Abstract
Questions with a quantificational subject have readings that seemingly involve quantification into questions (called ‘QiQ’ for short). In particular, in single-wh questions with a universal quantifier, QiQ-readings call for pair-list answers, similar to pair-list readings of multiple-wh questions. This paper unifies the derivation of QiQ-readings and distinguishes QiQ-readings from pair-list readings of multiple-wh questions. I propose that pair-list multiple-wh questions and QiQ-questions both involve a wh-dependency, namely, that the wh-/quantificational subject stands in a functional dependency with the trace of the wh-object. In particular, in a pair-list multiple-wh question, the wh-subject binds into the trace of the wh-object across an identity operator; in a QiQ-question, the quantificational subject binds into the trace of the wh-object across a predication operator. These operations give rise to distinct definedness requirements, which vary with the quantificational force of the wh-/quantificational subject. The proposed analysis explains a contrast in domain exhaustivity between the pair-list readings of multiple-wh questions and questions with a universal quantifier, while also doing justice to the intuitive similarities between these two types of questions. I further propose that the observed QiQ-effect in a QiQ-question is derived by extracting one of the minimal proposition sets that satisfy the aforementioned quantificational predication condition. The values of these sets determine whether the QiQ-reading is available and whether a QiQ-question admits pair-list answers and/or has a choice flavor.
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Notes
In \(\exists \)-questions, functional readings are only marginally acceptable. For example, the fragment functional answer (i-a) is under-informative; the identity of the boy who watched a movie has to be specified, as in (i-b). I leave this puzzle open.
One might wonder whether specifying the domain of quantification explicitly can sufficiently remove the confound with domain exhaustivity—could there be additional covert domain restrictions with the wh-phrases? In (9) and (10), for example, the confound would remain if the quantification domain of which one of the four kids were covertly restricted to a subset of the four kids, excluding the kid who will not sit on a chair. I argue that such covert restrictions are not possible once the quantification domain of a wh-phrase has been specified explicitly. As seen in (i), uniqueness is assessed relative to a domain containing all four contextually relevant kids, as in (i-a); if the phrase which one of the four kids could range over a subset of the four kids, the uniqueness inference would be as weak as (i-b), contrary to fact.
The reason why (15b) and (16d) marginally admit choice readings might be that \(\exists \)-quantifiers have more ways to take wide scope than \(\forall \)-quantifiers, such as through globally bound choice functions.
Besides these two general strategies, inquisitive semantics also avoids this type-mismatch problem because it defines declaratives and interrogatives uniformly as sets of classical propositions (of type \(\langle st, t\rangle \)) and generalized quantifiers as functions of type \(\langle \langle e,stt\rangle , stt\rangle \). For a recent account using inquisitive semantics, see Qing and Roelofsen 2021.
The core assumptions of these two approaches are compatible with each other. For example, Chierchia (1993) assumes a wh-dependency while defining a QiQ-question as a family of questions. For more details, see footnote 15.
One might wonder why we chose to treat pair-list readings as special functional readings, not vice versa. The reason is that pair-list readings are subject to more constraints than functional readings. As seen in (i), multiple-wh questions are congruent with fragment answers that are lists of pairs, but not with intensional functional answers (Kang 2012; Sharvit and Kang 2017). If pair-list readings were more general than functional readings, we wouldn’t expect such a gap.
Sharvit and Kang (2017) provide an explanation as to why pair-list questions do not admit intensional functional answers. However, the syntax of multiple-wh questions assumed by Sharvit and Kang is quite different from mine. This paper leaves this issue open.
Chierchia (1993) assumes that the wh-trace carries two indices, namely, a functional index i bound by the wh-phrase and an argument index j co-indexed with the non-interrogative quantifier. To bind the j-index carried by the wh-trace, the non-interrogative quantifier has to be moved to a position that c-commands this wh-trace. Thus in (i-b), unlike (i-a), when the quantifier every boy is moved from a position lower than the wh-trace, it inevitably moves across a co-indexed expression (viz., the wh-trace), causing weak crossover.
In contrast, competing accounts by Safir (1984) and May (1988) analyze the asymmetry and weak crossover in terms of separate syntactic constraints.
It might look appealing to analyze the subject–object/adjunct asymmetry in QiQ-questions and the superiority effects in multiple-wh questions uniformly. For example, Hornstein (1995) extends Chierchia ’s (1993) complex-trace analysis of wh-dependencies to superiority effects. He assumes that the in-situ wh-phrase contains a covert pro co-indexed with the fronted wh-phrase. Accordingly, in (i-b), moving the object what across the co-indexed pro causes weak crossover.
Relatedly, Shan and Barker (2006) argue that binding relations must be evaluated from left to right, and they use this single constraint to rule out crossover and superiority violations.
In contrast, I argue that superiority effects in multiple-wh questions and the subject–object/adjunct asymmetry in QiQ-questions have different origins: as seen in (ii), multiple-wh questions with which-phrases tolerate superiority violations and admit pair-list readings (Pesetsky 1987, 2000; Kotek 2014, 2019).
The analysis presented in this paper makes no prediction on the overt syntax of multiple-wh questions. Whatever its insufficiencies (see footnote 23 in Sect. 6.3), this analysis is exempt from the under-generation problem.
In contrast to the complex-trace approach, Jacobson (1999, 2014) develops a variable-free approach to functionality which does not make use of indices. In her analysis, functionality is derived by a type-shifting rule, called ‘the z-rule’, which closes off the dependency between the arguments of a predicate. (For example, \(\textbf{z}(\llbracket \textit{watched} \rrbracket ^{w}) = \lambda \varvec{f}_{\langle e, e\rangle }\lambda x_{e}.\llbracket \textit{watched} \rrbracket ^{w}(x, \varvec{f}(x))\).) This approach is especially advantageous in tackling cases where the wh-dependent is in situ or inside an island. For ease of comparison with existing works on composing complex questions, this paper follows the complex-trace approach.
Following Groenendijk and Stokhof (1984), I translate LF representations into the Two-sorted Type Theory (Ty2) of Gallin (1975). Ty2 differs from Montague’s intensional logic in that it introduces s (the type of possible worlds) as a basic type (just like e and t), and in that it uses variables and constants of type s which can be thought of as denoting possible worlds. For example, the English common noun boy is translated into \(\textsf {boy}_{w}\) in Ty2, where boy is a property of type \(\langle s, et\rangle \) and w a world variable of type s. With these assumptions, Ty2 can make direct reference to worlds and allows quantification and abstraction over world variables.
For simplicity, I assume that the extensions of wh-complements are evaluated relative to the actual world ‘@’.
Dayal (2016b) considers two ways to obtain the quantification domain of a wh-phrase. One way is to define a wh-phrase as an \(\exists \)-quantifier and extract out its quantification domain via the application of a Be-shifter (Partee 1986). The other way is to define a wh-phrase as a set of entities and derive its quantificational meaning via an \(\exists \)-shifter.
Live-on sets and witness sets are defined as follows (Barwise and Cooper 1981): For any \(\pi \) of type \(\langle et, t\rangle \), \(\pi \) lives on a set B iff \(\pi (C) \Leftrightarrow \pi (C \cap B)\) for any set C; if \(\pi \) lives on B, then A is a witness set of \(\pi \) iff \(A \subseteq B\) and \(\pi (A)\).
The approaches by Groenendijk and Stokhof (1984) and Chierchia (1993) are also family-of-questions approaches. They define a QiQ-question as a family of sub-questions ranging over a minimal witness set (mws) of the subject quantifier, as in (i). (‘\(\mathcal {P}_{\textsf {boy}_{@}}\)’ stands for a generalized quantifier ranging over the set of atomic boys. ‘\(\textsc {mws} (\mathcal {P}_{\textsf {boy}_{@}}, A)\)’ means that A is a minimal witness set of \(\mathcal {P}_{\textsf {boy}_{@}}\).)
However, the predictions made by these accounts are quite different from the predictions made by the non-flat semantics in (39). For example, Chierchia (1993) defines a sub-question as a set of propositions of the form \(\ulcorner \textit{boy-x watched movie-} \varvec{f}(x)\urcorner \), as schematized in (ii). The related \(\forall \)/\(\exists \)-questions are thus defined as in (iii).
Chierchia further assumes that answering a family of sub-questions means answering one of the sub-questions (in contrast to Fox’s assumption that answering a family of sub-questions means answering all of the sub-questions). Accordingly, since one of the boys has multiple minimal witness sets, the QiQ-reading of the \(\exists \)-question has a choice flavor. Although this account naturally extends to \(\exists \)-questions, it cannot explain the semantic effects in pair-list \(\forall \)-questions, such as domain exhaustivity and point-wise uniqueness.
One might propose to salvage the family-of-questions approach by arguing that pair-list multiple-wh questions, but not pair-list \(\forall \)-questions, permit covert domain restriction. This possibility has been ruled out by the discussion in Sect. 2.1. First of all, the contrast between the two types of pair-list questions in domain exhaustivity remains even if the domain has been explicitly specified, as seen in (10) and (12). Moreover, as argued in footnote 2, if the quantification domain of a wh-phrase has been explicitly specified, it does not take further covert restrictions.
In this paper, functions with a domain condition restricting the values of the inputs are represented in the form of \(\lambda v_{\tau } \!: P(v).\alpha \), where \(\tau \) is the semantic type of v, P(v) stands for the domain condition that restricts the value of v, and \(\alpha \) stands for the value description (Heim and Kratzer 1998). Functions without a domain condition are written in the form of \(\lambda v_{\tau }. \alpha \) or \(\lambda v_{\tau } [\alpha ]\), whichever is easier to read.
Following Fox (2013), Xiang (2016, 2020) assumes a weaker definition for complete true answers: a true answer to a question is complete as long as it is not asymmetrically entailed by any other true answer to this question. This answerhood is assumed to account for mention-some readings of questions and free relatives. Since mention-some is not the focus of this paper, for easier comparisons with competing theories of complex questions, I follow Dayal (1996, 2016b) here and define the complete true answer as the unique strongest true answer. For recent accounts on solving the dilemma between uniqueness and mention-some, see Fox 2018, 2020 and Xiang 2022. Also see Dotlacil and Roelofsen 2021 for an analysis using dynamic inquisitive semantics to account for both uniqueness effects and mention-some readings.
Instead of postulating a polymorphic restrictor, we can alternatively assume that wh-phrases are semantically ambiguous between either ranging over \(\llbracket \text {A}\rrbracket ^{w}\) or over a set of functions from individuals to \(\llbracket \text {A}\rrbracket ^{w}\). For example, Engdahl (1986) assumes a type-shifter that applies to the wh-complement that has the effect of turning a set of entities into a set of \(\langle e, e\rangle \)-type functions.
Crucially, \(\textsc {BeDom}(\pi )\) is type-flexible: it can combine with any function of a \(\langle \sigma , ...\rangle \) type where \(\sigma \) is the type of an element in \(\textsc {Be}(\pi )\). Type-flexibility makes it possible to compose a question regardless of whether the wh-phrase binds an individual or functional variable, and regardless of how many wh-phrases there are in this question. This assumption overcomes difficulties with traditional categorial approaches in composing multiple-wh questions with single-pair readings.
The following illustrates the contrast between the \(f_{\!\textsc {ch}}^{\textsc {min}}\)-operator and the \(\textsc {min}_{\textrm{S}}\)-operator:
For readers who are familiar with Boolean semantics, the \(f_{\!\textsc {ch}}^{\textsc {min}}\)-operator is roughly the same as the collectivity-raising operator in Winter 2001.
I use the subset symbol in \(\beta \subseteq \alpha \) since a function can be viewed as a set of ordered pairs. In standard mathematical terms, such \(\beta \) is called the restriction of \(\alpha \) to a set \(X'\) s.t. \(X' \subseteq \text {Dom}(\alpha )\), written as \(\beta = \alpha \mid _{X'}\).
Eagle-eyed readers might notice that here the wh-object is moved over the fronted wh-subject, which violates the generalization of ‘tucking-in’ (Richards 1997). Although violations of tucking-in are sometimes permitted for D-linked wh-phrases, it is certainly problematic to say that pair-list readings are only available in constructions that violate tucking-in. However, this problem does not stem from the specific assumptions involved in the composing of pair-list multiple-wh questions; it is a consequence of requiring covert/overt wh-fronting in question composition generally. This problem can be avoided if we assume a framework of composition that allows wh-in-situ. For example, in variable-free semantics (Jacobson 1999, 2014), abstractions can be passed up by type-shifting operations. Integrating my core proposal on composing pair-list questions into such frameworks allows us to create the wanted topical property without fronting the object wh-phrase.
I thank an anonymous reviewer of L &P for suggesting the LF in (72). I leave it open whether the answerhood operator used for obtaining the complete true answer of the matrix question is syntactically presented right below the predicate you-make-me-know or encoded within the lexicon of this predicate (see Xiang 2020: Sect. 4.2).
However, in an informal survey, I found significant individual differences among speaker judgments on whether (75) has a pair-list reading. For details, see footnote 37.
I thank an anonymous reviewer of L &P for pointing out a mistake in this analysis in an earlier version.
Binding with covert each is also marked in implicit binding. In the examples below, the ‘each boy’ reading is easily attested in (i) but quite unnatural in (ii). This contrast argues that the pronoun their cannot easily be bound by covert each.
Recent experimental work by van Gessel and Cremers (2021) shows that the distribution of pair-list readings forms a gradient, from \(\forall \)-questions with an every/each-phrase, which robustly allow for pair-list readings, to no-questions with a negative quantifier, which clearly do not. In particular, for matrix \(\exists 2\)-questions, pair-list readings were judged available in roughly half the cases in their experiment. One possible explanation of this variation is that some language users allow covert each to bind a covert variable.
Some speakers find (85a) slightly odd, which is probably due to the markedness of associating each with a non-specific indefinite.
I thank an anonymous reviewer of L &P for bringing this data to my attention.
As pointed out by a reviewer, it is syntactically permitted to interpret the \(\exists \)-quantifier within the embedded question and let its plural trace be associated with overt each (cf. (81) and (82)). This option, however, is semantically marked due to the reason outlined in footnote 29.
I thank an anonymous reviewer of L &P for pointing me in this direction for an explanation.
Here domain exhaustivity is trivially satisfied. For example, the set that the Montagovian individual ‘lift(the-boys)’ ranges over is a singleton set containing only the plural entity denoted by the boys.
I thank Bernhard Schwarz (pers. comm.) for bringing this issue to my attention.
Xiang 2020 also considers mention-some readings of questions, where a question can have multiple complete true answers. Once mention-some readings enter the picture, \(\textsc {Ans}^{S}(w)(\llbracket \text {Q} \rrbracket )\) needs to be defined as a set of entities/functions, not as a single entity/function.
Krifka (2001) assumes the structure in (i), where the quantifier scopes over a speech act operator quest. This analysis is exempt from the over-generation problem since Krifka assumes that speech acts cannot be disjoined. However, it also leaves the choice readings of \(\exists \)-questions unexplained.
Let me note, however, that in an informal survey, 7 out of 14 speakers judged (118b) as contradictory to the context. They reported that the use of which movie gives rise to the inference that two of the boys watched the same movie. I see two possible reasons for why some speakers found (118b) bad: (a) for these speakers, neither wonder-type nor find out-type embeddings allow a quantifier inside the embedded question to scope over the embedding predicate, or (b) these speakers do not actively use covert VP-each (for discussions on the distributional constraints of covert each, see Beghelli 1997). Regardless of the reason, the judgment is consistent with my claim that quantifying into questions cannot be analyzed as quantification into question-embeddings.
Rather than assuming covert movement of the quantifier, Szabolcsi (1997b) derives the wide scope reading by type-lifting the interrogative complements of extensional predicates. Combining the type-lifted question denotation (i) with an embedding predicate P yields a wide scope reading of the generalized quantifier \(\pi \) relative to P. Further, Szabolcsi argues that wonder-type predicates cannot select for lifted questions, and hence that quantifiers in intensional complements cannot take wide scope.
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Acknowledgements
This paper supersedes Chaps. 5 and 6 of my dissertation (Xiang 2016) and the proceedings paper of SALT 29 (Xiang 2019). The analysis, especially the parts on deriving domain (non-)exhaustivity and the distribution of pair-list readings, has been substantially reworked. For extremely helpful discussion and comments, I thank Chris Barker, Gennaro Chierchia, Veneeta Dayal, Danny Fox, Jess H.-K. Law, Haoze Li, Floris Roelofsen, Ken Safir, two exceptional reviewers of Linguistics and Philosophy, and the handling editor Bernhard Schwarz. I also thank the audiences at ILLC in Amsterdam, MIT, Rutgers, and SALT 29. I am especially grateful to Christine Bartels for her incredible editorial comments.
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Appendices
Appendix A. A partition-based approach
Section 3 mentioned that the following LF, repeated from (18), suffers type-mismatch in most frameworks of question semantics:
Partition semantics is exempt from this type-mismatch problem. Groenendijk and Stokhof (1984: Chap. 3) initially analyze the pair-list \(\forall \)-question (115) as a partition of possible worlds grouped in terms of which boy watched which movie. In the derivation of this denotation, the quantifier every boy quantifies into an identity operation (of type t), which says that x watched the same movies in w and in \(w'\).
World grouping yielded by (116)
However, Groenendijk and Stokhof themselves are not satisfied with this account since it does not extend to questions with a non-universal quantifier. For example, the predicted meaning for the corresponding \(\exists \)-question (116) is not a partition (see also Krifka 2001). Thus, they ultimately pursue another family-of-questions approach using witness sets (footnote 15).
For a concrete illustration, consider a discourse with two boys a,b and two movies \(m_1\),\(m_2\). The four worlds vary by which boy watched which movie. \(w_{1}\),\(w_{2}\),\(w_{3}\) are grouped in one shaded cell \(C_{1}\): a watched the same movie in \(w_{1}\) and \(w_{2}\), and b watched the same movie in \(w_{1}\) and \(w_{3}\). Likewise, \(w_{2}\),\(w_{3}\),\(w_{4}\) all belong to the shaded cell \(C_{2}\): b watched the same movie in \(w_{2}\) and \(w_{4}\), and a watched the same movie in \(w_3\) and \(w_4\). In addition, \(C_{1}\) and \(C_{2}\) are distinct cells because neither boy watched the same movie in \(w_{1}\) and \(w_{4}\). The world grouping in Fig. 5 is clearly not a partition: \(C_{1}\) overlaps with \(C_{2}\) — they both contain \(w_{2}\) and \(w_{3}\). Moreover, from this world grouping, we cannot identify which movie any of the boys watched. For example, if \(w_{1}\) is the actual world, then \(C_{1}\) is the cell which the actual world belongs to; however, based on \(C_{1}\), we cannot decide on whether a watched \(m_1\) (as in \(w_{1}\) and \(w_{2}\)) or he watched \(m_2\) (as in \(w_{3}\)).
In addition, this analysis inherits the theory-internal problems with partition semantics. For instance, since partition semantics cannot explain the uniqueness effects of singular-wh questions (Xiang 2020), a partition-based account cannot explain the point-wise uniqueness effects in pair-list \(\forall \)-questions.
Appendix B. A question-embedding approach
Another intuitive and framework-independent way to solve the type-mismatch problem in quantifying into questions is to analyze matrix questions as covertly embedded questions (Karttunen 1977; Krifka 2001). The LF assumed by Karttunen (1977) is given in (117). Basically, whatever the overt question denotes, it is embedded within a t-type expression which can be quantified into.
This analysis crucially requires the quantifier in the embedded question to scope over the intensional embedding predicate ask. However, drawing on the limited distribution of pair-list readings in matrix questions and intensional question-embeddings, I will now argue that this scoping pattern is not available.Footnote 36
As discussed in Sect. 3 and explained in Sect. 6.4, only every/each-phrases can license pair-list readings for matrix questions. As for question-embeddings, Szabolcsi (1997b) observes a contrast between intensional complements and extensional complements. In particular, in embeddings with an extensional predicate (e.g., know, find out), plural \(\exists \)-quantifiers such as two of the boys may also license a pair-list reading. For example, in a pair-list context where each boy watched a different movie, (118b) can be uttered felicitously and interpreted with the following scopal pattern: ‘\(\exists 2 \gg \textsc {each}\gg \textsc {V} \gg \iota \)’ where ‘V’ stands for the embedding predicate.Footnote 37 As argued in Sect. 6.4.2, this reading can be derived from the LF in (119) (see also (86)): the \(\exists \)-quantifier takes wide scope relative to the embedding predicate, and its trace in the matrix clause is associated with covert each.Footnote 38
However, embeddings with an intensional predicate (e.g., ask, wonder) behave like matrix questions—only every/each-phrases may license pair-list readings in these embeddings. For example, in (120a,b) the uniqueness inference triggered by which movie must be interpreted between the embedding predicate and the quantifier: \(\textsc {ask} \gg \iota \gg \exists 2\).
The lack of pair-list readings shows that the LF (119) is not available for (120a,b). Szabolcsi (1997b) argues that intensional predicates create weak islands, which prevent the quantifiers in the embedded questions from taking wide scope. If this explanation is on the right track, the embedding structure (117), which requires the quantifier in the embedded question to scope over ask, should be infeasible.
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Xiang, Y. Quantifying into wh-dependencies: multiple-wh questions and questions with a quantifier. Linguist and Philos 46, 429–482 (2023). https://doi.org/10.1007/s10988-022-09366-x
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DOI: https://doi.org/10.1007/s10988-022-09366-x