Monotonic Disambiguation and Plural Pronoun Resolution

U. Reyle

Institute for Computational Linguistics University of Stuttgart Azenbergstr. 12, 70174 Stuttgart

Abstract

The paper adresses the problem of representing ambiguities created by plural NPs in an underspecified way. It deals with the collective/distributive distinction, and also with generic and cumulative readings. In addition it provides a systematic account for an underspecified treatment of plural pronoun resolution.

1 Introduction

Recent work in computational and theoretical linguistics has made use of levels of semantic representation in which some aspects of the meaning of sentences or texts are left underspecified. Prominent among these aspects is the determination of scoping relations of quantifiers and the disambiguation of lexical ambiguities. The increasing interest in underspecified representations is motivated by the need to avoid computational explosion and by the desire to come to a theory of meaning that models human natural language understanding more adequately. Given the computational advantages it is surprising (i) that only very little work adresses the problem of ambiguities created by plural NPs^{1} and (ii) that there is no systematic account to deal with the aspect of anaphora resolution within such an underspecified framework. The present paper is a first attempt to fill this gap.

Whenever there are many possible interpretations of a single sentence there is a choice of approaches to semantic interpretation. Formal semantics was content with rigorously specifying the set of different readings of a sentence. Any practical system that is build according to the strategy to build up the set of possible representations of a sentence and let the pragmatics deal with the task of disambiguation in context will face a combinatorial explosion. As the disambiguation task can be seen as core problem of computational semantics, people working in this area have favored a different strategy. A single analysis (yielding an underspecified representation) of the sentence is postulated that determines all of its interpretations. Approaches that follow the latter strategy can be classified according to the way they relate the underspecified representation to its disambiguations. In most of them disambiguation proceeds through application of general rules that involve a destructive manipulation of the underspecified semantic representation. Only recently more stress has been put on the following monotonicity requirement. ([1], [4])

Monotonicity: Any additional information to an underspecified representation may reduce the set of its readings; and, in order to reduce the set of readings of an underspecified representation it is sufficient to add information.

Monotonicity guarantees that the transition from an underspecified representation r_{1} to a less underspecified (or even fully specified) representation r_{2} is achieved only by adding information. In [1] and [4] this requirement is met for the resolution of scope ambiguities and lexical disambiguation. The problems, however, that anaphora resolution imposes to the monotonic approaches have been ignored in the literature so far. Consider (1).

(1) Dov gave three roses to his secretaries. They liked them.

1We will not only adress the collective/distributive distinction, but also generic and cumulative readings.

Without further information the interpreter of (1) is not able to choose from among (1)’s (disambiguated) meanings. He does not know how many roses Dov bought, which roses are liked by which secretary, etc. Nevertheless he is able to establish the relation that links the pronoun they to the NP his secretaries and them to three roses and that remains invariant under the disambiguation process. Suppose we had an underspecified representation of (the set of meanings of) (1), then we must also have means to represent these links without being forced to make any decisions about collective and/or distributive readings of the antecedent phrase. This leads to the following requirement.

Linking: Identification of pronoun-antecedent pairs must be represented in an underspecified way.

Of course the relationship between coindexing and coreference should obey the monotonicity requirement. I.e. we must provide some means to specify the impact of possible disambiguations of the antecedent phrase to the set of readings of the phrase containing the pronoun. These means must, for example, explain why the pronoun they in (2) can be interpreted as semantically singular only if the antecedent NP is not read collectively.

(2) The lawyers hired a secretary they liked.

In this case the pronouns plurality is licensed, or even required by the plurality of its grammatical antecedent. they is semantically interpreted as singular and syntactically as plural.

In the next section we briefly show how scope ambiguities of quantifiers are dealt with in the theory of UDRSs. Section 3 will then show how this theory can be extended to deal with collective/distributive ambiguities of plural NPs as well as with generic and cumulative readings they may license. Section 4 deals with the problem to link a plural pronoun to its antecedent phrase such that the property of monotonic disambiguation is preserved. Section 5 mentions two further aspects that relate to the topic of this paper. On the one hand it deals with anaphoric relationships between a plural pronoun and a group of entities that is built by abstraction, and on the other hand it adresses the problem of consequence relations that hold between underspecified representations.

2 A Short Introduction to UDRS’s

The base for unscoped representations I proposed in [4] is the separation of information about the structure of a particular semantic form and of the content of the information bits the semantic form combines. In case the semantic form is given by a DRS its structure is given by the hierarchy of subDRSs, that is determined by ⇒, ¬, ∨ and ◇. We will represent this hierarchy explicitly by the subordination relation ≤. The semantic content of a DRS consists of the set of its discourse referents and its conditions.

Let us consider the DRSs (4) and (5) representing the two readings of (3).

(3) Everybody didn’t pay attention.

(4)  x ⇒ ¬ x pay attention

(5) ¬ x ⇒ x pay attention

The following representations make the distinction between structure and content more explicit. The subordination relation ≤ is read from bottom to top.

x ⇒

¬

x pay attention

¬

x ⇒

x pay attention (6)

Having achieved this separation we are able to represent the structure that is common to both, (4) and (5), by (7).

x ⇒  ¬

x pay attention (7)

(7) is already the UDRS that represents (3) with scope relationships left unresolved.

To be more precise, we express the structural information by a language with one predicate ≤ that relates individual constants l, called labels. The constants are names for DRS’s. ≤ corresponds to the subordination relation between them, i.e. the set of labels with ≤ is a upper semilattice with one-element (denoted by l_{⊤}). The constants are also used to position DRSconditions at the right place in the hierarchy. This is done by writing l:γ for an occurrence of a DRS-condition γ in a DRS named l. Thus the DRS (4) of (3) is represented by (8).

l_{⊤}:l_{11}⇒l_{12}

l_{11}:x  l_{2} ≤ l_{12}
l_{2}:¬l_{21}  l_{3} ≤ l_{21}

l_{3}:x pay attention (8)

(8) lists only the subordination relations that are neither implicitly contained in the partial order nor determined by complex UDRS-conditions. This means that (8) implicitly contains the information that, e.g., l_{21} ≤ l_{⊤}, and also that l_{21} ≤ l_{2}, l_{11} ≤ l_{⊤} and l_{12} ≤ l_{⊤}. To increase readability we will abbreviate the information in (8) by structures like (9), in which we will annotate DRSs with labels only if needed.

l_{⊤}: x ⇒

l_{2}: ¬

l_{3}: x pay attention (9)

The underspecified representation of the two readings of (3) is given by (10),

l_{1}:l_{11}⇒l_{12}  l_{1} ≤ l_{⊤}
l_{11}:x
l_{2}:¬l_{21}  l_{2} ≤ l_{⊤}
l_{3}:x pay attention l_{3} ≤ l_{21} l_{3} ≤ l_{12}

(10)

which is – if we do without annotating labels – the description of (7).

The construction of underspecified representations proceeds according to general principles such as the following.^{2}

(i) Suppose l_{i} is the label of the subDRS containing the verb and l_{j} the label of one of its arguments. Then a general principle ensures that the verb is in the scope of each of its arguments, simply by stipulating l_{i}≤scope(l_{j}). scope is a (partial) function defined on the set of labels of a given UDRS K. It associates with each node a scope of this node.

(a) scope(l) = l_{1}, if {l:¬(l_{1})} occurs in K.

(b) scope(l) = l_{2}, if {l:⇒(l_{1},l_{2})} occurs in K.

2An HPSG grammar for a fragment of German that realises these principles is presented in [2]. It deals with quantifier and scope ambiguities triggered by scrambling and/or movement.

(c) scope(l) = l, if no condition of the form l:X occurs in K, where X is a plural discourse referent.

According to condition (c) scope may not be defined only if we deal with plurals. We will come back to this in the next section.

(ii) Let l_{i} be the label of a generalised quantifier and l_{j} the label associated with the top of the clause in which the quantified NP occurs. Then conditions of the form l_{i}≤l_{j} ensure that the scope of the quantifier is clause bounded.

(iii) The scope potential of an indefinite description, labelled l_{i}, is not clause bounded. The fact that it may take arbitrarily wide scope, i.e. may get a more and more specific reading, is already captured by the condition introduced by (i). Of course it cannot exceed the toplevel DRS, i.e. l_{i}≤l_{⊤}.

(iv) Proper names, π, always end up in the top-level DRS, l_{⊤}. This is marked in the lexicon by l_{⊤}:π.

(v) If for some reason the subject, l_{i}, is to have scope over some other DRS, l_{j}, then this is achieved by adding the condition l_{j} ≤scope(l_{i}).

It is clear that the disambiguation of UDRSs obeys the monotonicity requirement. If we add l_{2}≤l_{12} to (10) we get a representation equivalent to (8). There is thus no need to restructure (parts of) a semantic representation if more information about scope restriction has become available. This process of enrichment is characteristic for the construction of UDRSs: Information from different sources (syntactic and semantic knowledge as well as knowledge about the world) may be incorporated in the structure by elaborating it in the sense just described.

3 Types of Ambiguities

Plural NPs bear a high potential for creating ambiguities. As they can be understood either to denote a collection of individuals or to quantify over the members of that collection they give rise to the well-known collective/distributive ambiguity. But there are further possibilities to interpret sentences with plural NPs. (11.a) and (11.b) are examples of so-called generic and shared responsibility readings, respectively.

a. The children in this city thrive.

b. The guys in 5b have been cheating on the exam again.

(11)

These readings differ from the distributive reading in that they can be accepted as true even if not all members of the set denoted by the subject NP are in the extension of the predicate expressed by the VP. To see that they differ from the collective reading for a similar reason consider (12).

(12) The girls gathered in the garden.

(12) has only a collective reading. It is true only if each of the girls goes to the garden with the intention to meet the others. The means that a predicate P is true of a group X, if every member of X contributes in some way or other to the fact that P is true of X. In (12) the contribution is the same for each girl and consists of having the property of intentionally going to the garden to meet the others. The generic and shared responsibility readings of (11) differ from the collective readings because they can be accepted as true even if not all members of the set denoted by the subject NP are in the extension of predicates that stand in such a relation to the VP. To specify the relevant relations is task of the lexical theory (and the specification of world knowledge).

Let γ be a verb with n arguments. We distinguish collective and distributive uses of γ by an n-place array taking values d(distributive) and c(ollective). If no particular use can be determined by the context it takes the value ’ ’, and singular arguments trigger the value s(ingular). A transitive verb, for example, that is used collectively with respect to its subject position and distributively with respect to its object is represented by γ^{<c,d>}. The interpretation function gives extensions only if all arguments differ from’ ’. For intransitive verbs it is subject to the following constraints (for verbs with more arguments the generalisation is obvious):

(i) The extension of a collective use γ^{<c>} of an intransitive verb contains only groups.

(ii) The extension of a distributive use of γ^{<c>} contains only atomic entities.

(iii) Lexical rules specify the relationship between collective and distributive uses of verbs on the one hand and between the role of other predicates and the collective use on the other hand.

The transition from ’ ’ to c, or to d, is determined by the choice of reading for the sentence containing the verb.^{3} If a collective reading is chosen for some argument, then the corresponding argument position is simply marked c. The UDRS in (14) represents the collective of (13),

(13) The lawyers hired a secretary.

X the lawers(X)

y secretary(y)

hired^{<c,s>}(X,y)

(14)

In case the distributive reading is chosen then in addition to d the quantification over the individual lawyers has to be added to the subject meaning, yielding (15).

\begin{matrix}x \\ x∈X \end{matrix} ⇒

hired^{<d,s>}(x,y)

(15)

X the lawers(X)

y secretary(y)

3We are not in the position to discuss the factors that determine the choice here.

In a similar way the choice of a generic or shared responsibility reading introduces a quantificational structure. The generic reading, e.g., for (11.a) may be represented by (16), in which GEN denotes the generic quantifier.

X the children(X)

\begin{matrix}x \\ x∈X \end{matrix} [figure: x; GEN]

thrive^{<d>}(x)

(16)

This method applies also to cumulative readings which are available when a verb is accompanied with two plural NPs, as in (17).

(17) Three breweries supplied five inns.

Under the cumulative reading (17) can be accepted as true if for each of the three breweries there is at least one inn the brewery supplies, and each inn is supplied by at least one brewery. The choice of this reading brings about the transition from (18) – which we may momentarily consider as the underspecified representation of (17) – to (19).

(18)

X three breweries(X)

Y five inns(Y)

supplied^{< , >}(X,Y)

three breweries(X)

Y five inns(Y)

\begin{matrix}x \\ x∈X \end{matrix} ⇒ y y∈Y supplied^{<d,d>}(x,y)

\begin{matrix}y \\ y∈Y \end{matrix} ⇒ x x∈X supplied^{<d,d>}(x,y)

(19)

Let us note that although (14) and (19) are not ambiguous any more the choice of the distributive reading (15) for (13) has created a further ambiguity. This ambiguity is due to the fact that the node representing the subject NP has been turned into a scope-bearing node by applying distribution to the lawyers(X). Thus the indefinite can be interpreted as being within the scope of the distribution, or not. In (14) and (19) the NP-nodes are not scopebearing, and, therefore, these UDRSs are equivalent to the DRSs that result from them by taking the union/merge of all their subDRSs.

As we regarded (18) to be underspecified with respect to the choice of possible readings, we need to mark nodes to which a distribution might be applied as potentially scope bearing. As every label comes with res and scope we define:

(i) l is scope bearing if scope(l) ≠ l.

(ii) l is not scope bearing if scope(l) = res(l) = l.

(iii) otherwise l is potentially scope bearing.

The discussion up to now suggests that the underspecified representation of (17) should have the shape of (18), where in addition the argument DRSs are marked as potentially scope bearing. But there is still a complication here that has to do with the discourse referents occurring as arguments of the verb: as long as we do not know if a distributive reading will be chosen, we do not want to use the referent standing for a group to occupy this position. Nevertheless we want to indicate the map between NP meanings and argument slots of the verb. This will be achieved by the syntax/semantics interface as follows.

Following [3] we assume that lexical entries of verbs are made up of a syntactic and a semantic component. The syntactic component specifies a set of θ-roles that is associated with the set of argument phrases accompanying its use. The semantic component of a verb consists of a so-called schematic DRS. The condition set of a schematic DRS consists of the concept argument specification of the verb together with the role specifications of its (non-referential) arguments. The universe of a schematic DRS consists of schematic discourse referents, i.e. the discourse referents occurring in the condition set. The correlation between θ-roles and schematic discourse referents is achieved by the syntax-semantics interface. Consider the schematic DRS (20) for marry (we ignore temporal information):^{4}

\begin{matrix}α β \\ marry(α,β) \end{matrix} (20)

Then according to [3] (21) is the DRS representing the sentence A man married.

x β ; man(x) ; marry(x,β)

(21)

During the construction α is instantiated by the discourse referent x introduced by the subject NP, whereas β is an argument of the conceptual structure that is not realized syntactically, i.e. an implicit argument. We will adopt this instantiation mechanism of [3] for syntactically singular NPs.

4We denote schematic discourse referents by greek letters taken from the beginning of the alphabet.

For plural NPs, however, we will not simply instantiate the schematic discourse referent by the referential argument x of the NP. The only thing that the syntax-semantics interface should do here is to indicate the map between a syntactically realized NP and the corresponding argument of the verb’s conceptual structure. So we write α(X) instead of instantiating α with X. This way we do not have to decide upon any of the semantic aspects that come into play when the conceptual structure is going to be evaluated, especially the aspect of distributivity. To be more precise, we represent the sentence The men married by the UDRS (22).

X the men(X)

α(X) β

marry^{< , >}(α(X),β)

(22)

marry^{< , >}(α(X),β) is underspecified with respect to the distributive/collective ambiguity. Now, the choice of the distributive reading (i) adds the quantification over the members of the group, (ii) marks the verb with d, and (iii) contributes a condition of the form α(X)=x. In general: let the UDRS condition l_{v}:γ^{<..., ,...>}(...,α_{n}(X_{n}),...) for the verb γ be underspecified w.r.t. its n^{th} argument, and l_{a_{n}} be the label of the conditions of this argument, in particular l_{a_{n}}:X_{n}. Then the choice of a distributive reading with respect to this argument involves the following operations:

(i) Add l_{a_{n}}:l_{a_{n}1}⇒l_{a_{n}2}, with l_{a_{n}1}:x_{n}, and l_{a_{n}1}:x_{n}∈X_{n}. Furthermore the scope of l_{a_{n}} is fixed: scope(l_{a_{n}}) = l_{a_{n}2}^{5}

(ii) Instantiate l_{v}:γ^{<..., ,...>}(...,α_{n}(X_{n}),...) to l_{v}:γ^{<...,d,...>}(...,α_{n}(X_{n}),...)

(iii) Add l_{v}:α(X_{n})=x_{n}.

We show the result for the case of applying distribution to (22) in (23).

5This automatically gives us l_{v} ≤ l_{a_{n}2} by the UDRS construction priciple (i) of the previous section.

X ; the men(X) ; \begin{matrix}x \\ x∈X \end{matrix} ⇒

α(X) β
marry^{<d, >}(α(X),β)
α(X)=x

(23)

For other readings the representation of which relies on the distributive use of the verb we have similar operations. In case of the generic reading (see (16)) the condition containing the generic quantification l_{a_{n}}:l_{a_{n}1}< x_{n}, GEN >l_{a_{n}2} is added instead of l_{a_{n}}:l_{a_{n}1}:⇒l_{a_{n}2}. In case of cumulative readings (as in (19) above) the introduction of quantifying structures does not take place at l_{a_{n}} but at some newly introduced label l ≤ l_{a_{n}}.^{6} Here as well as in the case of collective readings we have scope(l_{a_{n}}) = l_{a_{n}}. In the latter case the only further condition to add is l_{v}:α(X_{n})=X_{n}.

As conditions of the form l_{v}:α(X_{n})=X_{n}, or l_{v}:α(X_{n})=x_{n} are added iff a corresponding instantiation of ’ ’ takes place the information contained in the array superscribed to γ can be dispensed with.

4 On what ’they’ might depend

Plural pronouns are plural NPs and therefore share their ambiguity potential. In addition plural pronouns may be interpreted as ’individual variables’. This is shown by (24).

(24) Few lawyers hired a secretary they liked.

The NP few lawyers in (24) binds the they in the embedded clause. Being a quantifying NP it does not introduce a plural discourse referent as possible antecedent for they and, therefore, has only one reading. This situation changes drastically if we replace the subject by a plural non-quantifying NP, as in (25).

6This will be simplified at the end of the next section.

(25) The lawyers hired a secretary they liked.

(25) has five readings. To list them it is convenient to associate a pair of indices ⟨c, d⟩ with each of the plural NPs. We call c the collective index and d the distribution index. The distribution index will only be instantiated in cases where the NP is interpreted distributively; otherwise the single integer consisting of the collective index will do.

a.∃X(lawyer∗(X)∧∃y(secretary(y)∧hire(X, y)∧like(X, y))) ; b.∃X(lawyer ∗ (X) ∧ ∃y(secretary(y) ∧ hire(X, y) ∧ ∀x(x ∈ ; X → like(x, y)))) ; c.∃X(lawyer∗(X)∀x(x ∈ X → ∃y(secretary(y)∧hire(x, y)∧ ; like(x, y)))) ; d.∃X(lawyer∗(X)∀x(x ∈ X → ∃y(secretary(y)∧hire(x, y)∧ ; ∀z(z ∈ X → like(z, y)))) ; e.∃X(lawyer∗(X)∀x(x ∈ X → ∃y(secretary(y)∧hire(X, y)∧ ; like(x, y))) (26)

In (26a) to (26d) the pronoun is bound by the group variable X irrespective of the choice whether or not to quantify over the members of this group as in (26c) and (26d). Only in (26e) this quantification binds the pronoun. It is interpreted as singular bound variable, to which no further distribution is possible.

According to what we said in the previous section the underspecified representation of this set of five readings is (27).^{7}

7We ignore the differences arising from interpreting a secretary specifically.

the lawyers(X)

y secretary(y)

\begin{matrix}ζ \\ ζ=f(X) \end{matrix}

α(X) ; α(X) hired y  \begin{matrix}β(ζ) \\ β(ζ) liked y \end{matrix}

(27)

ζ is a neutral discourse referent, i.e. ranges over singular as well as plural entities. The anaphoric linkage between plural pronouns and their antecedent NPs is expressed by an equation of the form ζ=f(X), where X is the discourse referent introduced by the antecedent NP and ζ the one introduced by the pronoun. The function f in ζ=f(X) expresses the same underdeterminacy as α(X) does, i.e. it leaves the choice to add f(X)=x, or f(X)=X.

There are several possibilities to transform (27) in a couple of (partial) disambiguation steps into one of the fully specified readings given in (26). Let us start with the verb hired and choose its distributive reading first. This gives us (28).

X the lawyers(X)

\begin{matrix}x \\ x∈X \end{matrix} ⇒

y secretary(y)

\begin{matrix}ζ \\ ζ=f(X) \end{matrix}

α(X) ; α(X) hired y ; α(X)=x  \begin{matrix}β(ζ) \\ β(ζ) liked y \end{matrix}

(28)

Here, the equation ζ=f(X) still leaves open whether ζ is to be identified with x or X; a choice that is no longer available when the collective reading is chosen, i.e. when α(X)=X is added, as in (29).

l: X the lawyers(X)

y secretary(y)

l′: \begin{matrix}ζ \\ ζ=f(X) \end{matrix}

α(X) ; α(X) hired y ; α(X)=X  \begin{matrix}β(ζ) \\ β(ζ) liked y \end{matrix}

(29)

Having chosen the collective reading, we have deprived the subject UDRS of its scope bearing potential. We assume that together with α(X)=X also scope(l) = l is added to the description of the UDRS.^{8} This blocks distribution with respect to l. Consequently, only f(X)=X can appear in l′, and the only possibility to disambiguate (29) any further is to fix the interpretation of liked. Because ζ is set equal to a group discourse referent it is possible to distribute over it. This yields (30) (= (26.b)).^{9}

X the lawyers(X)

y secretary(y)

\begin{matrix}ζ \\ ζ=f(X) \\ f(X)=X \\ \begin{matrix}z \\ z∈ζ \end{matrix} ⇒ \end{matrix}

α(X) ; α(X) hired y ; α(X)=X  \begin{matrix}β(ζ) \\ β(ζ) liked y \\ β(ζ)=z \end{matrix}

(30)

If the subject of liked is a collection, we add β(ζ)=ζ to (29) and get the

8This piece of information is not contained in the graphical representation given in (29).

9Without further syntactic information (30) allows a disambiguation according to which the indefinite a secretary is within the scope of the distribution w.r.t. the verb liked. The problem of having more syntactic information available in order to restrict the disambiguation algorithm for UDRSs appropriately also shows up in the singular fragment and is discussed in [4].

reading in (26.a).

X the lawyers(X)

y secretary(y)

\begin{matrix}ζ \\ ζ=f(X) \\ f(X)=X \end{matrix}

α(X) ; α(X) hired y ; α(X)=X  \begin{matrix}β(ζ) \\ β(ζ) liked y \\ β(ζ)=ζ \end{matrix}

(31)

The second possibility to start disambiguating (27) is to leave the options with respect to the verb hire open and fix the possible interpretations of they first, i.e. we do the same thing as above with respect to the condition ζ=f(X). There are again two possibilities. First, they is interpreted as referring to the group X of lawyers, as in (32).

X the lawyers(X)

y secretary(y)

\begin{matrix}ζ \\ ζ=f(X) \\ f(X)=X \end{matrix}

α(X) ; α(X) hired y  \begin{matrix}β(ζ) \\ β(ζ) liked y \end{matrix}

(32)

This leaves the possibility of distribution open both with respect to hired and to liked. The different choices give us representations for (26.a) to (26.d), which we will not present here. Second, if they is interpreted as dependent we must assume that hired is read distributively.

the lawyers(X)

\begin{matrix}x \\ x∈X \end{matrix} ⇒ l:

y secretary(y)

l′: \begin{matrix}ζ \\ ζ=f(X) \\ f(X)=x \end{matrix}

α(X) ; α(X) hired y ; α(X)=x  \begin{matrix}β(ζ) \\ β(ζ) liked y \end{matrix}

(33)

As (33) stands, however, it is not guaranteed that the x in f(X)=x will be properly bound in all possible disambiguations of (33). In order to ensure this we must add the condtion l′≤l. Taking this into account (33) gets the shape in (34).

X the lawyers(X)

\begin{matrix}x \\ x∈X \end{matrix} ⇒

y secretary(y)

\begin{matrix}ζ \\ ζ=f(X) \\ f(X)=x \end{matrix} α(X)

hired(α(X),y)
α(X)=x

β(ζ)
liked(β(ζ),y)

(34)

To add a condition of the form l′≤l, that puts they under the scope of the distribution corresponds to the view that plural noun phrases are ambiguous between collective/referential and quantificational construals on the basis of a distribution operator that is not overtly specified (as in the case of obligatory distribution induced by each) but is covert and optional. From the point of view we adopt in this paper, however, we prefer to characterize the step from (27) to (34) as follows. The interpretation process involves a choice to be made among different concepts, lexicalized by some verb. In complex sentences more than one verb occurs and each of them will be subject to this choice. But there is the option of either choosing a specific concept type for all of these verbs simultaneously, or of allowing a free choice to be made for each of them. In case of (34) this means that we may interpret both verbs, hire and like, in parallel as regards the choice of concept type they express. The option between a simultaneous interpretation and a free choice interpretation requires a stage of representation that is fully decided with respect to the just mentioned option but is still underspecified with respect to the actual concept to be chosen. We represent this stage of interpreting (25) before the latter choice has been made by (35).

X the lawyers(X)

y secretary(y)

α(X) ζ β(ζ)
ζ=f(X)
hired(α(X),y)
liked(β(ζ),y)
α=β=f

(35)

Let l_{mv} and l_{rv} be the labels of the main verb and the verb of the relative clause, respectively; l_{p} the label of the pronoun they. Then the operations needed to transform a structure like (33) to (35) are the following.

1. add l_{mv} = l_{rv} = l_{p}

2. add l_{p}:α=β=f

The presence of α=β=f ensures that any disambiguation of (34) imposes the collective, distributive, generic and shared responsibility reading to both verbs, hired and liked simultaneously. Choosing the distributive reading expands (35) to (36).

the lawyers(X)

\begin{matrix}x \\ x∈X \end{matrix} ⇒

y secretary(y)

α(X) ζ β(ζ)
ζ=f(X)
hired(α(X),y)
liked(β(ζ),y)
α=β=f
α(X)=x

(36)

(36) represents (26.e) and if α(X)=X is chosen we get a structure that is equivalent to (31), representing (26.a). We think that we can do without explicitly discussing the last way of starting the disambiguation of (27), namely going for liked first.

We finish this section by showing how intermediate structures like (35) can be used to analyse dependencies of they in cumulative readings. Consider the reading of (37),

Three breweries supplied five inns with which they didn’t have a contract.

(37)

according to which for each of the three breweries there is at least one inn the brewery supplies and with which it didn’t have any contract, and each inn is supplied by at least one brewery which didn’t have any contract with it. The interpretation that doesn’t assume any choice w.r.t. the readings of the plural NPs, but assumes they to be ’coreferent’ with three breweries is (38).

three breweries(X)

Y five inns(Y)

α(X) ζ β(Y) γ(ζ)

ζ=f(X)

supplied(α(X),β(Y)) have-contract(γ(ζ),β(Y)) f=α=γ

(38)

And the coice of a cumulative reading would give (39).

X three breweries(X)

Y five inns(Y)

\begin{matrix}x \\ x∈X \end{matrix} ⇒ α(X) ζ β(Y) γ(ζ) ζ=f(X) supplied(α(X),β(Y)) have-contract(γ(ζ),β(Y)) f=α=γ α(X)=x

\begin{matrix}y \\ y∈Y \end{matrix} ⇒ α(X) ζ β(Y) γ(ζ) ζ=f(X) supplied(α(X),β(Y)) have-contract(γ(ζ),β(Y)) f=α=γ β(Y)=y

(39)

Let me say a few words about the relation between our treatment of cumulative readings and a treatment that uses polyadic quantifcation directly. Let us introduce a straightforward extension of (monadic) duplex conditions to ”polyadic” ones to represent the cumulative reading of, e.g., sentence (17) as shown in (40). The restrictor of the polyadic duplex condition in (40) consists of the pair of DRSs associated with the nouns, and the diamond is not only equipped with the quantifications over the corresponding variables, but also marks the polyadic quantification to be cumulative by means of the superscript cum.

(40)  < x brewery(x) , y inn(y) > [figure: three x; five y] ^{cum} supply(x,y)

But the representation in (40) cannot be what we want. It leaves no possibility to represent that reading of the extended sentence in (37) according to which the main clause is interpreted cumulatively, but – in contrast to (39) – a collective interpretation is given to the pronoun they.^{10} To accommodate for this reading we have to represent (17) by a structure like (41) which provides an antecedent for the group reading.

X Y

brewery*(X) |X|=3 inn*(Y) |Y|=5

< \begin{matrix}x \\ x∈X \end{matrix} , \begin{matrix}y \\ y∈Y \end{matrix} > [figure: ∀ x; ∀ y] ^{cum} supply(x,y)

(41)

Note that (41) restricts the set of cumulative polyadic quantifiers to those of type < ∀, ∀ >. It also shows that (41) may be used to abbreviate (39), or we may use the device of annotating the lower box in (34) by cum to indicate that its meaning corresponds to (39), or (41). Without such an annotation (34) would still allow for, e.g., collective and distributive interpretations. Since these can be expressed as iterations of monadic quantification it is possible to mark the intended meaning by annotating pairs of c and d (standing for ’collective’ and ’distributive’, respectively).

10Within the approach presented above there is no difficulty in representing this reading if we do not commit ourselves to the simultaneous interpretation, but allow for a free choice of concept types to be chosen for the two verbs.

5 Further Perspectives

As mentioned in the introduction this paper is only a first step towards an underspecified treatment of plurals and plural pronoun resolution. Let me mention only two aspects that are under investigation and will be published in forthcoming papers.

Providing Antecedents by Abstraction

Abstraction creates plural discourse referents by building the sum of discourse referents of DRS’s that are created out of duplex conditions. This is done by, first, building the union K of the left hand and right hand side boxes of the duplex condition, and second, adding a condition of the form Ση:K to the DRS in which the duplex condition occurs. To construct the antecedent of the they in (42), for example, we abstract over the discourse referent x in the DRS of the first sentence of (42) as shown in (43).

Every teacher showed a picture to some child (in the class). They were bored.

(42)

x teacher(x) [figure: x; ∀] y z picture(y) child(z)

give(x,y,z)

Z=Σz: x y z teacher(x) picture(y) child(z) give(x,y,z)

(43)

The discourse referent Z representing the group of children that have been showed a picture may now be picked up as antecedent for the pronoun. Note, however, that (43) represents only one reading of the first sentence of (42). There is also the reading where the indefinite a picture is interpreted specifically, as shown in (44).

y Z

picture(y)

x teacher(x) [figure: x; ∀] y z child(z)

give(x,y,z)

Z=Σz: x z teacher(x) child(z) give(x,y,z)

(44)

As a picture has wide scope over the universal quantifier its content doesn’t show up in DRS K used for Abstraction. When we define the operation of Abstraction for UDRSs we have to be careful that this feature is preserved. Consider the UDRS (45).

x teacher(x) [figure: x; ∀]

y picture(y)

z child(z)

give(x,y,z)

(45)

It not only represents the two readings (43) and (44) of (42), but also two other readings which we get out of (43) and (44) when we interpret a child specifically. Of course these readings are ruled out once we apply Abstraction. Abstraction, therefore, has disambiguating force. We implement this by requiring that l<l, i.e. consider (46) instead of (45).

x teacher(x) [figure: x; ∀] l:

z child(z)

l′: y picture(y)

give(x,y,z)

(46)

As this still allows for some scope bearing element to be located between l and l′ we must require l=l′.

x teacher(x) [figure: x; ∀] y picture(y)

z child(z)

give(x,y,z)

(47)

Now we apply Abstraction with respect to z in the following way. We introduce a group discourse referent Z into the universe of the DRS containing the duplex condition.

Z

x teacher(x) [figure: x; ∀] y picture(y)

z child(z)

give(x,y,z)

(48)

The UDRS in (48) leaves open if Z is to be interpreted as in (43) or as in (44).

5.1 Consequence Relation and Proof Theory

Suppose Γ is a premiss set consisting of representations of possibly ambiguous sentences, and let γ be the representation of a possibly ambiguous conclusion to be drawn from them; in symbols Γ ⊧ γ.

In order to determine the inferential properties of such ambiguous representations we have to decide about how to relate the readings of the conclusion to the readings of the premisses? There are several options.

In [4] a proof theory is provided that is complete with respect to the consequence relation one gets if ambiguities are treated as being equivalent to the disjunctions of their readings. The following examples show, however, that this consequence relation does not match intuitions.

a. ⊢ Everybody slept or everybody didn’t sleep. b. ⊢ Somebody slept and everybody didn’t sleep.

(49)

Both sentences are intuitively contingent, but would – according to the disjunctive interpretation be classified as tautologies. The next example shows that disambiguation is not an operation δ that takes (a set of) isolated sentences. Ambiguous sentences of the same type have to be disambiguated simultaneously.

If the students get £100 then they buy books. The students get £100. ⊧ The students buy books.

(50)

The meaning of the premises of (50) is given by (51b) not by (51a), where a_{1} represents the first and a_{2} the second reading of the students get £100.

{a. ((a_{1} → b) ∨ (a_{2} → b)) ∧ (a_{1} ∨ a_{2}) \atop b. ((a_{1} → b) ∧ a_{1}) ∨ ((a_{2} → b) ∧ a_{2})} (51)

Let us call sentence representations that have to be disambiguated simultaneously correlated ambiguities. Then the correlation can be expressed by coindexing. Any disambiguation δ that simultaneously disambiguates a maximal set of representations coindexed with i is a disambiguation that respects i, in symbols δ^{i}. A disambiguation δ that respects all indices of a given set I is said to respect I , written δ^{I}. Let I be a set of indices, then the consequence relation underlying ambiguous reasoning is

∀δ^{I}(Γ^{δ^{I}} ⊧ γ^{δ^{I}}) (52)

It can be shown (see [5]) that (52) is reflexive, transitive and monotone, and that it accounts for the correct interpretation of the examples discussed above.

References

[1] Hiyan Alshawi and Richard Crouch. Monotonic semantic interpretation. In Proceedings of ACL, pages 32–39, Newark, Delaware, 1992.

[2] Anette Frank and Uwe Reyle. How to cope with scrambling and scope. In Proceedings of KONVENS 92, Nürnberg, 1992.

[3] Hans Kamp and Antje Roßdeutscher. Remarks on lexical structure, DRS-construction and lexically driven inferences. Arbeitspapier 21, Sonderforschungsbereich 340, Institut für Maschinelle Sprachverarbeitung, Stuttgart, Baden-Württemberg, 1992.

[4] Uwe Reyle. Dealing with ambiguities by underspecification: Construction, representation, and deduction. Journal of Semantics, 10(2), 1993.

[5] Uwe Reyle. Underspecified discourse representation structures and their logics. Bulletin of the IGPL, 1994.