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Relativized Exhaustivity: mention-some and uniqueness


Wh-questions with the modal verb can admit both mention-some (MS) and mention-all (MA) answers. This paper argues that we should treat MS as a grammatical phenomenon, primarily determined by the grammar of the wh-interrogative. I assume that MS and MA answers can be modeled using the same definition of answerhood (Fox in Mention-some interpretations, MIT seminar, 2013) and attribute the MS/MA ambiguity to structural variations within the question nucleus. The variations are: (i) the scope ambiguity of the higher-order wh-trace and (ii) the absence/presence of an anti-exhaustification operator. However, treating MS answers as complete answers in this way contradicts the widely adopted analysis of uniqueness effects in questions of Dayal (Locality in wh quantification: Questions and relative clauses in Hindi, 1996), according to which the uniqueness effects of singular which-phrases arise from an exhaustivity presupposition, namely that a question must have a unique exhaustive true answer. To solve this dilemma, I propose that question interpretations presuppose Relativized Exhaustivity: roughly, the exhaustivity in questions is evaluated relative to the accessible worlds as opposed to the anchor/utterance world. Relativized Exhaustivity preserves the merits of Dayal’s exhaustivity presupposition while permitting MS; moreover, it explains the local-uniqueness effects in modalized singular wh-questions.

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  1. There is no clear consensus on what the lack of final fall contributes to meaning. I take it to mark the pragmatic imperfection of an answer, roughly read as ‘the best I can tell is ...’ For example, the lack of final fall in (i) (marked by ‘...’) indicates that A isn’t sure whether her answer is relevant to the question, not that the answer is possibly non-exhaustive.

    1. (i)
      figure b
  2. To be exact, Groenendijk and Stokhof (1984) assume that question interpretations are strongly exhaustive, namely, the extensional meaning of an interrogative not only affirms all the true answers but also rules out all the false ones. In this view, the interrogative who came denotes a function that maps a world w to the exhaustified proposition ‘only x came’ that is true in w. However, most recent works on question semantics take the weakly exhaustive meaning, which only affirms the true answers, as the basic meaning of an interrogative. For example, who came primarily denotes a function that maps a world w to the proposition ‘x came’ such that it’s true in w that only x came. Strong exhaustivity, then, is derived from weak exhaustivity via a separate operation.

  3. Not all question embeddings admit MS readings. For example, depends on-sentences (e.g., Where you can buy coffee depends on the time) only allow for exhaustive readings (Groenendijk and Stokhof 1984; Dayal 2016).

  4. Note that in this scenario it is infelicitous to ask “Do you know which three places on campus one can hold a large conference?”, because this question presupposes that there are only three such places on campus.

  5. One might give a pragmatic explanation as follows: (23b) is read exhaustively because it is over-informative, in contrast to (23a), which is just as informative as needed. However, it remains puzzling why over-informativity forces exhaustivity. One would need to assume two distinct answerhood operations: (i) a pragmatic answerhood that seeks a (possibly non-exhaustive) answer that is as informative as needed and (ii) a semantic answerhood that demands an exhaustive answer. Nevertheless, this hybrid analysis cannot explain (22): when the questioner’s goal is ‘mention-three’, a mention-one answer wouldn’t be permitted by any answerhood operations—it is neither informative enough nor exhaustive.

  6. This paper stays neutral on how topical properties are compositionally derived. I only specify the composition of the question nucleus. Besides categorial approaches, which define questions as functions, dynamic approaches (Dotlačil and Roelofsen 2019; Li 2021) also allow for extraction of short answers from question denotations.

  7. Unlike focus alternatives, variable alternatives are specific to variable-like expressions and do not require focus marking.

  8. For any meaning α of type τ, we have: the Montague-lifted meaning of α is \(\alpha ^{\Uparrow}\) (of type 〈τt,t〉) s.t. \(\alpha ^{\Uparrow} := \lambda m_{\langle \tau , t\rangle}. m(\alpha )\). Boolean conjunctions and disjunctions are defined in terms of set intersection and union, respectively. For any meanings α and β of type τ, we have: \(\alpha ^{\Uparrow} \cap \beta ^{\Uparrow}:= \lambda m_{\langle \tau , t\rangle}. m(\alpha ) \wedge m(\beta )\), and \(\alpha ^{\Uparrow} \cup \beta ^{\Uparrow}:= \lambda m_{\langle \tau , t\rangle}. m(\alpha ) \vee m(\beta )\).

  9. What is particularly interesting about treating dou as an anti-exhaustification operator is that it accounts for multiple uses of dou. As argued in Xiang 2020, the function of dou varies depending on what alternatives participate in anti-exhaustification. Assuming a non-vacuity presupposition on the existence of such alternatives, this analysis accounts for the universal distributor use, the FC-trigger use, and the even-like use. Alternatively, by giving dou an even-like semantics, Liu (2016) derives the anti-exhaustivity effect indirectly (reviewed in Xiang 2020, Appendix B). For an account that derives the FC-trigger use of dou without assuming anti-exhaustification, see Zhao 2019.

  10. I don’t call such alternatives ‘innocently (I-)includable alternatives’, a notion coined by Bar-Lev and Fox (2020), because I-includable alternatives include also the prejacent itself (see (72) in Sect. 5.2.1).

  11. The FC-trigger use of dou is also subject to modal obviation (Xiang 2020): when the existential modal keyi ‘can’ is dropped or replaced with a universal modal like bixu ‘must’, dou cannot be grammatically associated with a pre-verbal disjunction.

    1. (i)
      figure av
  12. To be exact, although RelExcl is treated as a definedness condition of anti-exhaustification here, the effects of RelExcl are independent of whether universal FC is derived by anti-exhaustification. The only prerequisite for assuming RelExcl is that the formal theory used for deriving universal FC allows us to extract the subdomain alternatives from a disjunctive sentence.

  13. For a set of possible worlds W and a sentence ϕ, we have:

    1. (i)
      figure aw
  14. In this definition, ϕ is a syntactic expression, not a semantic value. RelExcl is formalized as such because it concerns the interpretations of the alternatives evaluated relative to different modal bases.

  15. RelExcl is very similar to Dayal’s (2013) Viability constraint, which says that every exhaustified alternative is true relative to a subset of the accessible worlds. The following formulates Viability analogously to (50):

    1. (i)
      figure ax

    The main difference between RelExcl and Viability is the following: in (53a), where the ◊-disjunction is parsed without local exhaustification, Viability predicts that universal FC is possible as long as \(\Diamond O \phi _{j}\) and \(\Diamond O \phi _{m}\) are true, regardless of the truth or falsity of \(\Diamond (\phi _{j} \wedge \phi _{m})\). However, as we shall see in Sect. 6.3.4, to account for the universal local-uniqueness inferences in can-questions with a disjunctive MA interpretation, there has to be an interpretation that requires \(\Diamond (\phi _{j} \wedge \phi _{m})\) to be false.

  16. Figure 5b is labeled as ‘dis/con-junctive MA’, because the conjunctive answer at the top and the disjunctive answer at the bottom are truth-conditionally equivalent. They both express the MA answer that Andy alone and Billy alone can chair.

  17. A similar constraint applies if one follows Fox (2018, 2020) and assumes that a wh-domain may include higher-order pluralities: singular which-phrases cannot quantify over higher-order pluralities, otherwise (60b) would have an exhaustive true answer based on {{a},{b}}.

  18. Dayal (2016, Sect. 3.2.1) provides two reasons for keeping choice interpretations separate from MS interpretations. One revolves around morphological differences among wh-expressions, and the other is concerned with the availability of MS interpretations in questions with a universal quantifier. Let me add two additional arguments here. First, choice questions do not have FC-disjunctive answers. In (i), the disjunctive answer only has an ignorance reading. The unavailability of a FC reading argues that the machinery that makes a can-question congruent with FC-disjunctive answers is unavailable to choice questions. As argued in Sect. 4.4, the disjunctive MA interpretation of a can-question is derived by applying an anti-exhaustification operator to the local IP. Nevertheless, as commonly assumed, the indefinite in a choice question scopes fairly high at LF (Groenendijk and Stokhof 1984; Chierchia 1993; Szabolcsi 1997; Krifka 2001; Dayal 2016; Xiang 2022; a.o.) and therefore cannot interact with an anti-exhaustification operator.

    1. (i)
      figure bq

    Second, the two types of questions behave differently w.r.t. local uniqueness. In a singular can-question, the local uniqueness presupposition is universal—it applies to every accessible world that verifies a true answer (see (83)–(86) and (105)/(107)). In choice questions, however, the local uniqueness presupposition is existential. For example, the question in (i) implies that one of the boys watched exactly one movie, not that each boy watched at most one movie. This contrast argues that uniqueness is derived differently in these two types of questions.

  19. The analysis proposed by Fox (2018, 2020) is more complex than what is presented here. To account for negative island effects in higher-order interpretations, Fox argues for a non-vacuity principle: every proposition p in Q is such that the exhaustification of p is identical to a cell in the partition induced by Q. This principle predicts that non-modalized wh-questions do not have higher-order interpretations: in a higher-order interpretation, the answer space of a non-modalized wh-question contains plain disjunctions like \(\phi _{a} \vee \phi _{b}\), which cannot be paired with a partition cell by exhaustification. To account for the data in (77), Fox further assumes that the quantification domain of which children includes higher-order pluralities such as {{a,b},{c,d}}.

  20. Oddness arises even before the answer continuation “Chap. 3” is given, due to the problem we turn to next.

  21. RelExh is defined based on short answers because the meaning of a short answer is modal independent. In contrast, the meaning of a sentential answer to a modalized question varies as a function of the modal base.

  22. A reviewer points out that the definition of RelExh in (91) requires access to the modal base M, which is difficult to obtain from a compositional perspective. One way to solve this problem is to define modalized sentences as functions from modal bases to propositions, not as open propositions with a free modal base variable. This analysis allows the modal base to be retrieved as an argument of the sentential denotation. It also allows RelExh to be defined based on sentential answers, since sentential denotations are arrived at modal independently.

  23. In light of Uegaki 2021, to account for the projection of the uniqueness presupposition in embeddings, we can assume that the RelExh presupposition is carried by each max-informative true answer, rather than by the answer set as a whole.

  24. Relatedly, given the assumption in situation semantics that propositions can be characterized in terms of situations, we may also evaluate exhaustivity w.r.t. minimal situations. This option allows RelExh to be applied to non-modalized questions like Who has got a light?, which share many MS-related properties with can-questions (van Rooij 2004). For this question, MS is available if exhaustivity can be evaluated w.r.t. minimal situations that exemplify the proposition Someone has got a light.

  25. For example in (99b), ‘first-order (G)’ means that the first-order interpretation of (99a) yields global (G) uniqueness, and ‘narrow-scope higher-order (L)’ means that the narrow-scope higher-order interpretation of (99a) yields local (L) uniqueness.

  26. The discussion here is centered on have to-questions simply because the modal force of have to is unambiguously universal.


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This paper supersedes Chaps. 2 and 3 of my dissertation (Xiang 2016) and the proceedings papers of Amsterdam Colloquium 20 and NELS 46. The analysis, especially the part on resolving the dilemma between uniqueness and mention-some, has been substantially reworked. I thank Gennaro Chierchia, Danny Fox, and C.-T. James Huang for guiding me during the early development of this work, as well as two exceptional reviewers of Natural Language Semantics and the handling editor Clemens Mayr for extremely helpful reviews. I also thank the audiences at the New York Language of Philosophy Workshop, TaLK 2022 at Keio University, and UConn Logic Group for feedback on the recent versions of this paper. I am especially grateful to Christine Bartels for her incredible editorial comments, which have greatly reshaped the presentation of this paper.

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Xiang, Y. Relativized Exhaustivity: mention-some and uniqueness. Nat Lang Semantics 30, 311–362 (2022).

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  • Interrogatives
  • Questions
  • Answers
  • Mention-some
  • Uniqueness
  • Exhaustivity
  • Exclusivity
  • Free choice
  • Modality
  • Modal obviation
  • Higher-order interpretations