, \textrm{Pedro}(\R{x}),[_{2}\R{y}, \R{w}: \R{w} = \R{y}, \textrm{donkey}(\R{y}),\textrm{owns}(\R{x},\R{y})] \Rightarrow [_{3} :\textrm{beats}(\R{v},\R{w})] ]}\)
[7d]: \(\cb{[_{1} \R{x}: \textrm{Pedro}(\R{x}), [_{2} \R{y}:\textrm{donkey}(\R{y}), \textrm{owns}(\R{x},\R{y})] \Rightarrow [_{3}: \textrm{beats}(\R{x},\R{y})] ]}\)

3.2 Quantifiers vs. indefinites

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

[8a]: Every farmer who owns a donkey beats it.
[8b]: \(\cb{[_{1} : [_{2} \R{x}, \R{y}: \textrm{farmer}(\R{x}),\textrm{donkey}(\R{y}), \textrm{owns}(\R{x},\R{y})](\forall\R{x})[_{3} \ulR{v}, \ulR{w}: \textrm{beats}(\R{v},\R{w})] ])}\)

There are two ways of spelling out the interpretation of so-calledduplex conditions of the form \(K(\cb{\forall} x)K'.\) On its weak interpretation,[8a] means that every farmer who owns a donkey beats at least one of thedonkeys he owns; on its strong interpretation the sentence says thatevery farmer beats every donkey he owns. While the strong reading isthe most natural choice for [8a], other donkey sentences prefer a weakreading:

[9]: Every farmer who owns a tractor uses it to drive to church onSundays.

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

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

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

This can only be construed as saying that some donkey-owning farmersbeat at least one of their donkeys. More generally, the universallyquantified donkey sentences seem to be the only ones to license bothinterpretations. 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 \(K\) accessibleto \(K',\) and in this respect (which is the fulcrum of the DRTanalysis) conditionals and quantified sentences are the same.Consequently, the discourse referents \(\cb{\R{x}}\) and\(\cb{\R{y}}\) in [8b₂] are accessible to \(\cb{\R{v}}\)and \(\cb{\R{w}}\) in [8b₃], and the latter may be equatedto the former. The resulting representation is[8c], which is equivalent to[8d].

[8c]: \(\cb{[_{1} : [_{2} \R{x}, \R{y}, \R{v}, \R{w}: \R{v} = \R{x},\R{w} = \R{y}, \textrm{farmer}(\R{x}), \textrm{donkey}(\R{y}),\textrm{owns}(\R{x},\R{y})](\forall \R{x})[_{3} :\textrm{beats}(\R{v},\R{w})] ]}\)
[8d]: \(\cb{[_{1} : [_{2} \R{x}, \R{y}: \textrm{farmer}(\R{x}),\textrm{donkey}(\R{y}), \textrm{owns}(\R{x},\R{y})](\forall\R{x})[_{3} : \textrm{beats}(\R{x},\R{y})] ]}\)

The DRT analysis of quantified expressions like “all” or“most” is fairly standard. A quantifier binds a variableand delivers the truth conditions one should expect. Indefinites aredifferent. An indefinite like “a donkey” is treated not asa quantifier but simply as device for introducing a discourse referentand one or more conditions; on the DRT account, indefinites have noquantifying force of their own. What quantifying force they seem tohave is not theirs, but derives from the environment in which theyoccur. 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 thatthis DRS is verified in a model \(M\) iffthere is away of verifying it in \(M\). If the semantic material associated with“a donkey” is introduced in the antecedent of aconditional, as in[7d], the quantifying effect will be universal, owing to the fact that acondition of the form \(K\cb{\Rightarrow} K'\) is verified in \(M\)iffevery way of verifying \(K\) can be extended to away of verifying \(K'.\) This view on indefinites lies at the heart ofDRT.

DRT was one of the first semantic theories to go beyond the sentenceboundary, and take into account how the interpretation of anexpression may depend on the preceding discourse. Looking back at theexamples discussed in the foregoing, we see that, if the DRT approachis on the right track, sentence boundaries are not as important as theFregean (or Chomskyan) conception of language would have it. Inparticular, there is essentially no difference between the DRTanalysis of cross-sentential anaphora, as exemplified by themini-discourse[2]–[3a], and sentence-internal anaphora, as in[7a] or[9a]. In either case, the pronoun simply serves to pick up an accessiblediscourse referent. This raises the question of how DRT’snew-fangled notion of anaphora relates to the dichotomy betweenreferential and bound-variable pronouns. Curiously, this issuedoesn’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 thediscourse, as represented by a DRS, was obtained in two steps. In thefirst one, a sentence DRS was constructed in a compositional way; thispart of the construction process we took for granted, but itsimplementation wouldn’t be too difficult. In the second step,the sentence DRS was merged with the DRS representing the priordiscourse, and anaphoric references were resolved. This two-stageprocedure has become the industrial standard, but the original versionof DRT (Kamp 1981a; Kamp & Reyle 1993) was monostratal in thesense that one set of rules took care of both tasks at once. In themeantime, Kamp and his co-workers have adopted the two-step method,too (e.g., Kamp, van Genabith, & Reyle 2011).

In one-level versions of DRT, a single set of rules is used to obtainthe semantic contribution of a sentence. To take a simple example, ifthe sentence is “It is grey”, the first rule to apply willsay that the semantic correlate of the subject is an argument to thepredicate expressed by “is grey”. Then the two main partsof the sentence are analysed further, and the pronoun “it”triggers a rule which at once deals with the pronoun’s lexicalcontent and its context dependence; i.e., it says that we must selecta new discourse referent, link it to a discourse referent madeavailable by the preceding discourse, and update the DRS so as torecord these changes. In two-level versions of DRT there is no suchrule. Its duties are divided between two separate mechanisms. In thefirst stage, a separate DRS is constructed for the sentence, in whichpronouns prompt the introduction of new discourse referents that aremarked as being anaphoric (we use underlining for this purpose). Inthe second stage, there is a general mechanism for dealing withdiscourse referents thus marked.

The distinction between one- and two-level versions of DRT is ofinterest for at least two reasons. First, once we go beyond basic DRT,the two-stage system may actually be more economical overall. Forexample, if DRT is extended for dealing with presuppositions (seeSection 5), the one-pass approach becomes unwieldy. Secondly, and perhaps moreimportantly, two-stage theories may be viewed as implementing adistinction between semantics and pragmatics which has some primafacie plausibility: in the first stage a representation is computedwhich is projected from the sentence’s lexico-grammaticalstructure, while in the second stage, context-dependent aspects ofmeaning are dealt with. This division of labor is reminiscent of thetraditional dichotomy between compositional semantics andnon-compositional pragmatics, but it should be noted that the two donot coincide. Usually, the “semantical” part of atwo-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 DRSadmits of a truth-conditional interpretation, it will typically fallshort of capturing everything that is conventionally thought to belongto truth-conditional content.

4. The DRS language: syntax, semantics, accessibility

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

4.1 Syntax

DRSs are set-theoretic objects built fromdiscoursereferents andDRS-conditions. ADRS-condition is either atomic, in which case it consists of apredicate and a suitable number of discourse referents, or complex, inwhich case it embeds one or two DRSs. Hence, the definition of the DRSlanguage is by simultaneous recursion:

DRSs and DRS-conditions

A DRS \(K\) is a pair \(\langle U_K, \textit{Con}_K\rangle\),where \(U_K\) is a set of discourse referents, and \(\textit{Con}_K\)is a set of DRS-conditions.
If \(P\) is an n-place predicate, and \(x_1, \ldots, x_n\) arediscourse referents, then \(P(x_1, \ldots, x_n)\) is aDRS-condition.
If \(x\) and \(y\) are discourse referents, then \(x \cb{=} y\) isa DRS-condition.
If \(K\) and \(K'\) are DRSs, then \(\cb{\neg} K,\) \(K\cb{\Rightarrow} K',\) and \(K \cb{\lor } K'\) areDRS-conditions.
If \(K\) and \(K'\) are DRSs and \(x\) is a discourse referent,then \(K(\cb{\forall} x )K'\) is a DRS-condition.

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

DRS-merge

\(K\oplus K' = \langle U_{K} \cup U_{K'}, \textit{Con}_{K} \cup\textit{Con}_{K'}\rangle\)

The merge operation is also used for combining a sentence DRS with theDRS representing the preceding discourse, so the idea is that there isno principled distinction between (clausal) conjunction and sentenceconcatenation (see Muskens 1996 for discussion).

Officially, DRSs are set-theoretical objects, but in this article weuse a linear notation. Many sources, including Kamp’s own work,employ a graphical “box notation” which is sometimesclearer, but less parsimonious and more difficult to type. Toillustrate the various schemes, here are three ways of representingthe content of a conditional donkey sentence:

Official DRS:: \(\cb{\langle \{\}, \{\langle \{\R{x}, \R{y}\},\{\textrm{farmer}(\R{x}), \textrm{donkey}(\R{y}),\textrm{owns}(\R{x},\R{y})\}\rangle \Rightarrow \langle \{\},\{\textrm{beats}(\R{x},\R{y})\}\rangle \} \rangle }\)
Linear notation:: \(\cb{[: [\R{x}, \R{y}: \textrm{farmer}(\R{x}),\textrm{donkey}(\R{y}), \textrm{owns}(\R{x},\R{y})] \Rightarrow [:\textrm{beats}(\R{x},\R{y})]]}\)
Box notation:

The choice between these formats is one of convenience, and does nothave any theoretical significance.

4.2 Accessibility

Accessibility is a relation between DRSs that is transitive andreflexive, i.e., it is a preorder. More in particular, it is thesmallest preorder for which the following holds, for all DRSs \(K\),\(K'\), and \(K''\): if \(\textit{Con}_{K}\) contains a condition ofthe form …

\(\cb{\neg} K',\) then \(K\) is accessible to \(K'\)
\(K'\cb{\lor} K'',\) then \(K\) is accessible to \(K'\) and\(K''\)
\(K' \cb{\rightarrow} K''\), then \(K\) is accessible to \(K'\) and\(K'\) is accessible to \(K''\)
\(K'(\cb{\forall} x)K''\), then \(K\) is accessible to \(K'\) and\(K'\) is accessible to \(K''\)

To illustrate, in the following schematic representations, every DRSis accessible to all and only those DRSs whose number doesn’tprecede its own (so every DRS is accessible to itself):

\(\cb{[_{1} \ldots [_{2} \ldots ](\forall x)[_{3} \ldots \neg[_{4} \ldots ] ] ]}\)
\(\cb{[_{1} \ldots [_{2} \ldots ] \rightarrow [_{3} \ldots \neg[_{4} \ldots ] ] ]}\)

Theaccessible domain of a DRS \(K,\) \(A_{K},\) isthe set of discourse referents that occur in some \(K'\) that isaccessible to \(K\), i.e., \(A_{K} = \{x : K'\) is accessible to \(K\)and \(x \in U_{K'}\}.\) The main constraint which DRT imposes on theinterpretation of anaphora is this:

Accessibility constraint: A pronoun is represented by a discourse referent \(x\) which mustbe equated to some discourse referent \(y \in A_{K},\) where \(K\) isthe DRS in which \(x\) is introduced.

It is important to note that neither the notion of accessibility northe accessibility constraint need be stipulated. For, as we willpresently see, both follow from the way the DRS language isinterpreted (cf. the notion of variable binding in standard predicatelogic). Thus, we are not at liberty to modify the accessibilityrelation, should we wish to do so, unless we simultaneously revise thetruth conditions associated with the DRS language. The accessibilityconstraint is a semantic one.

As discussed inSection 2, the main syntactic constraint on bound-variable pronouns is that theybe c-commanded by their antecedents. The domain of the accessibilityconstraint overlaps with that of the c-command constraint, but it iswider. For one thing, sentence boundaries don’t affectaccessibility. For another, within the confines of a sentence, thesemantic correlate of an expression \(E_{1}\) may be accessible tothat of \(E_{2},\) even if \(E_{1}\) doesn’t c-command\(E_{2}.\) To illustrate this point, consider the conditional donkeysentence in[1a] and its sentence DRS in[1b]:

[1a]: \([_{S} [_{S} \textrm{ If} [_{S} [_{\textrm{NP}} \textrm{afarmer}] [_{\textrm{VP}} \textrm{owns a donkey}]] [_{S} \textrm{hebeats it}]\,]\)
[1b]: \(\cb{[_{1} : [_{2} \R{x}, \R{y}: \textrm{farmer}(\R{x}),\textrm{donkey}(\R{y}), \textrm{owns}(\R{x},\R{y})] \Rightarrow [_{3}\R{u}, \R{v}: \textrm{beats}(\R{u},\R{v})] ]}\)

Syntactically speaking, neither “a farmer” nor “adonkey” can bind (in the sense of Chomsky 1986) an expressionbeyond the smallest clause in which they occur. In particular, neithercan bind any of the pronouns in the main clause of [1a]. But thediscourse referents associated with these indefinites, i.e.,\(\cb{\R{x}}\) and \(\cb{\R{y}}\), are in the accessible domain of[1b₃], and therefore the anaphoric discourse referents inthis embedded DRS can be linked to \(\cb{\R{x}}\) and \(\cb{\R{y}}\).Put otherwise, they are bindable at a semantic but not at a syntacticlevel.

4.3 Semantics

The truth-conditional semantics of the DRS language is given bydefining when an embedding functionverifies a DRS ina given model. An embedding function is a partial mapping fromdiscourse referents to individuals. Given two embedding functions\(f\) and \(g\) and a DRS \(K\), we say that \(g\) extends \(f\) withrespect to \(K\), or \(f[K]g\) for short, iff \(\textit{Dom}(g) =\textit{Dom}(f) \cup U_{K},\) and for all \(x\) in \(\textit{Dom}(f):f(x) = g(x).\) Viewing functions as set-theoretic objects, this can beformulated more succinctly as follows:

\(f[K]g\) iff \(f \subset g\) and \(\textit{Dom}(g) = \textit{Dom}(f)\cup U_{K}\)

In the following we define what it takes for an embedding function toverify a DRS or DRS-condition in a given model. As usual, a model\(M\) is a pair \(\langle D,I\rangle,\) where \(D\) is a set ofindividuals, andI is an interpretation function that assignssets of individuals to one-place predicates, sets of pairs totwo-place predicates, and so on. To enhance the legibility of ourdefinition somewhat we omit the qualifier “in \(M\)”throughout:

Verifying embeddings

\(f\) verifies a DRS \(K\) iff \(f\) verifies all conditions in\(\textit{Con}_{K}.\)
\(f\) verifies \(P(x_{1}, \ldots , x_{n})\) iff \(\langlef(x_{1}),\) …, \(f(x_{n})\rangle \in I(P).\)
\(f\) verifies \(x\cb{=}y\) iff \(f(x) = f(y).\)
\(f\) verifies \(\cb{\neg }K\) iff there is no \(g\) such that\(f[K]g\) and \(g\) verifies \(K.\)
\(f\) verifies \(K\cb{\lor }K'\) iff there is a \(g\) such that\(f[K]g\) and \(g\) verifies \(K\) or \(K'.\)
\(f\) verifies \(K\cb{\Rightarrow }K'\) 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 \(K(\cb{\forall} x)K'\) iff, for all individuals\(d \in D_{M}\) 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 duplexconditions (cf.Section 3.2); the weak interpretation is obtainable as follows:

\(f\) verifies \(K(\cb{\forall }x)K'\) iff, for all individuals\(d \in D_{M}\) 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 embeddingfor it, as follows:

Truth: A DRS \(K\) is true in a model \(M\) iff there is an embeddingfunction \(f\) such that \(\textit{Dom}(f) = U_{K}\) and \(f\)verifies \(K\) in \(M\).

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

[2]: \(\cb{[\R{x}: \R{A}(\R{x}), [\R{y}: \R{B}(\R{x},\R{y})]\rightarrow [\R{z}: \R{C}(\R{x},\R{y},\R{z})] ]}\)

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

5. Applications and extensions

Having focussed thus far on the interpretation of anaphoric pronouns,the purpose of this section is to illustrate how the ideas lying atthe heart of DRT have been applied to other phenomena. We willrestrict ourselves to three topics: tense, presupposition, andpropositional attitudes. In the first two cases, we will encounterinterpretative dependencies reminiscent of the dependency relationbetween an anaphor and its antecedent, and we will describe, if onlyin outline, how basic DRT has been extended so as to capture thesimilarities. DRT’s representationalism comes into its own inits treatment of attitudes and attitude ascriptions, which is thethird topic on our list. Insection 6 we introduce SDRT, a significant extension of the DRT frameworkdesigned to model discourse structure more accurately.

5.1 Tense

Historically, the original ideas behind DRT grew out of Kamp’searlier work on tense logic (Kamp 1971) and the apparent inability ofsuch static systems to adequately capture the interpretation of tenseand aspect in natural language discourse (Kamp 1981b [2017]).Crucially, as Partee (1973) had already pointed out, there aresystematic parallels between tenses and pronouns, and, as Partee(1984) herself also showed, these parallels can be adequately capturedin a DRT framework. In the following we will briefly discuss some ofPartee’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 theopening sentence of[1a] serves to introduce an entity that is subsequently picked up by thepronoun “it” in the second sentence, and the DRT accountof 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 thesecond sentence as being anaphorically dependent on the content of thefirst: presumably, Juanita slapped Pedro right after he tried to kissher. In order to capture this idea, DRT adopts an event-basedsemantics, treating events as semantic values of a designated class ofdiscourse referents. Sentences are now construed like indefinitedescriptions in that they, too, serve to introduce discoursereferents. For example, the first sentence in [1b] introduces adiscourse referent of the event type that represents Pedro’sattempting to kiss Juanita, and the tense morpheme in“slapped” is construed as referring back, very much likean anaphoric pronoun, to that event: by default, the time ofJuanita’s slapping Pedro will be taken to be immediately afterthe time of him trying to kiss her.

Our second example involves what one might call “donkeytense”:

[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 ofquantification, this allows us to interpret the tense in“slapped” as referring back to the event introduced in thesubordinate clause, along the following lines (where \(\cb{\rar}\)stands for the “right after” relation,\(\cb{\ul{\R{e}}''}\) is the variable introduced by the tense in“slapped”, and its underlining indicates that it requiresan antecedent):

[3a]: \(\cb{[\R{x}, \R{y}: \textrm{Pedro}(\R{x}),\textrm{Juanita}(\R{y}),}\); \(\cb{[\R{e}: \textrm{try-to-kiss}(\R{e},\R{x},\R{y})](\forall\R{e})[\R{e}', \ul{\R{e}}'' : \R{e}'' \rar \R{e}',\textrm{slap}(\R{e}',\R{y},\R{x})] ]}\)
[3b]: \(\cb{[\R{x}, \R{y}: \textrm{Pedro}(\R{x}),\textrm{Juanita}(y),}\); \(\cb{[\R{e}: \textrm{try-to-kiss}(\R{e},\R{x},\R{y})](\forall\R{e})[\R{e}': \R{e} \rar \R{e}', \textrm{slap}(\R{e}',\R{y},\R{x})]]}\)

In[3a] the past tense in “slapped” introduces an anaphoricelement, which can be bound in the restrictor of the quantifier, justas an ordinary donkey pronoun might be bound. The result, after aminor simplification, is[3b], which says that every event \(\cb{\R{e}}\) of Pedro trying to kissJuanita is immediately followed by an event \(\cb{\R{e}'}\) of herslapping him. Note that, although this DRS captures the most naturalinterpretation of the past tense in “slapped”, it leavesthe past tense in “tried” unaccounted for. But this is asit should be, for if tense is anaphoric, then that tense, too, shouldbe linked to a salient time point, which seems to be right: in theabsence of a context that furnishes such a time,[2b] is simply infelicitous; it is like saying “He ishandsome” when it is not clear whom the pronoun is intended torefer to.

5.2 Presupposition

Presuppositions are chunks of information associated with particularlexical items or syntactic constructions. There are many such itemsand 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 thetruth of the respective [b] sentence. Of course, this does nothing todistinguish presuppositions from ordinary entailments, but thedifference becomes apparent when we embed presuppositional expressionsor 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 tothe truth of[5b] just as much as [5a] does. This behavior sets presuppositions apartfrom ordinary entailments.

Generally speaking, presuppositions tend to escape fromany embedded position in the sense that, if asentenceS contains a presupposition-inducing expressionP, an utterance ofS will usually imply thatP holds. This is only generally speaking, because this rule,though correct in the majority of cases, does not hold withoutexceptions:

[8a]: It may be that Pedro is a mean farmer, and that he beats hisdonkey.
[8b]: It may be that Pedro has a donkey, and that he beats hisdonkey.
[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 claimthat Pedro has a donkey, the same does not hold for the [b] sentences.It appears, therefore, that presuppositions are typically though notinvariably inherited by the sentences in which they occur. This is theso-called “projection problem” for presuppositions.

Considering that anaphora and presupposition were widely discussedfrom the late 1960s onward, it took a remarkably long time before itwas discovered that, in some respects at least, the two phenomena arevery similar (van der Sandt 1992; Kripke 2009). To see how, considerthe 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 adonkey}.
[10c]: Every farmer who owns a donkey knows {it / that he owns adonkey}.

It is true that the variants with the “that”-complementsare slightly odd, but actually the fact that they are proves the pointwe 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 ispresumably because the former does the job just as well.

While the factive verb “know” triggers the presuppositionthat its clausal complement is true, this presupposition isneutralized in[10b] and[10c]. If presuppositions behave similarly to anaphoric pronouns, it isclear why that should be so: in both cases, the presupposition isbound sentence-internally, just like the neuterpronoun is bound.

Presupposed information is information that is presented as given, andaccording to the so-called “binding theory” ofpresupposition this means that presuppositions want to have discoursereferents to bind to (van der Sandt 1992; Geurts 1999). However,whereas anaphoric pronouns are rarely interpretable in the absence ofa suitable antecedent, the same does not hold for allpresupposition-inducing expressions. For instance, a speaker mayfelicitously assert that she met “Pedro’s sister”even if she knows full well that her audience wasn’t aware thatPedro has a sister. In such cases, Karttunen (1974) and Stalnaker(1974) suggested, presuppositions are generallyaccommodated, which is to say that the hearer acceptsthe information as given, and revises his representation of thecontext accordingly. Accommodation, thus understood, is a form ofexploitation in Grice’s sense: the purpose of presuppositionalexpressions is to signal that this or that information is given, andif some information is new but not particularly interesting orcontroversial (like the fact that somebody has a sister) the speakermay choose to “get it out of the way” by presuppositionalmeans.

The binding theory incorporates the notion of accommodation asfollows. Presuppositions, according to the theory, introduceinformation that prefers to be linked to discourse referents that arealready available in the DRS, and in this respect they are likepronouns. However, if a suitable discourse referent is not available,a new one will be accommodated, and the presupposition is linked tothat. Generally speaking, accommodation is not an option in theinterpretation of pronouns, and a possible reason for this is that apronoun’s descriptive content is relatively poor. Being toldthat “she” is wonderful is not particularly helpful if itisn’t clear who the pronoun is meant to refer to. By contrast,if the speaker refers to “Pedro’s sister” there ismore to go on, and accommodation becomes feasible. Hence, the bindingtheory views pronouns as a special class of presuppositionalexpressions. 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 thereforecannot be construed by way of accommodation.

In addition to the classic examples of presupposition triggers, thebinding theory of presupposition in DRT has been applied to a varietyof constructions that are not traditionally classified aspresupposition triggers, including proper names (Geurts 1997),indexicals (Hunter 2013), and quotation (Geurts & Maier 2005).Arguments in each case involve pointing out examples of thecharacteristic presuppositional behaviors. For proper names, forinstance, Geurts gives[11] as a case of donkey binding:

[11]: If a child is christened “Bambi” and Disney Inc. findout about it, they will sue Bambi’s parents (Geurts 1997)

Geurts takes examples like these to support an analysis where a nameis analyzed as triggering the presupposition that there is someonenamed thus. This analysis amounts to a form of descriptivism (namesfunction like definite descriptions, in this case descriptions of theform “the person named so-and-so”). As such, it runs afoulof Kripke’s (1980) well-known anti-descriptivist argumentation,i.e., that we incorrectly predict the following two statements,uttered out of the blue, to be synonymous.

[12a]: Pedro is named “Pedro”.
[12b]: The person named “Pedro” is named“Pedro”.

As Kripke observes, while[12a] expresses a contingent truth (his parents might have changed theirmind),[12b] is, or at least has a reading that is, (weakly) necessary, whichshows that these sentences, and hence the name and the description,are not equivalent. Similar arguments apply to indexicals and theirdescriptive/presuppositional counterparts (“I” is notequivalent to “the speaker”). DRT solutions to thisKripkean challenge by Geurts & Maier (2013), Maier (2009a), andHunter (2013) all involve adding distinct layers or levels to the DRSlanguage in which certain discourse referents and conditions areevaluated with respect to the actual context of utterance (Kaplan1989) and thus effectively rigidified. Kamp (2015) provides a moredrastic overhaul of the DRT framework to capture the semantics andpragmatics of proper names in a broader mentalistic account ofattitudinal states, to which we turn now.

5.3 Attitudes and attitude ascriptions

Propositional attitudes (mental states like belief, desire, orimagination) and attitude ascriptions (sentences like “Johnthinks/hopes that Mary likes him”) have been a rich source ofphilosophical puzzles. DRT’s representationalism offers freshsolutions to some of them.

A central issue in the philosophical/semantic analysis of attitudes isthat of logical omniscience: humans are cognitively limited agents,and are not aware of all the consequences of their beliefs anddesires. On this basis, Kamp (1990) argues that the objects of thoughtshould not be seen as purely model-theoretic entities, for example aset of belief worlds, as is common in logics of belief from Hintikka(1962) onward. Rather, the logic of belief and other attitudes mustinvolve structured mental representations. The problem of logicalomniscience is then resolved in terms of the failure of human agentsto perform logically complete computations over these representations.Modeling the objects of attitudes in this way, as representationalentities, allows us to make the fine distinctions necessary todistinguish between believing that it is raining and believing that ifit is not raining then 37 is not prime. However, traditional versionsof such a sententialist or Language-of-Thought approach to omnisciencetypically run into the problem of intentionality: to be useful to us,our attitudes must somehow be about the external world we live in, sowe still have to explain how these syntactic representations connectwith reality. In a Hintikka-style possible worlds semantics the linkbetween world and attitude is hardwired into the semantics: believingthat John is happy means that the agent takes the actual world to bean element of the set of worlds where John is happy. The DRT approachcombines these two strategies: use the classic, model-theoreticsemantics of the DRS language to cash out intentionality in terms ofpossible worlds, but hold on to the representational level to modelresource-bounded computations in practical reasoning.

A special type of intentionality is that of singular orde reattitudes: beliefs and desires that directly involve objects in theexternal world that the attitude holder is acquainted with. To capturesuch attitudes, Kamp (1990) introduces anchors. An internal anchor isa descriptive mode of presentation, in the form of a DRS with adistinguished discourse referent, while an external anchor maps thatdiscourse referent to an actual individual. A full-fledged entityrepresentation comprises an internal and an external anchor, and is aconstruct closely related to the Mental Files of Récanati(2012).

A complete mental state representation then consists of

a set of internal anchors and attitude descriptions, eachconsisting of a DRS paired with a mode indicator (e.g.,\(\cb{\textrm{ANCH}},\) \(\cb{\textrm{BEL}},\) \(\cb{\textrm{DES}})\)and possibly sharing discourse referents, and
an external anchor mapping the internally anchored discoursereferents to objects of acquaintance.

By way of illustration, the mental state representation belowdescribes myde re belief that this cup \(c\) I’mholding now contains coffee, along with the desire that it is stillwarm:

[13a]: internal mental state: \[\cb{\begin{multline}\{ \langle \textrm{ANCH } \R{x}, [\R{x}: \textrm{cup}(\R{x}), \textrm{hold}(\R{i},\R{x})]\rangle ,\\ \langle \textrm{BEL}, [\R{y}: \textrm{contains}(\R{x},\R{y}), \textrm{coffee}(\R{y})]\rangle ,\\ \langle \textrm{DES}, [: \textrm{warm}(\R{y})]\rangle \}\end{multline}}\]
[13b]: external anchor: \(\cb{\{\langle \R{x},c\rangle \}}\)

Compartmentalized mental state representations with entityrepresentations like in[13] have been turned into semantic theories of attitude ascriptions andused to solve a variety of familiar philosophical puzzles, such asthose involving Kripke’s (1979) Pierre who believes Londres ispretty but London is not (i.e., puzzles involving “doublevision”, see Asher 1986; Kamp 2015), Geach’s (1967) Hobwho believes a witch killed his cow while Nob believes she, the samewitch, blighted his mare (i.e., puzzles of intentional identity, seeAsher 1987), and Karttunen’s (1973) examples of presuppositionstriggered under non-doxastic attitude verbs but filtered out bydoxastic ones (i.e., puzzles involving parasitic attitudes, see Maier2015). Other applications of what Kamp calls Mental State DRT (MSDRT)include semantic analyses of specific indefinites (Kamp &Bende-Farkas 2019) and reference to fictional entities (Maier 2017;Kamp 2021). All of the semantic analyses listed above exploit the dualnature of DRT, i.e., they use both the representational level(featuring (in)accessible discourse referents, embedded DRS’s,presupposition accommodation, etc.) and the model-theoretic semantics(featuring entities, possible worlds, contexts etc.)

6. Discourse relations in SDRT

We’ve seen insection 3 andsection 4 above how anaphora resolution is constrained by accessibility. Butthere are many other factors that play a role in the resolutionprocess, from local, grammatical factors like subject antecedentsbeing generally more prominent than object antecedents in pronounresolution, to more global, discourse structural factors. Focusing onthe latter, Hobbs (1979) argues that the interpreter’s desire tocreate a maximally coherent interpretation will often play a decisiverole in choosing between multiple accessible antecedents for apronoun, 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 thecombination soon.

In[1a] the pronoun binds to John, because that delivers the most coherentinterpretation of the two-sentence discourse as a whole. In[1b], by contrast, the pronoun is bound to Bill. This differencecorresponds to a difference in the kind of coherence relationsexemplified by the two discourses: [1a] exemplifies a coherencerelation known as Elaboration, while [1b] is an instance of Result.That discourse relations play an important role in discourseinterpretation 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 thanwhat you can get in France, which shows to what lengths we go toaccommodate 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 andthe inference of coherence relations between discourse units areclosely related processes as the starting point for their SegmentedDiscourse Representation Theory (SDRT), combining DRT with insightsfrom discourse coherence theories like Hobbs’s. SDRT assumes aset of discourse relations (Narration, Result, Contrast, Explanation,Elaboration, Correction, to name a few) that we may infer to connectthe propositions expressed by the individual sentences (or clauses),in order to create a maximally coherent interpretation of thediscourse as a whole. Different types of discourse relations may leadto discourse structural configurations that impose differentconstraints on anaphora resolution over and above DRT’saccessibility. Take the example in[3] (Asher & Lascarides 2003).

[3a]: John had a great evening last night.
[3b]: He had a great meal.
[3c]: He ate salmon.
[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 explainsthis observation in terms of a discourse structural constraint knownas the Right Frontier Constraint (Polanyi 1988). This constraint isbest illustrated by drawing a discourse structure graph like the onebelow. A horizontal arrow in this graph indicates a coordinatingdiscourse relation (where neither relatum dominates the other), inthis case Narration. A vertical arrow indicates a subordinatingdiscourse relation (where a nucleus dominates a satellite), in thiscase Elaboration.

a diagram: link to extended description below

Figure 1 [Anextended description of figure 1 is in the supplement.]

SDRT’s version of the Right Frontier Constraint says that thepronoun in[3f] can only find an antecedent introduced in the node that was justintroduced ([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 toa DRS. A common SDRS notation extends the familiar box notation of DRT(seesection 4) with labels for elementary and complex discourse units:

a SDRS notation diagram: link to extended description below

Figure 2 [Anextended description of figure 2 is in the supplement.]

Anaphora and presupposition resolution rely on a modified notion ofaccessibility that incorporates the Right Frontier Constraint (Polanyi1988). The model-theoretic interpretation of an SDRS is built around aset of special interpretation rules for the various discourserelations. Narration for instance comes with a semantic interpretationrule that states that the both relata hold and moreover the mainevents they introduce stand in a relation of temporal progression.Some notational shortcuts: \(K{\pi}\) is the DRS associated with label\(\pi\) (i.e. the \(K\) such that \(\pi : K\) occurs somewhere in theSDRS); \(e_{\pi}\) is the unique event discourse referent that occursin \(U(K_{\pi})\) (for readability we left out event referents fromthe example SDRS above, but for the purpose of modelling narrativetemporal progression in SDRT we should treat every verb phrase asintroducing an event discourse referent).

\(f\) verifies \(\cb{\textrm{Narration}(\pi_{1},\pi_{2})}\) iff thereis an embedding function \(g\) such that \(f[K_{\cb{\pi_1}}]g\) and\(g\) verifies \(K_{\cb{\pi_1}},\) and there is an embedding \(h\)such that \(g[K_{\cb{\pi_2}}]h\) and \(h\) verifies \(K_{\cb{\pi_2}}\)and moreover \(h(e_{\cb{\pi_1}}) \rar h(e_{\cb{\pi_2}})\)

Although we’re treating SDRT here as an extension of DRT, thetheory of discourse structure it provides is entirely compatible withother forms of dynamic semantics providing the contents of theelementary discourse units. In fact, SDRT does not even require thoseunits to be linguistic in nature—as long as we’reinterpreting sequences of discourse units that express propositionalcontents and introduce discourse referents for events and individuals.The framework has thus proved useful for studying multimodal discourselike speech-plus-gesture (Lascarides &Stone 2009), text-plus-emoji(Maier 2023), or even primarily visual narratives like comics and film(Bateman &Wildfeuer 2014).

7. Representationalism and compositionality

DRT is a representational theory of interpretation. In the DRTframework, sentences have meanings, if at all, only in a derivativesense: a sentence prompts the hearer to modify his DRS, and it is DRSsthat have a truth-conditional interpretation. Thus, a sentence causesa DRS \(K_{i}\) to be transformed into a DRS \(K_{i+1}\) (usually,though not invariably, the process will consist in adding informationto\(K_{i}\)), and at a remove there is anassociated transition at the semantical level, from \(K_{i}\)’sinterpretation to \(K_{i+1}\)’s. Hence, if we could cut out therepresentational middleman, we would have a theory that definessentence meaning in dynamic terms, as a transition from one semanticobject to another. Thus arises the question of whether or not therepresentational level can be dispensed with. Kamp (1981a) claimedthat representations were essential to hisur-versionof DRT, which was a subset of our basic DRT, but in the meantime hewas proved wrong by a slew of dynamic theories that recast classicalDRT in purely semantical terms: Barwise (1987), Rooth (1987),Groenendijk and Stokhof (1990, 1991), Muskens (1996), and many more.However, Kamp’s claim still holds for various extensions ofclassical DRT, like the theories of presupposition and attitudeascription discussed in the foregoing.

These theories are allegedly “non-representational”, butwhat exactly does that mean? Consider, for example, Groenendijk andStokhof’s (1991) Dynamic Predicate Logic (DPL), which provides adynamic semantics for the language of classical predicate logic, inwhich “Pedro owns a donkey and beats it” may be renderedas follows:

[1]: \(\exists x [\textrm{donkey}(x) \amp\textrm{own}(\textrm{Pedro},x)] \amp\textrm{beat}(\textrm{Pedro},x)\)

DPL’s semantics ensures that, unlike in classical logic, thelast occurrence of \(x\) is bound by the existential quantifier. Theway this works is that DPL analyzes sentence meanings in terms ofvariable assignments, which are manipulated by the logical constantsin such as way that[1] is true in a model \(M\) iff \(M\)’s domain features anindividual which, when assigned to the variable \(x,\) makes each ofthe atomic sentences in [1] come out true in \(M\). Formally, this isimplemented by stipulating that the meaning of a sentence is arelation between variable assignments.

A variable assignment is a mapping from variables to individuals, andvariables are linguistic expressions, so it can hardly be claimed thatassignments are non-representational entities. Therefore, it is a bitpuzzling that, in contrast to DRT, DPL is widely seen as anon-representational variety of dynamic semantics. True, DRT’srepresentationalism goes quite a bit further than DPL’s, and thetwo theories are obviously very different in this respect, but boththeories import representational elements into semantics, and thisholds for dynamic theories of semantics generally (though see Dekker2012 for exceptions).

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

One way in which DRT’s representationalism manifests itself asbeing more radical than other theories’ is that it is anon-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 linkedto another discourse referent. This is a non-deterministic process,but even if it were not, it is clear that the anaphoric link is notpart of the meaning of the pronoun. A standard statement ofcompositionality would say that the meaning of a compound expressionmust be a function of the meaning of their components and their modeof combination. But if some of the components, like pronouns, do notintroduce into the DRS any object that can naturally be described asthe meaning of that object, then it is clear that we do not have acompositional system.

How bad is it that we lose compositionality? It is generally agreedthat some of the standard arguments for compositionality are notparticularly compelling. In particular, we don’t needcompositionality to explain why people can understand an indefinitenumber of sentences, and we don’t need it for explaining howlanguages can be learned. To explain such facts, it suffices thatthere is some procedure for establishing the meanings of sentences,but it is not strictly necessary for this procedure to becompositional. Compositionality is often viewed as a methodologicalprinciple (Groenendijk & Stokhof 1990, 1991; Janssen 1997; Dever1999), since any model of interpretation can be made compositional ifwe are sufficiently relaxed about the nature of lexical meaningsand/or the syntactic structures over which the compositional theory isdefined. In the case of DRT, the standard approaches to making thesystem compositional involve dispensing with Kamp’s constructionalgorithm, and replacing it with a system in which DRSs can becomposed either through suitable merge operations (Zeevat 1989b), or,in dynamic semantic systems like those of Groenendijk and Stokhof(1990) and Muskens (1996), through use of function application in alambda calculus. But while most such systems capture meanings forsentences that are closely related to those of original DRT, they donot capture every aspect of DRT. In particular, whereas anaphoraresolution is an integral part of DRT, in compositional theoriesanaphora resolution is assumed to be performed by a separate componentof the theory.

Representationalism and compositionality are issues that have producedgreat disagreement amongst semanticists and philosophers of languagein the 1980s. The methodological view of compositionality has reducedwhat appeared once to be a substantive argument to a matter of taste,and as regards DRT it has been shown that researchers who care aboutcompositionality are free to adopt a variant of the original systemwith many of its features, but none of its non-compositionality. Butrepresentationalism may be a more serious issue, for if it is true, aswe have suggested, that dynamic theories generally rely on a notion ofmeaning that contains at least some form of representation, then thatnotion could do with a bit more philosophical attention than it hasreceived so far.

8. Further reading

General introductions: Kamp and Reyle (1993),Kamp (1995), Blackburn and Bos (2005), van Eijck (2006), van Eijck andKamp (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 (2001a), Kampand Roßdeutscher (1994), Krahmer (1998).
Tense and aspect: Kamp and Reyle (1993), Kamp etal. (2011), Partee (1984).
Quantification and plurality: Kamp and Reyle(1993).
Attitude reports: Asher (1986, 1989), Zeevat(1996), Kamp (1990), Geurts (1999), Maier (2009b).
Discourse structure and SDRT: Asher (1993), Asherand Lascarides (2003), van Leusen (2007).
Inference systems for DRT: Kamp and Reyle (1996),Saurer (1993).

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