Presupposition Projection and Anaphora in Quantified Sentences

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1 Introduction Presupposition Projection and Anaphora in Quantified Sentences Yasutada Sudo December 17, 2012 Quantified sentences constitute a recalcitrant problem for theories of presupposition projection, and a great number of attempts have been made to explicate the projection properties of quantificational expressions (Karttunen & Peters 1979, Cooper 1983, Heim 1983, van der Sandt 1992, Beaver 1994, 2001, Fox 2008, 2012, George 2008a,b, Schlenker 2008, Charlow 2009, Chemla 2009b, Schlenker 2009, Fox 2010, Sudo, Romoli, Hackl & Fox 2011). Schematically, the main problem is how to predict the presuppositions of sentences of the form QpRqpS q from the meaning of Q and the meanings of R and S, where Q is a quantificational determiner and R and S are its restrictor and nuclear scope, that are potentially presuppositional. For the moment, I will concentrate on cases where the nuclear scope S has a presupposition and the restrictor R does not, and will discuss presupposition projection out of the restrictor in depth in 5. The main challenge comes from the fact that different quantifiers have different projection properties. Consider, for instance, the sentences in (1), where the the predicate criticized herself has a presupposition that the subject is female. (1) a. Every student criticized herself. b. No student criticized herself. c. A student criticized herself. Intuitively, one can fairly reliably infer from (1a) that every student is female. I assume that the assertive meaning of criticized herself is simply, λx. x criticized x, and the presupposition is λx. x is female (Cooper 1983, Heim & Kratzer 1998, Sudo 2012). On this assumption, the gender inference of (1a) that every student is female should be a purely presuppositional inference, rather than an entailment in the assertive meaning (see Sudo 2012 for more sophisticated arguments for this analysis, and its related theoretical implications). 1 Similarly, one can infer from (1b) that every student is female. Again, this cannot be an entailment in the assertive meaning, and should be attributed to the projection property of the quantifier no student. On the other hand, (1c) has a [acknowledgments to be added] 1 If one is skeptical about this analysis of the predicate criticized herself, and insists that the assertive meaning to be λx. x is female and criticized x, consider the negation of this predicate, didn t criticize herself. The assertive meaning will beλx. either x is male or x didn t criticize x. Now consider the sentence Every student didn t criticize herself under the reading where every student takes scope over the negation. Again, the gender inference cannot be an entailment in the assertive meaning, but nonetheless this sentence still suggests that every student is female. Thus, we arrive at the same conclusion: Every student gives rise to an inference that every student satisfies the presupposition of the predicate. 1

weaker inference that at least the student who criticized herself is female, and says nothing about other students. I will call an inference that every individual in the restrictor of the quantifier satisfies the presupposition of the nuclear scope, a universal inference. The judgments of these sentences are rather delicate, and conflicting opinions have been expressed by various authors (cf. Cooper 1983, Heim 1983, van der Sandt 1992, Beaver 1994, 2001, George 2008a,b, Charlow 2009). Recent experimental works, most notably Chemla (2009a) and Sudo, Romoli, Hackl & Fox (2011), observe that the following pattern holds at least as a general tendency: Quantifiers like none of the 5 students are likely to have a universal inference, while quantifiers like some of the 5 students are much less likely to have a universal inference. Furthermore, the latter type of quantifiers optionally give rise to a universal inference. 2 Concretely, the sentences in (2) are associated with a universal inference in the decreasing order of robustness. (2) a. None of the 5 students stopped smoking. b. Some of the 5 students stopped smoking. c. The tallest of the 5 students stopped smoking. To summarize the observations so far, universal quantifiers like every student and negative quantifiers like no student and none of the 5 students give rise to universal inferences fairly reliably, while indefinites like a student do not. Partitive existential quantifiers like some of the 5 students, on the other hand, optionally give rise to universal inferences. It is not a purpose of the present paper to justify the empirical status of these generalizations, and I take them to be the main data to be accounted for. I refer the skeptical reader to the references cited above. The main goal of this paper is to present a new theory of presupposition projection in quantificational sentences. The theory is motivated by a hitherto unnoticed parallelism between the projection properties and anaphoric properties of quantificational determiners. Specifically, I claim that a universal inference arises with quantifiers that support what is calledmaxsetanaphora across sentences, and a non-universal inference arises with quantifiers that support refset anaphora across sentences. Maxset anaphora is exemplified by (3a), where the pronoun they picks out all the individuals in the restrictor of the quantifier No students majoring in linguistics in the first sentence. Refset anaphora, on the other hand, involves reference to the intersection of the restrictor and the nuclear scope of the quantificational determiner, as illustrated by (3b). 3 (3) a. No students majoring in linguistics got an A in the mid term exam. They are all interested in phonology. (they=all the students majoring in linguistics) b. Some students majoring in linguistics got an A in the mid term exam. They are all interested in phonology. (they=the students majoring in linguistics who got an A in the mid term exam) 2 Sudo, Romoli, Hackl & Fox s (2011) results indicate that some speakers insisted on a universal inference with quantifiers like some of the 5 students even under the experimental bias against this interpretation, i.e. the participants were asked to indicate whether a non-universal inference was possible at all. Furthermore, in an ongoing work, Sudo, Romoli, Hackl and Fox show that such quantifier are more likely to be associated with a universal inference in comparison to singular definite descriptions like the tallest of the 5 students. 3 This might not be the only reading of (3b), but at least it is a possible reading. Such an ambiguity seem to obtain more robustly with partitive quantifiers like some of the students majoring in linguistics. This ambiguity plays a crucial role in accounting for the projection properties of partitive existential quantifiers, as we will see in later sections. 2

The parallelism between presupposition projection and cross-sentential anaphora is schematically summarized as follows. (4) Let QpRq be a quantifier with a restrictor R. Consider a discourse of the form: QpRqpFq. pro G. for some predicates F and G. pro is some pronoun with appropriate phi-features, e.g. they. a. If pro can be resolved to R (i.e. QpRq supports maxset anaphora), then sentences of the form QpRqpλx.S pxq ppxq q can have a universal inference that @x P R : ppxq; and b. If pro can be resolved to R X F (i.e. QpRq supports refset anaphora), then sentences of the form QpRqpλx.S pxq ppxq q can have an inference weaker than a universal inference. Combined with the earlier observations about presupposition projection in quantified sentences, we expect the following correspondence. (5) a. Every student, no student, and none of the students only support maxset anaphora. 4 Consequently they always give rise to a universal inference in presuppositional sentences. b. A student only supports refset anaphora. Consequently it always gives rise an inference weaker than a universal inference in presuppositional sentences. c. Some of the students supports both maxset anaphora and refset anaphora. Consequently it optionally gives rise to a universal inference. As I will show in greater detail later in this paper, these predictions are borne out. Adopting a two-dimensional theory of presupposition (Karttunen & Peters 1979; see Sudo 2012 for empirical motivation for a two-dimensional theory over a partial function or trivalent theory), I will offer a theory that directly captures the correlation between the presupposition projection and anaphoric properties of quantifiers summarized in (4). The underlying idea is that the assertion and presupposition of a quantificational sentence stand in a relation similar to two independent sentences occurring in a discourse like (3). It will be demonstrated that the proposed theory derives different projection properties of different quantifiers from essentially the same presupposition and their independently motivated anaphoric properties. Furthermore, it will be shown that the same idea can be straightforwardly extended to presupposition projection from the restrictor argument. The paper is organized as follows. In 2, I first introduce the core ideas of the proposal in a simplified form. Then I will closely examine the proposed correlation between presupposition projection and cross-sentential anaphora with concrete examples for a number of quantifiers in 3. In 4, the theory introduced in 2 will be extended to account for generalized quantifiers, and in 5, the resulting theory will be shown to capture presupposition projection out of the restrictor without further ado. Finally 6 summarizes the discussion, and addresses further issues. 4 In the case of universal quantifiers, the maxset reading is equivalent to the refset reading, i.e. everyprqps q asserts that R pr X S q. Therefore, strictly speaking, it is not possible to know whether they support maxset anaphora or refset anaphora, or both. This is not a problem for the analysis put forward here, since the prediction depends on the referent of the pronoun, rather than the type of the anaphora, as we will see. For the sake of brevity, I use refset anaphora to mean anaphora to a set strictly smaller than the maxset. 3

2 The Idea in a Simplified Form Before delving directly into the empirical aspect of the proposed parallelism between presupposition projection and cross-sentential anaphora, I will first introduce the formal theory put forward here in a simplified form using a simple sentence. The main purpose of this section is, therefore, to present the core idea in an accessible way, and the proposed formal theory will be enriched to deal with more complicated cases in 4. Firstly, I assume a two-dimensional theory of presupposition where the assertive meaning and presupposition of a sentence are treated as independent propositions (Karttunen & Peters 1979). This is a rather crucial assumption, but it is beyond the scope of this paper to present a full justification (see Dekker 2008, van Rooij 2005 and Sudo 2012 for discussion; see also Chemla 2009b for another multi-dimensional theory of presupposition). (6) is a simple example where the presupposition trigger is herself that presupposes that the referent is female. (6) Jesse criticized herself. a. Assertivemeaning: Jesse criticized Jesse. b. Presupposition: Jesse is female. We are not concerned with the compositional derivation of atomic sentences in the present section (see 4.3 below). Adopting this framework, I claim that a quantifier introduces an anaphoric term in the presupposition that is anchored to the quantifier in the assertion. This is illustrated by the example in (7). (7) A student criticized herself. a. Assertivemeaning: There is a student x such that x criticized x. b. Presupposition: x is female Here, the existential quantifier a student introduces an existential quantifier binding a variable x in the assertion, and the same variable x is reused in the presupposition. That the same variable is used is crucial here, and I will show shortly how the occurrences of x in the assertion and presuppositions are ensured to co-vary, i.e. to be bound by the same quantifier. To informally state the underlying intuition, the assertive meaning says There is a student x who criticized x, and the presupposition says That same x is female. In other words, the two dimensions of meaning are treated as a small discourse involving a pronoun anchored to a quantificational antecedent. 5 Notice importantly that the resulting presupposition is non-universal in the sense that only the student introduced in the assertion is presupposed to be female, rather than every student is. As 5 Thus, I am offering a solution to the binding problem of a two-dimensional theory (Karttunen & Peters 1979, Cooper 1983, Beaver 2001). Karttunen & Peters (1979) pointed out that if the presupposition of a sentence like (7) is analyzed as involving existential quantification as well, the resulting presupposition is going to be too weak. That is, (7) is predicted to be felicitous whenever there is a female student, and to be true when any student criticized themselves, regardless of their gender. In other words, such an analysis will fail to capture that the two dimensions of meaning are about the same individual. This was once considered to be a fatal problem for multi-dimensional theories of presupposition (Karttunen & Peters 1979, Cooper 1983), but as I demonstrate below, this conclusion is not warranted. Dekker (2008) and van Rooij (2005) also offer solutions to the binding problem in a multi-dimensional setting, but their theory do not capture the correlation between presupposition projection and cross-sentential anaphora in an obvious way, unlike the theory put forward here. 4

will be shown below, this will amount to an existential presupposition. To repeat, the idea that the relation between the quantifier in the assertion and the anaphoric term in the presupposition is analogous to that of cross-sentential anaphora involving a quantificational antecedent. More concretely, consider the small discourse in (8). (8) A student is singing. He is loud. Here, the pronoun he can refer back to the student that the quantifier a student in the first sentence introduces. As a result, the overall meaning of the sentence will be equivalent to that of A student is singing and is loud. My main proposal is that the same mechanism that ensures the anaphoric link between a student and he in (8) is at stake in the case of presupposition projection in quantified sentences like (7). Cross-sentential anaphora is a well studied topic in formal semantics/pragmatics, and I will borrow technical machinery from previous studies for the purposes of formalization. Approaches to cross-sentential anaphora can be broadly classified into two kinds: E-type approaches (Evans 1980, Heim 1990, Elbourne 2005) and dynamic semantics (Heim 1982, Kamp 1981, Groenendijk & Stokhof 1990, 1991, Kamp & Reyle 1993, van den Berg 1996, Nouwen 2003a,b, 2007b, Brasoveanu 2007, 2010). Since the latter offers a rigorous treatment of plural cross-sentential anaphora and quantificational dependencies that will prove to be useful for our purposes, I will use a version of dynamic semantics as my metalanguage. However, the choice of the approach here is in principle arbitrary, and it is left open how the same idea can be formalized under the E-type approach. For the rest of this section, I will give a formally simple theory of presupposition projection that deals with sentences like (7) that involve singular indefinite quantifiers, which will be generalized to account for other kinds of quantifiers in 4. Although the extension proposed in 4 involves a number of complications, the core idea stays the same, which the simplified fragment presented below will hopefully clearly illustrate. 2.1 Dynamic Predicate Logic A principled account of singular cross-sentential anaphora with an existential antecedent is one of the earliest achievements of dynamic semantics. In this section, I will use Groenendijk & Stokhof s (1991) Dynamic Predicate Logic (DPL) as my tentative metalanguage, primarily because of its simplicity. In particular, it has a very simple syntax, which is identical to that of Predicate Logic (PL), and hence it allows us to keep the exposition of the formal details minimum (there are a number of accessible references, such as Chierchia 1995, Nouwen 2003b, Brasoveanu 2007, as well as the original paper by Groenendijk & Stokhof 1991). In order to see how DPL works, consider the example in (8), which is repeated here, under the reading where he refers to the student who is singing. (8) A student is singing. He is loud. In DPL, it is assigned the following formula as its meaning, where the same variable x is used throughout. (9) Dxrlinguistpxq ^ singingpxqs ^ loudpxq In PL, the last occurrence of the variable x is not bound by the existential quantifier Dx, and fails 5

to capture the intuition that he refers to the student who is singing. By contrast, in DPL x will be bound by the quantifier Dx, and co-vary with the other occurrences of x. This is ensured by the semantics of DPL, where a formula like (9) is interpreted as a relation between pairs consisting of an assignment function f and a possible world w. 6 More specifically, DPL sentences are interpreted in the following manner with respect to a first-order modelm xd, W, Iy, where D is a non-empty set of individuals, W is a non-empty set of worlds such that W X D H, and I is an interpretation function. Definition 1 DPL Semantics x f, wy} Dxrφs }xg, vy 1 iff there is h such that f and h differ at most in the value of x, and xh, wy}φ }xg, vy 1 x f, wy} Ppx 1,..., x n q }xg, vy 1 iff w v and f g and x f px 1 q,..., f px n qy P I w ppq x f, wy}φ ^ψ }xg, vy 1 iff there are h and u such that x f, wy}φ}xh, uy 1 and xh, uy}ψ}xg, vy 1 What is crucial for our purposes is that this semantics validates the so-called donkey equivalence : } Dxrφs ^ψ } iff } Dxrφ ^ψs }. 7 As a consequence, all the occurrences of x are bound by Dx in (9), capturing the reading of (8) that we are after. I will use this very mechanism to enable an anaphoric link between the assertion and presupposition of a quantified sentence. 2.2 Two Dimensional Semantics and the Update Rule Using DPL as part of our meta-language, let us analyze the meaning of the sentence in (7), repeated in (10) below. The assumption is that a sentence S is assigned as its meaning a pair of DPL formulae,αps q andπps q, representing the assertive and presuppositional meanings respectively. (10) A student criticized herself. a. αp10q: Dxrcriticizedpx, xqs b. πp10q: femalepxq Crucially, the same variable x is used in both dimensions, which I assume is ensured in syntax (see 4). Here is how such a sentence is interpreted in a discourse. Following Heim (1982, 1983) among others, I assume that a conversational context (or common ground) is modeled as a set C of pairs x f, wy of an assignment function f and a possible world w. The key is that the update by the assertive meaning is performed before the presupposition is processed, so that the anaphoric link can be established. More precisely, the update of a context C with a sentence φ proceeds in two steps. 8 I use the following notations here: fncpcq : t f : Dwrx f, wy P Csu and wldpcq : tw: D f rx f, wy P Csu. 6 For Groenendijk & Stokhof (1991), DPL formulae are interpreted as relations between assignment functions. I include possible worlds here, anticipating an application to presupposition, which is essentially an intensional phenomenon. 7 This is actually one side of the donkey equivalence. The other side says, } rdxrφss ñψ } iff } @xrφ ñψs }, which doesn t concern us here. Also as a corollary of the donkey equivalence, there is no need to assign a scope to Dx in DPL and we can regard it as a formula in itself: x f, wy} Dx }xg, vy 1 iff w v and f and g differ at most in the value of x. 8 I thank Philippe Schlenker (p.c.) for pointing out a problem of an earlier formulation. 6

Definition 2 2D Update 1. Update C with αpφq: C 1 tx f 1, w 1 y: Dx f, wy P Crx f, wy}αpφq }x f 1 w 1 ysu 2. Check if the worlds of the initial context wldpcq satisfy the presupposition, i.e. check if for each w P wldpcq, there is an updated assignment function f 1 P fncpc 1 q that satisfiesπpφq: if for all w P wldpcq, there is f 1 P fncpc 1 q such that x f 1, wy}πpφq }x f 1, wy then return C 2 tx f 1, w 1 y P C 1 : Dw P wldpcqrx f 1, wy}πpφq }x f 1, wysu presupposition failure otherwise. The context is updated with the assertive meaning first in Step 1, and then the presupposition is checked in Step 2 with the assignment functions f 1 that have been updated with the assertive meaning in Step 1, and hence encode the anaphoric information of the assertive meaning. Note that the presupposition is evaluated against the worlds of the original context C, which captures the intuition that the presupposition needs to be already part of the common ground when the assertion is made. More specifically, in all the worlds of the original context C, the presupposition should already be true, with respect to some assignment function f 1 P fncpc 1 q. If the context satisfies the presupposition, then the update of the context will terminate, and C 2 will be returned; otherwise presupposition failure will ensue. Let us take (10) for an illustration. Let C be the initial context, in which the utterance of (10) is made. I assume that it is not known whether any students criticized themselves in C so that the assertion is informative, i.e. in some of the worlds of C, some students criticized themselves, and in others none did (Stalnaker 1978). After the update with the assertive meaning (10a), the latter worlds are eliminated. Thus, in each of the worlds of the intermediate context C 1, at least one student x criticized x. Each assignment function f 1 of C 1 assigns x some particular student who criticized himself or herself. At this point, there is no constraint on the gender. Now using such assignment functions f 1, we check whether the original context C satisfies the presupposition. That is, for each world w P wldpcq, there needs to be an assignment function f 1 P fncpc 1 q such that x f 1, wy satisfies the gender presupposition. Assuming that it is known that there are some female students and at least one of them could have criticized herself, there must be such an f 1 for each world w. Then, in the final output context C 2, we discard those assignments f 1 of C 1 that assign x a male individual. In other words, (10) is a presupposition failure just in case there is no female student, or it is already known in C that no female student criticized herself. Notice that the presupposition is effectively existential: for each w P wldpcq, it requires there to be f 1 P fncpc 1 q that assigns x an individual who is female and who criticized herself, which does not require every student to be female. This captures the intuitive meaning of (10). 2.3 Summary To repeat the main point, the core idea of the analysis put forward here is that a quantifier in natural language introduces an anaphoric term in the presupposition that stands in the relation of crosssentential anaphora with the quantifier in the assertion. Informally put, a sentence like A student criticized herself presupposes that individual is female, which due to the update rule amounts to the presupposition that there is a female student who could have criticized herself. In this section, I 7

presented a simple fragment using DPL, and demonstrated how this idea can be formalized. I will claim in the following sections that the same idea accounts for the projection properties of various quantifiers from their cross-sentential anaphoric properties. 3 Presupposition and Cross-sentential Anaphora In the present section, I will show with concrete examples with several quantifiers that presupposition projection and cross-sentential anaphora are related in the manner suggested in the generalization repeated here from (4). (4) Let QpRq be a quantifier with a restrictor R. Consider a discourse of the form: QpRqpFq. pro G. for some predicates F and G. pro is some pronoun with appropriate phi-features, e.g. they. a. If pro can be resolved to R (i.e. QpRq supports maxset anaphora), then sentences of the form QpRqpλx.S pxq ppxq q can have a universal inference that @x P R : ppxq; and b. If pro can be resolved to R X F (i.e. QpRq supports refset anaphora), then sentences of the form QpRqpλx.S pxq ppxq q can have an inference weaker than a universal inference. Before proceeding, one caveat is in order: Since the theory sketched in the previous section is not expressive enough to deal with most of the quantifiers discussed here, the exposition will be kept at an informal level. A full formal theory will be offered in the next section. Also, I will first focus on atomic quantified sentences, and will discuss more complex cases involving embedded quantified sentences in the second half of this section. Let us begin with universal quantifiers. As briefly discussed at the outset, they give rise to universal inferences. For instance, (11) suggests that every student is female. (11) Every student criticized herself. Recall that the gender inference of criticized herself is purely presuppositional (cf. fn.1). Therefore, the universal inference that every student is female should be due to the projection property of every student. According to the analysis put forward here, the presupposition of (12) involves an anaphoric term, as in (12). (12) a. Assertive meaning: Every student x criticized x b. Presupposition: x is female How is the variable x in the presupposition resolved? We again draw an analogy with crosssentential anaphora. That is, we can liken the two dimensions of meaning in (12) to two separate sentences occurring in a discourse like (13). (13) Every student passed the test. They were happy. The pronoun they in this discourse refers to all the students, or in other words, it supports maxset anaphora. I claim that the same happens in the case of (12). That is, assuming that the variable x in (13b) is number neutral, the predicted presupposition will be that all of the students are female. Notice that the resulting presupposition is universal in that it requires every individual in the 8

restrictor to satisfy the presupposition of the predicate criticized herself. The analysis of (12) here does not depend on the form of the quantifier, but only on its meaning, in particular its anaphoric property. Therefore, a universal presupposition is predicted with other universal quantifiers like each of the students and all the students, which also support maxset anaphora, as shown in (14). (14) Each of the students/all the students passed the test. They are happy. That these universal quantifiers also give rise to a universal inference is a correct prediction, as demonstrated by (15). (15) a. Each of the students criticized herself. b. All the students didn t stop smoking. While (15a) is exactly parallel to (13), (15b) merits some discussion. It contains negation, and we are interested in the surface scope reading where all the students takes wide scope. Under this reading, (15b) is associated with a universal inference that all the students used to smoke. This universal inference is not due to its assertive meaning, which is that all the students are either still smoking or never smoked (Sudo 2012; see also fn.1 for relevant discussion). Together with the universal presupposition that all the students used to smoke, we infer from (15b) that all the students are still smoking. These observations show that universal quantifiers in general give rise to universal inferences, and they also support maxset anaphora cross-sententially. Now let us turn to no student. As remarked at the outset, a negative quantifier like no student tends to give rise to a universal inference as well. 9 This is demonstrated by the sentence in (16), which suggests that every relevant student is female. (16) No student criticized herself. According to the present account, this universal inference should be due to the cross-sentential anaphoric property of no student. More precisely, (16) is analyzed as (17). (17) a. Assertive meaning: No student x criticized x b. Presupposition: x is female What does x resolve to in this case? As demonstrated by (18) below, no student only supports maxset anaphora cross-sententially, i.e. anaphora to its restrictor (cf. Kamp & Reyle 1993, Nouwen 2003a,b, 2007a,b). (18) No student passed the test. They were unhappy. The pronoun they in the second sentence of (18) is read as all the relevant students. Just as in the case of a universal quantifier, therefore, (17b) amounts to the universal presupposition that every 9 I do not exclude the possibility of local accommodation of the presupposition at the VP level (Heim 1983, van der Sandt 1992, Beaver 2001, Kadmon 2001), in which case the universal inference does not obtain. That is, the resulting reading will be: No students are female self-criticizer. This is arguably harder to obtain, but I believe it is not impossible. Notice that with a universal quantifier, a reading with local accommodation still entails the universal inference. This accounts for the fact that negative quantifiers are associated with universal inferences relatively less robustly than universal quantifiers are (see Chemla 2009a for experimental data). 9

student is female. The exact same pattern obtains with other negative quantifiers like none of the students. Recall that unlike every student and no student, a student does not give rise to a universal inference. As we have already seen in the previous section, the present analysis captures this. That is, the cross-sentential anaphoric property of a student is such that a pronoun in a later sentence cannot be resolved to its restrictor, i.e. it does not support maxset anaphora. Rather, it supports refset anaphora, i.e. anaphora to the intersection of the restrictor and nuclear scope. This is shown in (19). (19) A student passed the test. a. He was happy. (he=the student who passed the test) b. #They were unhappy. (they=all the students) Correspondingly, in a sentence like the following, the presupposition is predicted to be about the individual in the intersection of the restrictor and the nuclear scope, rather than all the individuals in the restrictor. (20) A student criticized herself. a. Assertivemeaning: there is a student x who criticized x b. Presupposition: x is female As we saw in the previous section, this amounts to an existential presupposition, due to the update rule. Therefore, the resulting inference is weaker than a universal inference. To repeat the prediction of the theory, those quantifiers that support anaphora to the individuals in the restrictor, or maxset anaphora, cross-sententially can have a universal inference, and those that support anaphora to the intersection of the restrictor and the nuclear scope, or refset anaphora, cross-sententially can have a non-universal inference. Notice in the case of universal quantifiers, maxset anaphora and refset anaphora are identical, as they assert the equivalence of the restrictor and the intersection of the restrictor and nuclear scope. For our purposes here, we do not need to say that universal quantifiers support either one of the two kinds of anaphora, or both. What is important is that the value of the anaphoric term in the presupposition in this case will be all the individuals in the restrictor, and hence a universal presupposition is predicted. So far, we have only seen those quantifiers that only allow, or at least strongly prefer, either maxset anaphora or refset anaphora, but there are also quantifiers that are ambiguous. For example, unlike a NP, partitive existential quantifiers generally can give rise to universal presuppositions but can also have a weaker inference, as witnessed in the experiments reported in Chemla (2009a) and Sudo, Romoli, Hackl & Fox (2011). For example, the following sentences have a universal presupposition more robustly than a non-partitive sentence like (19). (21) a. Some of the students stopped smoking. b. One of the students stopped smoking. c. At least three of the students stopped smoking. d. At most five of the students stopped smoking. Correspondingly, these quantifiers seem to support maxset anaphora, as well as refset anaphora, unlike in the case of a student in (19). 10

(22) One of the students passed the test. a. He was happy. b. They were all happy. Thus, according to the present analysis, partitive existential quantifiers are predicted to optionally have a universal inference, as desired. Additionally, other kinds of partitive quantifiers such as the following are also predicted to have ambiguous presuppositions. (23) a. Most of the students stopped smoking. b. Few of the students stopped smoking. c. Many of the students stopped smoking. d. Less than 30% of the students stopped smoking. As shown below for (23a), they support both types of cross-sentential anaphora. (24) Most of the students criticized John. They are not so smart. a. They=the students who criticized John (Restrictor Anaphora) b. They=all the students (Nuclear Scope Anaphora) This prediction seems to be on the right track as well, as suggested by Chemla s (2009a) experimental data. We have seen so far basic cases where the predicted correlation between presupposition projection and cross-sentential anaphora nicely holds. However, there are several places where it breaks down, three of which I will discuss below. Importantly, they are merely complications essentially stemming from the pragmatic flexibility of pronominal anaphora, and do not undermine the proposed analysis of presupposition projection. Specifically, they are all cases where cross-sentential anaphora involves some extra discourse move which I claim is not available in the case of presupposition resolution. The idea is that the relation between the assertive meaning and the presupposition of a single sentence is not completely identical to the relation between two independent sentences in a discourse. In particular, in the latter case, several additional pragmatic inferences can be drawn in processing the second sentence after the first sentence is processed, which give rise to more possibilities for anaphora resolution. 3.1 Complication 1: Complement Anaphora Some sentences with certain determiners such as few are known to allow so-called complement anaphora cross-sententially, where a pronoun in the second sentence is resolved to the individuals in the restrictor who do not satisfy the nuclear scope. Firstly, as a baseline, consider the sentence in (25). (25) Most of the students showed up today. They were all sick. The pronoun they in the second sentence of this example can be read as the students who showed up today, or all the relevant students, but it is impossible to construe it as the students who did not come today. Now, consider the following sentences. (26) a. Few of the students showed up today. They were all sick. 11

b. Only one student showed up today. They were all sick. In these sentences, they can be read as the students who did not show up, unlike in the case of (25). On the other hand, presuppositions can never be about the complement of the nuclear scope, even if the quantifier is one that supports complement anaphora. More concretely, (27) cannot be read as presupposing that those students that did not criticized themselves are female. (27) Only one of the students criticized herself. Therefore, the parallelism between presupposition projection and cross-sentential anaphora seems to fail here. However, given the assumption that an extra pragmatic inference is necessary for licensing complement anaphora, the observation made here is not a problem for the proposed theory. Nouwen (2003a,b) claims that complement anaphora involves a mechanism that amounts to inferring a possible salient antecedent that is similar in nature to what is going on in the following example (see also Hardt 2004). (28) John kept on staring at the newly-wed couple. She resembled a childhood sweetheart of his. (Nouwen 2003a:105) The antecedents of the pronouns she and his in the second sentence of (28) are not explicitly mentioned anywhere, but they are inferable from the first sentence, i.e. the bride and the groom, respectively. Nouwen (2003a,b) extensively argues that complement anaphora is only licensed when such an inference is available. Adopting Nouwen s analysis of complement anaphora, I suggest that an inference that gives rise to a possible antecedent for complement anaphora is inherently unavailable in the case of presupposition resolution. That is, the relation between the assertion and presupposition does not completely mirror that of two independent sentences occurring in a real discourse in that no extra pragmatic inference that would license complement anaphora is allowed between the assertion and presupposition. As a consequence, the predicted presupposition for a given quantificational sentence is solely determined by the anaphoric potential of the quantifier (see 4 for more on this). 10 10 Something similar be said about complex quantifiers like not every student and almost no student. It seems that not every student is associated with a universal inference very robustly. For instance, (i) suggests that every student is female. (i) Not every student criticized herself. However, this statement is equivalent to the existential statement that some students didn t criticized themselves, and indeed these individuals can be referred to by a pronoun in a later discourse. (ii) Not every student came. They stayed home. If this is a possibility, (i) should allow an inference that the students who did not criticized themselves are female, contrary to fact. It seems to me that refset anaphora like (ii) involves an inference similar to that of complement anaphora. If this analysis is on the right track, this discrepancy between presupposition projection and cross-sentential anaphora can be attributed to the pragmatic inference that is only available in the latter case. I thank Philippe Schlenker (p.c.) and Benjamin Spector (p.c.) for raising this issue. 12

3.2 Complication 2: Projection out of Questions Another place where the correlation between presupposition projection and cross-sentential anaphora seems to fail is questions. Here, I will only discuss matrix questions, as embedded questions involve further complications regarding embedding predicates whose projection properties are ill understood. Presuppositions of wh-questions are not well discussed in previous studies (but see Chemla 2009a, Fox 2010, Schlenker 2009), but the current analysis can be straightforwardly extended to projection through wh-phrases, given that they are a species of quantificational noun phrases. However, there is a complication concerning the corresponding cross-sentential anaphora. Firstly, it seems that wh-phrases generally give rise to universal presuppositions. For instance, all of (29) presuppose that all the relevant people are female. 11 (29) a. Which student criticized herself? b. Which of the students criticized herself? c. Who criticized herself? What does my analysis predict for (29)? Notice that these sentences involve one additional level of embedding, i.e. embedding under the question operator. Thus, in order to analyze them, we need to know how the question operator interacts with the presupposition of its argument. Fortunately, its projection properties are rather simple. It is uncontroversial that presuppositions triggered in the body of the question simply project out (cf. Beaver 2001). For example, the factive presupposition triggered by realize projects out in both wh- and polar questions. Thus, all of the following examples presuppose that it was raining. (30) a. John realized that it was raining. b. Who realized that it was raining? c. Did John realize that it was raining? On the assumption that wh-phrases are quantifiers of some sort, there the present analysis assigns the presupposition that x is female to the wh-questions in (29). Again, what x can be resolved to depends on the anaphoric properties of the quantifier in the assertive meaning. Thus, the prediction depends on the cross-sentential anaphoric properties of wh-phrases and how they interact with the question operator. To the best of my knowledge, cross-sentential anaphora involving wh-antecedents are not well discussed in the literature. In general, maxset anaphora is possible, as demonstrated by (31). (31) A: Which students smoke? B: They don t smoke. (they=all the students) Therefore, universal presuppositions are predicted to be possible for (29). However, there also seem to be cases of refset anaphora with wh-antecedents. For example, them in the second sentence of (32) can be interpreted as the students who smoke. (32) Which students smoke? I will go smoke with them. 11 As in the case of negative quantifiers like no students, I do not exclude the possibility of local accommodation which will give rise to readings that do not entail universal inferences. 13

If refset anaphora is available, non-universal inferences are wrongly predicted for sentences like (29). I suggest here that the relevant reading of (32) requires a pragmatic inference of the kind that wh-phrases in general independently trigger, and assuming that such an inference is only available in a real discourse and not in processing the presupposition, the data like (32) does not pose a problem for the theory proposed here. The key observation is that wh-questions are generally associated with an existential inference that there are some individuals who satisfy both the restrictor and nuclear scope. In (32), for example, one can infer that the questioner assumes or at least considers it fairly likely that there are some smokers among the students. It is conceivable that the pronoun them in the second sentence is resolved to these students whose existence is entailed by this inference. Although the existential inference of wh-questions is sometimes considered to be a presupposition (Belnap 1969, Dayal 1996, Rullmann & Beck 1998, Lahiri 2002), there is good reason to doubt that it is part of the conventional meaning of a wh-question. Firstly, it does not have to be fully endorsed even by the speaker. For instance, (32) can be felicitously uttered even if the questioner considers it possible (though not so likely) that there are no students who smoke. Thus, the relevant existential inference is weaker than a presupposition. Secondly, this existential inference is cancelable as demonstrated by (33). (33) Which students smoked after the talk? Was there any of them? Here the existence inference should be in conflict with the second question. As shown in (34), a question that does have the same existence inference as a genuine presupposition results in an infelicity. (34) Who knows that some students smoked after the talk? #Was there any of them? Thirdly, one can easily answer a wh-question with an answer that entails the non-existence of individuals in the restrictor and nuclear scope, which amounts to the denial of the existential inference. (35) A: Which students smoked after the talk? B: None of them did. Although presuppositions can generally be denied by another discourse participant, the denial in (35B) seems to have a different pragmatic status from a typical denial of a presupposition which is exemplified by (36). 12 (36) A: Mary knows that some students smoked after the talk. who. 12 It should be remarked that a negative answer like (35B) is more marked in comparison to a question involving (i) A: Who smoked after the talk? B: No one did. That is, (35B) is associated with a feeling of canceling the speaker s assumption, which (ib) lacks. Thus, the contrast is actually three-way here: (i) is more felicitous than (35), which in turn is more felicitous than (36). Although the contrast between (i) and (35) requires further investigation, what is crucial for my purposes is the contrast between (i) and (36). I thank Philippe Schlenker (p.c.) for helpful discussion on this point. 14

B: No, that can t be right, because none of them did. For these reasons, I think it is reasonable to consider the existence inference of a wh-question as a pragmatic inference of some sort, rather than part of the conventional meaning like a presuppositional inference. Coming back to the example in (32), I assume that the refset anaphora is mediated by this pragmatic inference. More concretely, it can be paraphrased by (37). (37) I assume that there are some students who smoke. Which ones are they? I will go smoke with them. Although I leave open exactly how this inference is generated, but I assume that this pragmatic inference is an inference of a sort that is unavailable between the assertion and presupposition. Assuming that refset anaphora is unavailable otherwise, therefore, the only available antecedent for the anaphoric presupposition in a wh-question is the set of individuals who satisfy the restrictor. Thus, (29) are all associated with universal inferences, as desired. A similar remark applies to polar questions with existential quantifiers. Sudo, Romoli, Hackl & Fox (2011) provide experimental evidence suggesting that sentences of the following sort give rise to universal inferences. (38) Did any of the students criticize herself? This sentence fairly reliably suggest that all of the students are female. As in the case of wh-questions, refset anaphora seems to be available with existential quantifiers in polar questions. For instance, consider (39). (39) Do any of the students smoke? I will go smoke with them. However, refset anaphora in (39) seems to involve an extra existential inference, similar to the case of wh-questions. An intuitive paraphrase of (39) involves an implicit conditional with a yes answer, as in (40). (40) Did any of the students smoke? If yes, I will go smoke with them. Although a detailed analysis of this existential inference is not offered here, it is sufficient for our purposes to assume that the relevant pragmatic mechanism is not available in the case of presupposition resolution. As a result, a universal inference is predicted for a sentence like (38). 3.3 Complication 3: Projection out of Conditional Antecedents Lastly, let us consider cases where a quantificational sentence is embedded in the antecedent of a conditional. When an existential sentence appears in a conditional antecedent, a universal presupposition obtains, as exemplified by (41). I use the quantifier any of the NP to force the narrow scope reading with respect to if. (41) If any of the students criticizes herself, I will be surprised. Just as in the case of questions, it is uncontentious that if passes up the presupposition of the embedded clause. According to the analysis put forward here, therefore, the presupposition of (41) 15

is that x is female, with x being anaphoric to the existential quantifier embedded in the antecedent clause. Since what is observed is a universal presupposition, maxset anaphora should be the only option for cross-sentential anaphora in the same configuration. However, contrary to this expectation, refset anaphora is possible in sentences like (42). (42) If any of the students show up, let me know. They will get their homework back. Here, they can be read as referring to the students who show up. Such anaphora is called modal subordination and extensively studied in the literature (for example, see Roberts 1989, Kadmon 2001, Brasoveanu 2007, 2010). Although I will not go into the details here, modal subordination arguably involves some extra pragmatic inference. In fact, refset anaphora is not freely possible in this configuration. (43), for example, only has a maxset anaphora reading. (43) If any of the students show up, let me know. They have been absent lately. Thus, existential sentences embedded in if -conditionals only support refset anaphora in special contexts where modal subordination is licensed. Again, assuming that whatever licenses modal subordination is not available between the assertion and presupposition, the only available antecedent for the anaphoric term in the presupposition in (41) is the restrictor individuals. Consequently a universal inference is predicted. 4 Incorporating Generalized Quantifiers In the previous section, we saw that presupposition projection in quantified sentences and crosssentential anaphora with quantificational antecedents are closely related. The main purpose of this section is to explain this in a formal way. The core idea is the same as in 2, but the theory presented there is only capable of singular indefinite quantifiers like a student, since the metalanguage we adopted there, i.e. DPL, does not contain plural (selective) generalized quantifiers. Luckily, generalized quantifiers have been implemented in dynamic semantics by a number of authors, and their cross-sentential properties are also formalized (van Eijck & de Vries 1992, Chierchia 1995, van den Berg 1996, Nouwen 2007a,b, Brasoveanu 2007, 2010). In this section, I will demonstrate how the theory proposed in 2 can be extended to deal with various quantifiers, while keeping the core idea intact: A quantifier introduces an anaphoric term in the presupposition, which refers back to the relevant individual(s) in the assertion. 4.1 Generalized Quantifiers in Dynamic Semantics DPL as formulated by Groenendijk & Stokhof (1991) does not contain (selective) generalized quantifiers, but later authors offered several ways to implement them in dynamic semantics (van Eijck & de Vries 1992, Chierchia 1995, van den Berg 1996, Nouwen 2007a,b, Brasoveanu 2007, 2010). 13 I will adopt the analysis along the lines of van den Berg (1996) and Brasoveanu (2007, 2010) here. The main advantage of this analysis of generalized quantifiers is that both the restrictor and nuclear scope individuals are explicitly represented, which is useful for formalizing the phenomenon we are after. 13 Unselective generalized quantifiers that are relations between sets of assignment functions rather than sets of individuals were available in the earlier works of dynamic semantics (Kamp 1981, Heim 1982, Groenendijk & Stokhof 1990, 1991). 16