On Dependency and Quantification in Dynamic Semantics

This paper considers how the interaction of quantification and dependent anaphora may be analysed in a dynamic semantics. It discusses a simple theory of the creation and accessibility of dependencies based on a dynamic semantics for distributivity and some basic assumptions on number agreement in discourse. This theory forms a partial defence of the line of semantics set out by van den Berg 1996. I argue, however, that it is essential to quantified anaphora to contextualise the labelling of antecedents. 1 Quantification, Accessibility and Dynamic Semantics Anaphora involving quantified statements is constrained in many ways. Given a reading of the first sentence in (1) where ‘a book by Astrid Lindgren’ receives narrow scope with respect to the subject quantifier ‘every child’, the continuation in (1-a) is infelicitous, while the continuation in (1-b) is fine. In case the object indefinite receives wide scope, the intuitions are reversed, such that the continuation with the plural pronoun becomes infelicitous and the one with the singular pronoun becomes fine. (1) Every child read a book by Astrid Lindgren. a. It was very good. b. They were very good. Intuitively, this difference may be explained by noticing that in the narrow-scope reading the indefinite object co-varies with the quantification over children and that, consequently, a plurality of books by Astrid Lindgren is described. This co-variation is absent in the wide-scope reading, and so only a single book is presented. In discourse, the plural pronoun is capable of accessing the potentially plural antecedent, while the singular pronoun may only access the semantically atomic one. One of the main goals of the programme of dynamic semantics is to explain constraints on anaphora, like those exemplified above, as resulting from the compositional interpretation process. Under the slogan ‘meaning is context-changepotential’ the classical truth-conditional meaning of a form and its ability to ? The work which lead to this paper was funded by a grant from the Niels Stensen Stichting, which I gratefully acknowledge. change the context by potentially introducing antecedents for future (pronominal) anaphora in the discourse are reconciled in a single framework. With respect to quantified statements like (1), however, it is not at all straightforward how to model the dynamic semantics. This is due to the fact that plural pronouns with quantified antecedents do not in any trivial sense co-refer with their antecedent, nor are they bound by it, as was pointed out by Evans’ famous example (2). (2) Few senators admire Kennedy; and they are very junior. [1] 6= Few senators admire Kennedy and are very junior. 6= Few senators admire Kennedy and few senators are very junior. The discourse representation theory (DRT) of [2] gives an explicit model of the interaction between quantification and anaphora. Surprisingly, the analysis of this interaction relies on the assumption that quantifiers like ‘every child’ have no dynamics at all – they are themselves not (directly) responsible for making any antecedents for future anaphora available. To account for examples like (1) and (2), DRT stipulates a set of inference principles. One of these principles gives rise to the so-called abstraction procedure. Basically, DRT allows for the abstraction of pluralities from quantified statements. If Qx(φ)(ψ) is a quantificational sentence, then the values of any variable occurring free in φ or ψ (including x) may be accumulated into a new plural discourse referent. For ‘Every x(child(x))(book(y) ∧ read(x, y))’ this means that we may infer a plural antecedent containing the y’s that satisfy child(x)∧book(y)∧ read(x, y), where, as is usual in DRT, the free variables in this form are existentially closed. Further data, however, make a second inference principle necessary. In (3), the singular pronoun ‘it’ co-varies with the quantification over the subject pronoun ‘they’. (3) Every student wrote a paper and they each sent it to L&P. That is, the second sentence can only be understood to mean that each student sent the paper he or she wrote to L&P. The dependency between students and papers is preserved and accessed in discourse. If, however, as DRT has it, quantificational sentences may only play a part in anaphora after application of the abstraction procedure, then only the pluralities described by the sentence will be available, not the relation between the atomic parts of these plural antecedents. Kamp and Reyle propose that the second conjunct of (3), may be analysed using a second inference principle. Once quantification takes place over an abstracted domain of entities (basically, a plural antecedent made available by a quantificational sentence), then details of how this antecedent was constructed are accessible as well. So, say that X is the abstracted referent for the students who wrote a paper and Y for the papers written by students, then instead of the faulty analysis ‘Each x(x ∈ X)(sent to L&P(x, Y ))’, one may infer a form ‘Each x(student(x) ∧ paper(y) ∧ wrote(x, y))(sent to L&P(x, y)).’ The inference principles posited by DRT go counter the ideas of dynamic semantics in two interrelated ways: (i) they necessarily involve a level of representation and (ii) they are independent of the interpretation of quantificational expressions. One may clarify these shortcomings in terms of the notion of antecedent accessibility. Whether or not an antecedent is accessible in DRT depends not only on the result of interpreting some form, but moreover on the application of inference principles defined on representational structures. This is in sharp contrast to dynamic predicate logic (DPL, [3]). The goal of DPL was to devise a formalism which describes anaphora in model-theoretic terms. In a dynamic semantics based on DPL, the notion of the accessibility of an antecedent can be said to be completely semantic, obscuring the need of any representational level. Predicate logical forms in DPL are interpreted as relations between assignment functions. The values of the variables specified in these functions are the (potential values of) accessible antecedents. Resolved pronouns correspond to free variables, taking their value from a contextual assignment. Accessibility is thus reduced to a non-representational notion which is solely governed by semantics. DRT’s inference principles are incompatible with both DPL’s lack of a representational level and its purely semantic notion of accessibility. Within DPLstyle dynamic semantic theories, it has therefore been argued that examples like (1), (2) and (3) may be accounted for without stipulating principles on top of the anaphora mechanisms that are directly available from the semantics (see especially [4] and [5] but also, more recent, [6].) It is evident that the complexity of the semantic analysis will increase when turning to quantificational sentences. The example in (3), for one, shows that not only antecedents, but also relations between antecedents are stored in context. This undeniably calls for some sort of structured form of context (cf. [5, 4, 8]). In this paper, I will review some design choices of dynamic analyses of quantification and argue for a semantics which is minimal in the sense that only one type of operation is responsible for introducing and accessing dependencies (roughly along the line of the work of van den Berg [4]). The van den Bergian notion of context as a set of partial assignment functions offers a direct and simple explanation of the distribution of dependent pronouns. However, I will argue that the interaction of anaphora with quantification (and especially with distributive predication) calls for a careful handling of variable names. I therefore present an incremental dynamic analysis of distributivity and dependency. The structure of the paper is as follows. In section 2, I will introduce some assumptions on how number interacts with anaphora and review the accounts of dependency of [9] and [4]. Then, in section 3, an alternative to van den Berg’s semantics will be presented within the framework of incremental dynamics [10]. Section 4 presents a application of the proposal to the problematic phenomenon called telescoping. 1 A different alternative is presented in [7], where an abstraction procedure is made an explicit part of the (dynamic) semantics of quantificational sentences. 2 Dependency and Distributivity In what follows I will make the following basic assumption concerning the role of number agreement in the process of pronominal anaphora resolution in discourse. (4) a. Singular pronouns take antecedents which are both syntactically and semantically singular. b. Plural pronouns take antecedents which are syntactically or semantically plural (or both). Whereas singular pronouns are strict about the number restrictions they impose on their antecedent, requiring both semantic and syntactic agreement, plural pronouns may take all remaining kinds of antecedents. The examples in (5) support this assumption. (5) antecedent: sem syn Most boys think { they are *he is wise. sg pl Most boys wrote a paper. { They weren’t *It wasn’t very well written. pl sg Most boys wrote a paper. { They *He read a book as well. pl pl Every boy thinks { *they are he is wise. sg sg Since dynamic predicate logic has no way of representing pronouns – it is capable only of representing resolved utterances – the goal will be to devise a semantics which covers the attested accessibility patterns bearing in mind the above assumptions on agreement. So, ‘∀x(φ)(ψ)’ should not output a state in which x yields a singular value, since the corresponding ‘Every N VP’ does not allow for subsequent singular pronouns to pick up the antecedent introduced by the subject quantifier. We will also use the agreement assumptions to make generalisations concerning dependent readings. Consider the following contrasts: (6) Three students each wrote a paper. a. #It wasn’t very well-written. b. They weren’t very well-written. 2 In fact, it seems as if plural pronouns might actually be more loosely constrained then exposed here. One example is generally referred to as gender bias, where the gender-neutral plural pronoun is preferred over use of a singular pronoun which would enforce an explicit choice of a gender feature. (For instance, ‘Every passenger is responsible for their luggage.’) Other examples exist as well. In general, plural pronouns may be thought of as severely under-constrained with respect to the antecedent they take (see [11] for discussion). What is important for what follows is the strict requirement of both semantic and syntactic agreement for singular pronouns. We can safely ignore extra-ordinary cases of plural pronoun use. c. #Together, they sent it to L&P. d. They each sent it to L&P. We relate the fact that the continuation in (6-a) is infelicitous while the one in (6-b) is fine to the observation that (distributively) quantified indefinites yield semantically plural antecedents. Similarly, from (6-c) we may conclude that collectivity does not affect the plurality of this antecedent. The continuation in (6-d), however, strongly suggests that distributive quantification, however, makes accessible the atomic (singular) parts of the plurality in question. Apart from finding a semantics of (distributive) quantification which accounts for the introduction of plural antecedents, the goal is now to find a semantics of distributive quantification which entails the accessibility contrast in (6-c) and (6-d). The relationship between the semantic number of an antecedent and the mode of predication it occurs in is intuitively clear. Consider (7). (7) Three students wrote a paper. a. It wasn’t very well written. b. They weren’t very well written. Distributivity brings about a co-variation of the values for the object indefinite with the values in the restriction predicate. The result is a plurality of atomic values for the object indefinite each of which is related to an atomic part of the plurality of values brought about by the subject. In case (7) is understood collectively, no co-variation takes place. The result is a single value for the indefinite object. (Compare with (1).) The mode of predication in a sentence is determined by a range of semantic and pragmatics factors. For instance, certain predicates may only take collections (e.g. ‘be a good team’), while certain quantifiers enforce distributive quantification (hence ‘*Most students form a good team’). In their account of plurality in discourse, Asher and Wang [9] propose that the mode of predication should be represented as a type of transition, where a collective transition passes on values in the form of collections and a distributive transition passes on values ‘one at a time.’ Moreover, a special kind of transition is used for the storage of dependencies in discourse. So, φ;dep(x,y) ψ expresses that respective atomic values for x and the corresponding depending values for y (as given by φ) are considered one at a time for the interpretation of ψ. Asher and Wang’s proposal is based on the argumentation that the two traditional lines of analysis of the distributivity/collectivity distinction cannot be maintained. The so-called term-ambiguity approach where the term predicated over is ambiguous is untenable since the same term may combine with a coordinated predication consisting of both a distributive and a collective part. Moreover, terms occurring in a distributive sentence, may be picked up anaphorically and subsequently used collectively. If terms are really ambiguous, then such cases are hard to explain. The second approach Asher and Wang argue against, dubbed the predicate ambiguity approach, claims that it is the predication rather than the term which is ambiguous. This approach comprises analyses which make use of a distributivity operator (as in [4] and the analysis to follow.) Asher and Wang argue against such proposals on the basis of the persistence of modes of predication in discourse. In their own proposal, the mode of predication becomes a discourse-level interpretation mode. They argue that such an analysis is desired since distributivity, collectivity, dependency and cumulativity tend to persist in discourse. In a predicate ambiguity approach, the argument goes, this persistence is hard to explain. While I acknowledge the existence of a tendency towards persistence of modes of predication in discourse and appreciate the power of Asher and Wang’s solution, I doubt whether it is actually an argument against the predicate ambiguity line of thinking. All that is wrong is that the persistence of a mode of predication does not follow from the semantics. That is, the predicate ambiguity approach needs an extra-semantic explanation about why distributive sentences seem to come in pairs. At the same time, however, Asher and Wang’s approach needs an extra-semantic explanation of why discourse sometimes breaks with the persistence tendency. In fact, I believe there are reasons why, semantically, the persistence of a mode of predication is undesirable. We have seen from the examples in (7) that the mode of predication influences the way antecedents are introduced in discourse. For instance, dependencies may be seen as a side-effect of distributive predication. It has often been overlooked, however, that the mode of predication in which a pronoun is involved does not determine the pronoun’s choice of antecedent. In (8), the pronoun ‘they’ is embedded in the explicitly distributive predication. Still, its antecedent is the collection John and Mary, not the varying atoms of that pair. (8) John and Mary both think they make a great couple. Cases comparable to (8) occur in discourse as well. Notice the contrast between (9-a) and (9-b). (9) a. Three students each wrote a paper. They then each sent it to a (different) journal. b. Three students each wrote a paper. They then each sent them to a (different) journal. In both examples, both sentences are made explicitly distributive by using the distributor ‘each’. The singular pronoun in (9-a) enforces the dependent reading. The example in (9-b), however, has a reading were each of the three paper-writing students sent (copies of) the three written papers to a journal. (So, three packages containing the same three papers reach three (different) editors.) This illustrates that the distributive reading of the sentence containing the pronoun does not enforce the pronoun to be interpreted in a dependent way. The kind of antecedent a pronoun in a distributive sentence takes is decided on by the resolution process. From a semantic perspective both collective and dependent antecedents are available. In Asher and Wang’s proposal, however, the dependent transport of values in discourse (as caused by the ‘dependent’ first sentence) calls 3 Compare with ‘John and Mary both think they are being cheated on.’ for a correction in the logical form once the resolution process decides to choose a collective antecedent for a pronoun, since the dependent transition renders the collective antecedent inaccessible. In what follows, I intend to show that such a representation of anaphora at the level of logical form is unnecessary. The contrast between (9-a) and (9-b) is not a counterexample against Asher and Wang’s approach using modes of transition. But it does show that the choice between dependent readings and independent readings of pronouns is merely due to anaphora resolution. The ‘mode’ of the predication containing the pronoun or the antecedent is not a sufficient condition for whatever type of anaphora. This suggests that the fact that persistence does not follow directly from a predicate ambiguity approach might be a virtue rather than a vice. In van den Berg’s dynamic analysis of plurality [4], dependency is modelled in a rather straightforward way, yielding a simple but powerful formalism to describe the interaction of quantification and dependency. Let me abstract away from the details of his analysis and present – what I feel – is the core of his proposal. Recall, first, that we concluded from (7) that distributivity is a necessary condition for both the storing and the accessing of a dependency, since without distributivity there is no source of the co-variation necessary for dependency effects. Dependency and the semantics of distributivity are therefore essentially linked in van den Berg’s semantics. Let X be a set of variables. An information state is a set of assignments defined on X. If FX is such a state, then FX(x) = {f(x)|f ∈ FX} (the collected values for x in FX) and FX |x=d = {f ∈ FX |f(x) = d} (the substate of FX where x’s value is d.) In a state FX , variable y ∈ X is dependent on x ∈ X if and only if: ∃d, e ∈ FX(x) : FX |x=d(y) 6= FX |x=e(y) Interpretation of predicate logical forms with respect to a state FX may now proceed in either of two ways. The standard way results in collective interpretations across the board. FX [P (x)]GY is true if and only if GY = FX and FX(x) ∈ I(P ). What is considered here is the collection of values for x, FX(x). Dependency information (information at the level of substates) is not accessed. Using a distributivity operator δx the distributive reading may be obtained. FX [δx(P (x))]GY is true if and only if for each atomic entity d ∈ FX(x) it holds 4 This is most clear from examples where the antecedent is plural. Consider (i): (i) Three students each wrote exactly two papers. They each sent them to L&P. The second sentence in (i) is now truly ambiguous between a resolution for the pronoun to co-vary with the distributive quantification or one wherein the pronoun takes the collection of (six) papers written. In Asher and Wang’s proposal the difference between these readings is not one between antecedent choice, but rather one between a combination of antecedent choice and a relating correction of logical form. that FX |x=d[P (x)]GY |x=d. That is, the predication P (x) is evaluated step by step with respect to substates FX |x=d. In general for forms δx(φ) this has the following consequences. In case φ contains a variable dependent on x, then the values for this variable will co-vary with those of x during interpretation, since they are interpreted with respect to FX |x=d (for some d) instead of FX . Moreover, in case φ contains the introduction of a new discourse entity, then the values for that discourse entity will co-vary with those of x in the output state. In other words, accessed variables in φ are interpreted dependently and introduced variables in φ are stored dependently in van den Berg’s proposal. With respect to our discussion above, however, this does not suffice. While we saw that the storing of entities under distributivity inevitably leads to dependency, the accessing of antecedents within the scope of a distributivity operator could either co-vary with the running variable or not. Given a state FX with a dependency of y ∈ X on x ∈ X, the independent reading of a pronoun indexed with y is simply the interpretation of y in the global state FX . The dependent reading of such a pronoun is an interpretation with respect to a varying state FX |x=d, where d ranges over the atoms in FX(x). The problem is that both the dependent reading (interpretation with respect to a global perspective on context) and the independent reading (interpretation with respect to a local perspective on context) are associated with the same label (y, in this case). We observed, however, that the resolution process has a genuine choice between the two. So, in context, both perspectives should be present. Unfortunately, there is no straightforward way of combining FX and FX |x=d into one anaphoric resource pool, especially since the latter is a (most often proper) subset of the former. What is needed is a means of contextually determining with what label a certain anaphoric option is to be associated. The next section presents an account of distributivity and dependency in a formalism which fully contextualises the index of an antecedent. 3 Incremental Dynamics of Distributive Quantification The framework of incremental dynamics (ID, [10, 12]) differs from DPL-like formalisms in assuming that the introduction of a new discourse entity results in an incremented context, whereas in DPL it results in the random assignment of a value to a pre-determined variable. A context in ID is a stack. The action ∃ represents a push operation defined on stacks. The label of the newly introduced value depends on the size of the input stack. Performed on a stack of n entities, ∃ pushes a new entity to slot n (given that the first slot is labelled 0, the second 1, etc.) In ID, the labels of antecedents are not decided upon by the logical form, but rather by the context in which interpretation takes place. As a direct result, there is a straightforward way of combining contexts, namely the append 5 Unfortunately, this is a serious oversimplification, since it puts far too few restrictions on the output state G. An extra condition stating FX(x) = GY (x) should be added (cf. (10)). See [8] for discussion. operation, without the risk of a name clash. In the light of the discussion in the last section, this means that ID is a promising formalism for the simultaneous representation of global and local perspectives on context. In [8, 13], an incremental dynamic account of distributive quantification, dependency and plurality in discourse is presented. Let s, s′, . . . range over stacks. Stacks are functions from {0, 1, 2, . . . , n} to the domain of entities De. The function {〈0, d0〉, 〈1, d1〉, 〈2, d2〉, . . .} is notated [d0, d1, d2, . . .]. The size of a stack is the cardinality of its domain. We write s∧s′ for appending stack s′ to s, i.e. s∧s′ = s∪{〈i+ |s|, d〉|s′(i) = d}. Let S, S′, . . . range over sets of stacks of the same size. We write S(i) for {s(i)|s ∈ S} and S|i=d for {s ∈ S|s(i) = d}. Two sets of stacks may be combined as follows: SuS′ = {s∧s′|s ∈ S & s′ ∈ S′}. The size of a set of stacks S, |S|, is the common size of the individual stacks. The semantics is presented as a set of functions. That is, operators, and predicates are represented as functions on indices and sets of stacks. This effectively does away with any meaningful representational level. (10) ∃ := λS.λS′.∃d ∈ De : S′ = Su{[d]} P ? := λi.λS.λS′.S′ = S & M |= P (S(i)) δ i := λP.λS.λS ′.S(i) = S′(i) & ∀d ∈ S(i) : SS|i=d ∈ P (|S|+ i)(SS|i=d) φ · ψ := λS. ∪ {ψ(S′′)|S′′ ∈ φ(S)} The functions in (10) have polymorphic types. ‘∃’ is a state transition taking a set of stacks of some size n and returning sets of stacks of size n+1. The ∃quantifier is a push operator on sets of stacks. It increments the context with a single atomic individual. A predicate-function P ? takes an index and returns a test on states with respect to P and that index. The function δ takes a predicate and an index to return a state transition. It quantifies over the set associated with the index. In the scope of a distributivity operator, the functional perspective on an input state (S|i=d) and the global perspective (S) are combined. The set of accessible labels is thus doubled, but, since the label of an entity in a stack is determined by its position, there is no clash of labels. Outside the scope of δ, a global state S′ remains. So, in a state S with some salient set of students in slot i, the following set of states represents the processing of a sentence ‘each student wrote a paper.’ (11) (δ i (λn.λT.(∃ · paper(|T |) · write(n, |T |))(T )))(S) States S′ in the set described by (11) are such that S′(i) returns the same set of salient students as S(i) did. The plurality S′(|S|) is a set of papers written by these students. Moreover, each substate S|i=d for d ∈ S(i) returns a single student d at i and a paper written by d at |S|. If we subsequently take states like S′ as the input context for the processing of a sentence ‘they each submitted PRO to L&P,’ where PRO is some third person pronoun (say, either ‘it’ or ‘them’), the resolution process has a choice between the following two sets of states (ignoring the choices for the subject pronoun). (12) a. (δ i (λn.submit to L&P (n, |S|)))(S′) b. (δ i (λn.submit to L&P (n, |S′|+ |S|))(S′) The set of states in (12-a) represents the reading where the set of papers written by students is accessed by the object pronoun (which will have to be plural in this case). The set in (12-b) collects output states for a reading where the pronoun is read dependently. In sum, the incremental dynamics frameworks open up the possibility of describing dependent and independent anaphora using information states wherein the labels for antecedents are themselves contextualised.