A discourse model for überhaupt

The German particle überhaupt exhibits a variety of uses with seemingly unrelated meanings. Correspondingly, only partial and non-unified theoretical accounts have been proposed. I show how the various intuitions and ostensibly different meanings can be derived from a unified characterization of überhaupt as a move to a higher-level question under discussion. The account explains how überhaupt could correspond to a single word in German, and it provides additional support for questions under discussion as an important aspect of contexts.


Introduction
There has been great interest in recent years in studying the way that information is organized in conversation. New theories of discourse structure (Ginzburg 1994, 1995a, 1996, 2012, Groenendijk 1999, Groenendijk & Roelofsen 2009, Roberts 1996/2012, van Rooij 2003, and notions such as that of a discourse strategy or a question under discussion (Roberts 1996/2012, Ginzburg 1995a have led to advances in our understanding of focus (Büring 2003, Beaver & Clark 2008, anaphora resolution (Roberts 2003, Schoubye 2009, and speech acts (Roberts 2004). Ideas from discourse structure have also led to new insights for the analysis of discourse particles (McCready 2006, Davis 2009), as some particles are better understood if we move beyond the level of a single utterance and take into account the structure of the surrounding context. 1 In this paper I develop a unified account of the meaning and distribution of the German discourse particle überhaupt based on discourse structure. This particle has defied a unified analysis by other means in the past (Thiel 1962, Becker 1976, Harden 1983, and some have even claimed that überhaupt has no unified meaning (König 1983: 160). I argue that, if we treat the particle as engaging directly with discourse issues, then a unified analysis can be achieved.
The paper is organized as follows: Section 2 describes the various uses of überhaupt and their meanings. I then address problems with previous accounts of überhaupt in Section 3, in particular with the domain widening account of Anderssen 2006. In Section 4, I propose a unified account of überhaupt that can account for those various uses. Sections 5 and 6 explain links to syntactic issues and to focus theory, respectively. Section 7 shows that the intuitions of widening underlying Anderssen's account, in those situations where they arise, can be understood within my framework.
2 The use of überhaupt in German Native speakers of German, including myself, tend to ascribe to the particle überhaupt several distinct uses and meanings. What these various uses and meanings are is described in this section. I will restrict attention to sentences without embedded clauses, 2 and will consider only questions which are polar. In the rest of the paper, my goal is to demonstrate that all the meanings described can be captured in a formal, unified account based on discourse structure.

How überhaupt is used
2.1.1 Focused and unfocused überhaupt plus negative determiner in statements The most basic distinction between uses of überhaupt is given by its focus. When focused, überhaupt is used in a statement to indicate that it generalizes previous statements in the dialogue: Here the final sentence can be paraphrased as I actually don't drink alcohol. As in the focused case, it has the effect of terminating a line of inquiry by generalizing prior statements. However, in the unfocused case it plays an additional role: that of invalidating a presupposition. The presupposition 3 The focus in (C2.iv) falls on the main verb trinke. In the presence of unfocused überhaupt, this sentence does not allow any other focus structure. I will not put much emphasis in my exposition of the überhaupt usages on showing where the main stress/focus falls when überhaupt is not focused. This is because the puzzle of überhaupt has standardly been presented simply as one of understanding the relationship between the stressed and unstressed überhaupt. I therefore choose to present the issue starting from the same departure point. A tentative explanation of the larger focus issues is given in Section 6.

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Tania Rojas-Esponda in question here is not a logical presupposition of any particular utterance, but rather an implicit working assumption by conversant A that B drinks alcohol. 4 On the other hand, by uttering (C1.iv) interlocutor B does not signal whether or not he thinks that A held some particular assumption about whether he sells cake. B answers the questions about each type of cake in one swoop, but does not comment on what he thinks A's beliefs are. This is why (C1.iv) is permissible both in situations where B thinks A had some assumption and where he thinks A did not. The crucial thing is that, by uttering (C1.iv), B does not take any explicit stance on this point.
Unfocused überhaupt could have been employed in sentence (C1.iv) as well, with the same presupposition-challenging meaning (paraphrasable as We actually don't sell cake) that is illustrated in (C2). Conversely, focused überhaupt could be used in (C2.iv) as well. In that case, B would no longer be challenging an assumption or presupposition, but would simply be resolving the line of questions about alcohol a, alcohol b, alcohol c, all at once.
In what follows, I will mostly stick to (C2) when illustrating the use of unfocused überhaupt in declaratives and to (C1) to illustrate the use of focused überhaupt in declaratives, just to keep a simple association between uses of überhaupt and specific examples.

Focused and unfocused überhaupt in questions
In the above examples überhaupt appeared in statements resolving a line of inquiry. It can also appear in questions, both in focused and unfocused form, as in the following dialogues: 5 5 In this paper, dialogue glosses are omitted to preserve the flow of information. However, the glosses can be reconstructed straightforwardly from the translations provided. In the declarative utterances containing überhaupt the translations can be read directly as word-by-word glosses. The German interrogatives containing überhaupt differ from the English translations just in that they have the order verb-subject-überhaupt as opposed to the English subject-verb-überhaupt (überhaupt not being an English word, of course). In addition, the German sentences distinguish between formal and informal address in the second person. However, this distinction is not important for and does not interact with the usage restrictions on überhaupt. As an illustration, the glosses for (C3.v) and (C4.v) are provided below: A careful review of the uses of überhaupt must make mention of its discourse requirements. In fact, there are precise conditions that must be met in order for a sentence with überhaupt to be felicitous. One such condition is that überhaupt cannot occur in an utterance that constitutes a direct reply to a question. Observe that in none of the conversations in 2.1, the utterance containing überhaupt directly answered the immediate (last) question asked. In conversation (C1), the answerer does not respond directly to the question about chocolate cake, but opts to make a more general statement instead. In conversation (C2) again the statement that B does not drink alcohol is not a direct answer to any question asked. And finally, in conversation (C5) the last direct question asked is about the people in Rome, and, while B answers this question directly (Die Leute waren sehr nett.), the utterance containing überhaupt (Es war überhaupt F (alles) sehr schön in Rom.) is a more general statement that does not constitute a direct answer to the question asked by A. If one uses a definition of relevance wherein a reply to a question q is relevant exactly when the reply helps resolve q without providing any additional information not requested by q, then the utterances involving überhaupt are not strictly relevant to the last question asked. This is a general restriction on überhaupt sentences, a point which will be made more formal in 4.2.

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Section 2.1 also illustrated that überhaupt is used only when the speaker is settling a sequence of questions. The word 'settling' is used here both for the case that a speaker answers an entire sequence of questions in one swoop (resolves the questions), as in (C1), and for the case that a speaker rejects or challenges the validity of a sequence of questions (by calling into question necessary preconditions for these questions), as in (C2). If we entertain the idea that the sequence of questions serves to address some overarching larger question, then we could say that utterances with überhaupt turn directly to this superordinate question, answering it or challenging it depending on whether überhaupt is focused or unfocused, respectively.
Due to the requirement that utterances with überhaupt must settle a sequence of questions, conversation-initial uses of überhaupt usually have the effect that the hearer must fill in the requisite context and accommodate the use of überhaupt by supplying a suitable background of implicit questions. How difficult this is depends on the nature of the überhaupt utterance and on the amount of common ground between the speakers (as it relates to the überhaupt utterance).

Examples for contextual restrictions on überhaupt
In Section 2.1 we saw felicitous uses of überhaupt in conversation. The examples in 2.1 satisfied the discourse requirements of 2.2.1: that an utterance involving überhaupt must settle a sequence of questions, and that it must not be a direct reply to a question asked. In this section, we give examples of infelicitous uses of überhaupt alongside felicitous ones to illustrate that these discourse conditions are in fact necessary.
It was mentioned in 2.2.1 that out-of-the-blue uses of überhaupt usually are cases where the requisite background is being accommodated. Thus, with no special common ground or accommodation the following utterance is odd: Example 6 (At the beginning of a conversation): (i) #Trinkst du überhaupt/überhaupt F Alkohol?
#Do you drink überhaupt alcohol?
If, however, a group of friends were looking at the different wines available at a wine tasting, it may be natural for friend A to ask (E6.i) of friend B. This would convey that A has realized that the issues of whether B wants to taste different wines can be shortcut in two different ways. If A uses focused 1:7 Tania Rojas-Esponda überhaupt it is to see if all answers to the questions about different wines (Do you want to taste wine a? Do you want to taste wine b? etc.) might be No, in other words, by using focused überhaupt A is seeking to find an exhaustive answer to all these questions at once. By using unfocused überhaupt, A signals in addition that she is challenging some prior assumption (by A or someone else) that B drank alcohol.
Below we give a number of example conversations, followed by both felicitous and infelicitous uses of überhaupt. We will see that the general rule is that überhaupt is felicitous whenever the sentence containing it settles a sequence of questions.
Consider the following conversation: The follow-ups using überhaupt that are felicitous are the last two, (C7.v)-e and (C7.v)-f. These utterances include instances of unfocused überhaupt. Utterance (C7.v)-f states that B does not even drink alcohol. Because this was a necessary condition for B wanting to drink wine, beer, or any number of different alcoholic drinks, the uttering of (C7.v)-f serves to settle the line of interrogation begun by A in (C7). In this case, B terminates the line of interrogation by showing that it may have been based on an incorrect 1:8 A discourse model for überhaupt assumption. 6 The corresponding question (C7.v)-e with unfocused überhaupt (paraphrasable as Do you even drink alcohol?) serves to elicit the statement that B does not drink alcohol, which would likewise settle the sequence of questions in (C7), again by showing that a necessary precondition was not met. Other types of utterances, such as questions continuing the line of interrogation, like (C7.v)-c, or direct answers to an uttered question, like (C7.v)-a, are infelicitous if they include überhaupt, although they are felicitous without überhaupt-see (C7.v)-d and (C7.v)-b. 7 Similarly, in the following conversation only those uses of überhaupt that occur in a sentence that settles a sequence of questions are allowed. The utterance (C8.v)-f, which contains focused überhaupt, settles the sequence of questions of whether B sells a-cake, b-cake, c-cake, . . . , while the matching question (C8.v)-e is geared towards eliciting an utterance that settles whether B sells a-cake, b-cake, c-cake, . . . . 8 Once again, simple followup questions that continue the same line of interrogation-like (C8.v)-c-or direct answers containing no extra information-such as (C8.v)-a-cannot felicitously include überhaupt. 9 Again, in the conversation below, the utterances that may include überhaupt are those that settle the line of interrogation pursued by A in (C9) regarding Rome's weather, Rome's food, Rome's people, and so on. Among the choices listed, those are the utterances (C9.vii)-f, which says that everything was nice in Rome, or the corresponding question (C9.vii)-e that tries to elicit the answer that everything was nice in Rome. 8 These uses of focused überhaupt are possible both when there was an assumption that B sells cake and when there was no such assumption. Contrary to unfocused überhaupt, the use of focused überhaupt therefore does not signal that an assumption is being challenged.
9 I used the follow-ups to (C8) to illustrate the use of focused überhaupt, while I used the follow-ups to (C7) to illustrate the use of unfocused überhaupt. However, the setup of these conversations is identical, they are kept apart just to maintain an easy phenomenon-toconversation correspondence.

A discourse model for überhaupt
The examples above show that the particle überhaupt is used only within very specific discourse structures and that usually a context of a very particular kind is required. Since überhaupt does not make any truth-conditional contribution, I take these strict distributional requirements of überhaupt to be the key to the meaning of überhaupt. This is rigorously formulated in 4.4, the central part of this paper.

Two uses of überhaupt: Resolving and Doubting
I stated that all felicitous sentences with überhaupt in the examples above have the property that they settle a sequence of questions. For instance, the sequence of questions of whether or not B sells marble cake, chocolate cake, or any number of other cakes is resolved when B utters Ich verkaufe überhaupt F keinen Kuchen. B resolves the whole sequence of questions regarding a-cake, b-cake, c-cake etc. in one swoop. I will call this kind of settling a resolving use of überhaupt. Now consider the felicitous statement Ich trinke überhaupt keinen Alkohol in (C7). This statement also settles the sequence of questions regarding alcohol a, alcohol b, alcohol c, . . . . However, this time B settles this sequence of questions by pointing out that a certain precondition, namely the condition that B drinks alcohol, does not hold true. I will label this kind of settling move a doubting use of überhaupt. The corresponding resolving and doubting questions, i.e., Verkaufen Sie überhaupt F Kuchen? and Trinkst du überhaupt Alkohol? are the natural question counterparts to the previously mentioned declaratives and I will call them resolving and doubting, respectively.
Notice that, in our examples, the focused uses of überhaupt correspond to resolving moves, while the unfocused uses correspond to doubting moves. This distinction between doubting and resolving moves will be made explicit in Section 4.

Summary
A summary of the main types of überhaupt just discussed is given in Table   1, which also indicates the presupposition 10 behavior of each type.
10 The term 'presupposition' will be used in a wide sense here, both for presuppositions in the classical sense, as well as for an implicit assumption from previous discourse. What we call There is previous work describing the uses and meanings of überhaupt. König (1983) classifies them into: group A, which encompasses focused überhaupt plus universal quantifier or scalar predicate; group B, which includes other uses of focused überhaupt, such as focused überhaupt plus negation or uses in polar questions; and group C, unfocused überhaupt. 11 In the examples that König provides for group A, the particle überhaupt could be translated by English generally; he describes the meaning as one of generalization, which is in accord with my description of (C5). König's group B features examples where überhaupt can be captured by English at all, see conversation (C1). Finally, English actually or even are suitable paraphrases for the group C uses of überhaupt. König characterizes this use as one that challenges a precondition, just as I claimed for (C4). He posits, using etymological arguments, that the use of überhaupt plus universal quantifier (or scalar predicate) is the primary one. He likens groups A and B by saying that they presuppositions here will not always be a syntactically visible presupposition; however, in the tree structures that I will define in 4.2, these will be true presuppositions in the classical sense.
11 König also mentions a "surprise" use of überhaupt, but considers it marginal. He describes it as being used when someone has a sudden insight. One of his examples is the utterance Ach überhaupt, da fällt mir noch was ein (roughly: Oh überhaupt, I have an idea).

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both are used in an utterance that represents a generalization. König further points out that groups B and C occur in the same syntactic environments, after negation or in polar questions, although they place different restrictions on the previous discourse. König stops short, however, of providing a unified meaning for überhaupt, and in fact claims that the meaning of these different groups cannot be derived from a single basic meaning (König 1983: 160). Like König, Thiel (1962) and Becker (1976) identify one particular usage of überhaupt (and the corresponding meaning for this usage) as primary, and the other meanings as derivative. They also find no single meaning to be applicable to all uses of überhaupt. Interestingly, these authors vary in terms of which usage they regard as the basic one. For König and Thiel, a certain meaning conveying generalization or exhaustivity, corresponding roughly to König's group A, is the primary one. On the other hand, Becker regards the use of überhaupt to challenge a claim (König's group C) as the primary use. Harden (1983) offers a characterization of überhaupt in terms of features (such as extension, casual, aggressive, emphasis, . . . ), which are either present or absent in different syntactic or conversational environments. While it is possible to collect all the possible überhaupt features (not all of which are active in all contexts) in a single matrix, this corresponds more to taking the union of all the possible meanings rather than representing a single meaning for überhaupt that unifies the different uses. Anderssen (2006) develops a broader theory of überhaupt in terms of domain widening. However, the usages of überhaupt that he addresses still form just a proper subset of the types described by König. For instance, Anderssen (2006) remarks that his domain widening theory does not rigorously account for the use of überhaupt that challenges a claim or assumption (Anderssen 2006: 7).
In the remainder of this section I will discuss the meaning that Anderssen proposes to account for and η, two of the überhaupt types I laid out in Section 2. Anderssen points out that his domain widening account cannot fully explain types α and β, namely unfocused überhaupt (Anderssen 2006: 7). 12 Anderssen's account (Anderssen 2006) of überhaupt incorporates elements from a group of theories of English any that claims that any consists of a domain widener plus existential quantifier. Under these accounts, any is thus semantically complex. In Anderssen's analysis, the German überhaupt is an overt manifestation of this semantic complexity. He claims that it 12 Anderssen does not refer to this type of überhaupt as the unfocused one, but it is clear that these are the uses he has in mind.

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Tania Rojas-Esponda corresponds just to the first part of any, namely the domain widening part, and argues that this is evidenced by the fact that überhaupt, unlike any, can combine with a determiner. To describe when überhaupt is licensed, Anderssen relies on a group of theories for NPIs that argue that domain widening is allowed only when it results in strengthening (Kadmon & Landman 1993, Krifka 1995, Lahiri 1998, Chierchia 2004. Below are the expressions that Anderssen, based on Chierchia 2004, provides as analyses for the following two sentences. I have added focus marking on überhaupt because sentence (E10.ii) below without focus has a different sense-namely, one paraphrasable as I actually don't have potatoes., which would not be adequately captured by the formulas in (E10). In what follows, I will be distinguishing focused from unfocused überhaupt even though Anderssen did not make this distinction.
Example 10 Anderssen argues that überhaupt is licensed in this case because the second of these expressions entails the first. He uses a similar strengthening argument to explain why überhaupt can occur with a universal quantifier. The argument here is that a positive sentence with überhaupt alle (überhaupt all) will entail the corresponding sentence without überhaupt if überhaupt is analyzed as a domain widener. I believe that Anderssen's account is problematic in two respects: its distributional predictions for überhaupt, and its characterization of überhaupt in terms of domain widening. I begin by discussing the distributional predictions.
Anderssen's theory predicts that the sentence Ich habe überhaupt F keine Kartoffeln is always felicitous, anywhere in discourse. This follows from the fact that it entails the (in his account) weaker sentence Ich habe keine Kartoffeln and that thus überhaupt satisfies the strengthening requirement. But I illustrated in Section 2 that this is not correct. The reader may refer to Section 2 for the distributional requirements for sentences involving focused überhaupt plus negation. I limit myself to one illustrative example here. Consider the following exchange in English, which roughly captures the intuitions behind Anderssen's definition (E10.ii): Anderssen points out "distributional similarities" between phrases with überhaupt and any (Anderssen 2006: 1), but ultimately does not discuss the distributional requirements of überhaupt in their own right. As a result, important distributional differences are missed. For instance, Anderssen seems to suggest that in negative sentences überhaupt kein and not any pattern in the same way. I believe this to be incorrect. The stringent restrictions of überhaupt on the preceding discourse were outlined in the previous section. Without going into a lengthy discussion of English any, I claim that there are many cases where any in a negative sentence can be used, but where überhaupt plus negation would not be appropriate. An important reason for this is that very often negated sentences with any represent the canonical, or least marked, way to negate a given positive statement. A sentence like I don't have any friends is the canonical negation of I have friends, whereas Ich habe keine Freunde (without überhaupt) is the canonical negation of Ich habe Freunde. The corresponding sentence with überhaupt, i.e., Ich habe überhaupt keine Freunde has much more stringent requirements on the preceding discourse, as outlined in Section 2. A minimal difference is that Ich habe überhaupt keine Freunde cannot be used as a direct answer to the question Hast du Freunde? (Do you have friends?), whereas the corresponding sentence with any could be used in such a situation. Anderssen's analysis, however, is restricted to the scope of a single sentence. It does not take into account the distributional requirements of überhaupt, but in fact makes the incorrect prediction that focused überhaupt plus negation or universal quantifier can occur at any point in a discourse.
The second problem with Anderssen's account is its range of applicability: I claim that the domain widening account only applies to a specific subclass of examples, even once we have restricted our attention just to types and η. Here, the notion of "Kind" has not been widened-in part because it is a fairly precise notion, but also because it is not necessary in order to make sense out of the conversation. The interlocutor is simply stressing that she has no child. The use of überhaupt does not produce a widened or loosened notion of the word child. However, that is exactly what is expressed by the formula in (E10).
In the next section I will account for the type and type η uses of überhaupt using a discourse approach. In addition, I will show how the discourse approach can explain types α, β and δ, which Anderssen does not explain. In particular, I will demonstrate how the gap between focused and unfocused überhaupt can be bridged, something that König and Anderssen indicated their accounts did not do, at least not rigorously. Finally, in Section 7, I will show that the domain widening intuition, in those cases where it is in fact appropriate (which is a proper subset of where Anderssen claims domain widening applies), can be recovered using the discourse account of this paper. The advantage of the discourse-based approach is that it continues to work even in non-domain-widening instances of the usages discussed by Anderssen and, contrary to domain widening, it also covers the other types of überhaupt described in Section 2, most importantly unfocused überhaupt.

A discourse model for überhaupt
The central claim of this paper is that the inclusion of discourse structure allows a unified account of überhaupt. The study of discourse structure begins with the assumption that discourse serves a purpose, and is organized to serve that purpose. The simplest manifestation of this is the idea that, at any given point in a conversation, there is some issue being addressed or 1:16 A discourse model for überhaupt pursued. If we think of this issue as a question that we are trying to answer, we arrive at the notion of a question under discussion, or QUD for short.
As a simple example, if Sam says Jenny ate a salad and Mary ate pasta, he might be addressing the QUD What did your sisters eat for lunch? This QUD is a simple question that Sam can answer using a single utterance. For the most part, however, the goal of a conversation cannot be articulated in terms of a simple question, and even if the QUD has been articulated it may not be easily answerable. Such complex issues can be addressed in stages, via sub-issues manageable using the vocabulary available. These ideas have been formalized by various authors, e.g., Groenendijk (1999) and Roberts (1996Roberts ( /2012, and will play a role in the formalism of Section 4.4.
Section 4.1 lays out the proposal for überhaupt informally. Afterwards, in Sections 4.2 and 4.3, we develop the formalism we will use to make our formal proposal for the meaning of überhaupt in Section 4.4.

Informal description
The core of our proposal is that überhaupt marks an utterance as a move to a higher question under discussion. In a conversation, we may be addressing questions concerning sub-aspects of a larger question Q. If we want to doubt the validity of the larger question Q, or if we want to resolve Q in one swoop, we can use überhaupt to do so. Let us consider the following example: , particular questions about possible ways to solve the problem. If many methods have failed, at some point an interlocutor might ask whether the original question Q was valid at all or, in particular, whether the presupposition that there was some way the problem could be solved is valid. When we address this higher-level concern, it is the use of überhaupt that signals that we are abandoning the current level (regarding whether method a works, method b works, method c works, . . . ) and moving to a higher question under discussion. This example used unfocused überhaupt, so it corresponds to what we dubbed a doubting move: i.e., it terminated a line of conversation about what methods work to solve the problem by raising a higher-level issue of whether or not the line of inquiry is even sensible. 13 (If the problem has no solution, then going through various methods and asking if they will work for this problem is pointless.) Here is another example, this time featuring focused überhaupt: In this conversation, the larger question being addressed might be What were things like in Rome? The actual uttered questions address only aspects of this larger question, namely if the weather was nice, the food was nice, etc. The sentence with focused überhaupt moves from this level to a higher level question, What were things like in Rome?, and then immediately resolves it. Another way of saying this is that the question with überhaupt resolves all 13 Magdalena Kaufmann makes the intriguing remark that in this conversation überhaupt can be used in its focused form as well, while still producing the same conversational doubting effect, which she suggests may be due to the binary nature of the adjective lösbar (solvable). This is a fruitful question for further investigation. Interestingly, the conversational effects seem to remain distinct in the corresponding declarative utterances. That is, in the sentence Die Aufgabe ist überhaupt F /überhaupt nicht lösbar only the version with unfocused überhaupt signals explicitly that some previous assumption is being challenged.

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A discourse model for überhaupt the possible questions regarding subaspects of Rome (was aspect a nice?, was aspect b nice?, . . . ), by answering each one with Yes. But by resolving all subquestions of the higher question we get the answer to the higher question.
The notion I will require for my treatment of überhaupt is that of a hierarchy of questions. I will represent these hierarchies as trees, which allows me to talk about higher (more general, complex) questions and lower (more specific, simple) questions under discussion. Here I will motivate the basic idea behind my account of überhaupt, putting off the formal definitions until Section 4.2.
There are two properties it is important for such a hierarchy to have.
(a) Daughter questions are intended to investigate parts of their parent question. As such, any possible answer to a daughter question should narrow down the possible answers to the larger question, reducing them to a strict subset.
Since in structuring discourse an interlocutor may make working assumptions (which may or may not be valid), we allow daughter questions to make assumptions not present in the parent question. Thus the parent question may have possible answers which are not consistent with the assumptions of a daughter question.
For example, if A meets B on the street and is interested in knowing about the person (the larger issue), A may ask the question What is your job? Yet B might turn out to be unemployed. A need not be making the assumption that B has a job; more likely, he finds it more efficient than asking What is your job, if you are employed?, and accepts the risk of being told that B does not actually have one.
Conversely, an interlocutor can make a working assumption, but that working assumption need not be reflected explicitly in the uttered question. For example, in Conversation (C7) speaker A may be making the working assumption that B wants alcohol, yet for the sake of brevity he would not bother to specify he is making that assumption. Therefore we only wish to require the possible answers to the subquestion to narrow down the answers to the parent question given the assumptions of the parent question.
(b) In this model, the purpose of daughter questions is to get at all the information necessary to answer their parent question by breaking it down into manageable pieces. However, if the daughter questions are not

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Tania Rojas-Esponda capable of providing a complete answer to the parent question then they will be unable to satisfy their goal of resolving it. Therefore, in a good strategy the daughter questions, once they have all been answered (or found to make assumptions inconsistent with the facts), must provide a unique answer to the parent question.
In other words, the answers to every daughter question must combine to answer the parent question uniquely.

Developing the formalism
Before embarking on a characterization of überhaupt we will require a formal model for discourse strategies that allows us to capture a hierarchy of questions under discussion and felicitous ways of moving between them. Various formal discourse representations have been proposed (Roberts 1996/2012, Carlson 1983, Groenendijk 1999, Büring 2003. The ones that have most influenced my treatment here are those of Groenendijk (1999) and Roberts (1996Roberts ( /2012). Groenendijk described a formal framework for discourse wellformedness that is, in a sense, a formalization of Grice's conversational maxims. He used the characterization of a question as a partition on the set of worlds, as used in (Groenendijk & Stokhof 1982, 1984, Lewis 1988. This approach models a question as the set of its possible answers, and identifies each answer with the set of possible worlds in which it holds true. Groenendijk formalized the Gricean maxims in terms of relationships between such partitions. In (Groenendijk 1999), every utterance, whether it be a question or statement, is incorporated into the interlocutors' common ground. These update requirements are loosened in Roberts' paper on interrogation strategies (Roberts 1996(Roberts /2012, where utterances (or, as she calls them, moves) may or may not be accepted by the interlocutors. Roberts puts forward the notion of interrogation strategies, i.e., strategies that try to address some superquestion Q in terms of more specific questions, which in turn may be addressed via subquestions, and so on. In what follows, I will define what I call an S-tree, a specific type of strategy tree that roughly satisfies some of Groenendijk's formalized Gricean principles and also has some resemblance to Roberts' strategies. 14 14 A certain type of strategy tree (called d-tree) was defined by Büring (2003). Büring constrains the family of admissible d-trees in two types of ways. The first type of constraints is based solely on the propositional content, while the second type of constraints, Givenness and the CT-Condition, concern the exact word choice and focus marking of an utterance.

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A discourse model for überhaupt Let W be the set of possible worlds. We will think of a question as a partition of some subset of W . More formally, we can define a question to be a binary relation on W which is symmetric and transitive. Equivalently, a question is an equivalence relation on a subset W ⊆ W . A statement corresponds to a set of worlds, which in this framework can be viewed as a partition with just one cell (equivalence class). Thus we can model a statement as a question q such that, whenever w, w ∈ q and w , w ∈ q, then w, w ∈ q. Below we will use the word 'question', but it is understood that a question with just one cell (equivalence class) is in fact a statement. The question definition used here corresponds to what is called a question relativized to an information set in Groenendijk & Stokhof 1984, or later a structured context in Groenendijk 1999 In order to capture the discourse structures that license the use of überhaupt questions or überhaupt statements, a few additional definitions are needed: (1)q, the completion of q, is the question obtained from q by adding a cell containing all worlds not contained in q. All preexisting cells are left intact.
Definition 1 (completion). The completionq of q is When applied to a statement (a question with a single cell), completion has the effect of turning it into a polar question. For instance, if p is the statement The sun is shining, thenp is the question Is the sun shining? 15 When completion is applied to a proper question (one that is not a statement), the effect is that of lifting any presuppositions of this question. For instance, the question q = What presents did you buy for Büring's tree restrictions that are based just on the propositional content of an utterance are formulated in terms of probabilities and are rather more permissive than the conditions for S-trees, which are formulated in terms of partitions on the set of possible worlds (Büring 2003: 20-21). The set of possible d-trees is thus constrained primarily via restrictions on form, whereas S-trees are defined solely based on information content, using a possible worlds model.
15 For statements with presuppositions, this is a slight simplification. We could define completion with respect to a subset of W to keep track of presuppositions in polar questions. For the present purposes, this simpler version where the completion is always taken with respect to the whole set W is sufficient, as we do no further computation with it.

Tania Rojas-Esponda
Sam's birthday? could be said to presuppose that the addressee obtained at least one present for Sam. Completing this question-i.e., forminḡ q-lifts this presupposition and yields the question What presents, if any, did you buy for Sam's birthday?.
(2) is a binary operation that is similar to question intersection, except that some cells lying outside the intersection are also included.
Definition 2 (combination). The combination q 1 q 2 of q 1 and q 2 is The combination Q of the n questions q 1 , q 2 , q 3 , . . . , q n , written n i=1 q i , corresponds to the most general question whose answer can be determined by answering each of the questions q 1 , q 2 , . . . , q n . 16 Take for instance the two questions, q 1 = 'What are your girls' names?' and q 2 = 'What are your boys' names?' The first question q 1 presupposes that the addressee has daughters, and the second question q 2 presupposes that the addressee has sons. When we form the combination Q = q 1 q 2 we obtain the question What are your children's names? This latter question still has a presupposition, namely that the addressee has children, but it does not presuppose anything about their gender. Moreover, if the person answers the first two questions (by either providing their children's names or by indicating that they in fact don't have sons or don't have daughters), then the answer for the combination Q (or the fact that Q has no answer) will be fully determined.
(3) ≺: q ≺ Q if the cells of q are unions of cells of Q; this is similar to saying that q is a subquestion of Q, but entire cells can be thrown out (i.e., additional presuppositions can be made).
Definition 3 (narrowing). If q and Q are questions in W × W , then we say that q narrows Q (q ≺ Q) if and only if, for every pair of worlds v, w ∈ W × W , 16 While is defined as a binary operation, it is associative and symmetric so the order in which we combine a sequence of questions is irrelevant.

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A discourse model for überhaupt The fact that q narrows Q tells us the following: Firstly, any answer to q provides a partial answer to Q (that is, some cells of Q are eliminated). 17 Secondly, if we know the full answer to Q, then we know the full answer to q. This intuitively means that q cannot raise issues that are not raised by Q.
(4) The smashq of q is the statement that q is answerable, i.e., it is the question with one cell containing all worlds contained in q. Intuitively, it is the presupposition implicit to q.
Definition 4 (smash). The smashq of q is defined bŷ (5) Therefore,q is the question is q answerable? It contains two cells: one with all the worlds contained in q, and another with all the worlds not in q.
Now that the relevant relations between questions are defined, I turn to the definition of the strategy trees that will allow a formalization of überhaupt. These trees, let us call them S-trees, will be trees of questions in which the children of each node combine to form that node and each node narrows down its parent node (see terms above). The leaves are traversed from left to right. At any point, the ancestors of the uttered leaf are the questions under discussion.
Definition 5 (S-tree). A strategy tree, or S-tree, is a question tree 18 satisfying the following: i. Every child q of a node Q satisfies the relation q ≺ Q.
ii. The children q 1 , . . . , q k of Q satisfy Q = k i=1 q i ∩Q.
The S-trees just defined represent a hierarchy of questions that capture a possible organization of a conversation. Some nodes in the tree may correspond to questions that were actually uttered, while others correspond 17 Or implies that Q has an invalid presupposition, if the answer lies outside of Q. This can happen because the definition of narrowing does not prevent a question q that narrows Q from containing worlds not contained in Q, in other words, from having fewer presuppositions than Q.
18 I assume that the questions in the tree are proper questions, not statements.

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Tania Rojas-Esponda to implicit or tacit questions. It is not claimed that a collection of explicit questions forms part of a unique strategy tree, and in fact there is usually more than one strategy tree that is compatible with the uttered questions. We also do not update the context by the implicit questions, but interlocutors can have different degrees of commitment towards the presuppositions of implicit questions in the tree. Note that a child node may lack cells from the parent node or include cells not in the parent node. This is desired, as we want to allow daughter nodes both to have additional presuppositions that the parent question does not have or lack presuppositions that the parent question has (see (T19) in Section 4.4).
For later reference, it will be useful to give a name to all the S-trees compatible with a given set of utterances U . Let us denote this set by Stra(U ). Different interlocutors participating in a conversation with utterances U may have different S-trees (corresponding to different views of how the conversation is organized), but each of their S-trees must be in Stra(U ).
The definitions made above are further motivated and exemplified in 4.3.

Relation between strategy trees and discourse
It is important to emphasize that trees satisfying the properties of the previous subsection form a very natural structure. The principle underlying them is simple: discourse has a hierarchy of issues which proceed from more complex at a higher level to simpler at a lower level, with the questions at a lower level providing partial answers to the questions which subordinate them.
In the course of a long and involved conversation the interaction between strategy and discourse will be complex, but there is a simple model that can be used to generate discourse from strategy trees which will suffice in many exchanges. This model characterizes discourse as uttering a sequence of nodes within the tree, with the sequence of traversal determined by the answers the interlocutor receives after each question.
Definition 6 (Rules of traversal). An S-tree can be traversed as follows: We may proceed from a node to a sister node or to a child node. 19 However, 19 As the idea is that the daughter nodes q 1 , q 2 , . . . , q n of a node Q serve to break down Q into less complex questions, a move from a mother to a child node happens only if we have not received a full answer to the mother question yet.

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A discourse model for überhaupt we may only move to a parent or ancestor node if we are doing one of the following: (i) Resolution: We are resolving the parent node Q by providing a full answer to it. Or we are requesting a full answer by raising a possible full answer as a polar question. (ii) Doubting: We show that the parent node Q is not answerable, by stating the negation of its presupposition, ¬Q. Or we raise the presupposition as a polar question, utteringQ.
The four types of utterances just mentioned, a full answer to Q , the polar question version of a full answer to Q, ¬Q andQ, can allow us (in the case of the questions, only if they elicit an answer that is an ascending move) to move upward in the tree. These are the only ways that we may move upward in the tree. I will term these ascending moves.
We also impose one additional requirement, namely that an ascending move not be a relevant reply (whether in proposition or question form) to the immediately preceding question. Relevance for propositions is defined here as in Groenendijk 1999: 115, and expanded to questions.
Definition 7 (Relevance). A proposition or question q 2 is said to be a relevant reply to a question q 1 if the intersection of q 1 and q 2 can be obtained from q 1 only by removing an integer number of cells (0 or more).
Definition 7 captures the intuition that an ascending move represents a move away from the current level of inquiry in the tree to a higher level.
Thus the full definition of an ascending move is: Definition 8 (Ascending move). An ascending move is a move (or, in the case of questions, an attempt at such a move) up an S-tree that (i) respects the rules of traversal.
(ii) is not a relevant reply to the immediately preceding question traversed in the tree, in other words violates the requirement of relevance. Suppose q 1 , q 2 , . . . , q n are the questions at our current level in the tree, and that the question Q is their parent node. Then at this point the possible ascending moves can be classified into four logical possibilities: ¬Q,Q (the doubting moves) and n i=1 ans(q i ) , n i=1 ans(q i ) (the resolving moves)

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Tania Rojas-Esponda where for each i, ans(q i ) represents some choice of answer to the question q i and n is the number of sibling questions at the current level. 20 It is important to understand that S-trees are a natural discourse structure, which have no a priori relation to überhaupt. In an S-tree, each question is divided into a collection of related questions which, when answered, provide just enough information to answer the original question. This reflects a common situation in discourse where, for various reasons, a primary question of interest is investigated by dividing it into smaller pieces. 21 An example of this principle is provided by the following tree:

Tree 16
Who brought what to the party?
What did John bring?
Did John bring pizza?
Did John bring soft drinks?
What did Lisa bring?
Did Lisa bring pizza?
Did Lisa bring soft drinks?
The intuition for an S-tree is that the daughter nodes of a question Q are questions that are simpler to answer as they are more coarse-grained than Q. In tree (T16) each cell of the question What did John bring? is a union of (one or more) cells of the question Who brought what to the party?. In tree (T16) the daughter nodes are subquestions of the mother nodes in the sense of Groenendijk 1999. General S-trees are somewhat more permissive, however. Thus, in an S-tree a daughter question may contain a presupposition not contained in the mother question or it may lack a presupposition that was present in the mother question.
If a person offers someone a range of alcoholic drinks, each time by asking a simple polar question with no presuppositions (e.g. Do you want wine?), they may be doing this in pursuit of the question What is the (one) alcohol you want?. That is, even though none of the uttered questions actually express any presupposition, a line of interrogation may be serving a larger question 20 In other words, ans(q i ) is a proposition corresponding to one of the cells denoted by the question q i .
21 Which particular strategy people use for this subdivision, i.e. which S-tree is ultimately used, is partly influenced by factors such as contrastive accent. See Büring 2003, for example.

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A discourse model for überhaupt that does have a presupposition or assumption. Using an S-tree, we can make these presuppositions overtly visible in the hierarchy.

Tree 17
What is the (one) alcohol you want?
Do you want wine?
Do you want beer?
Do you want vodka?
Similarly, it may happen that we pursue a question by asking daughter questions that have presuppositions not present in the mother question. This is exemplified below:

Tree 18
What are the names of your children?
What are the names of your daughters?
What are the names of your sons?
In an S-tree, daughter nodes are questions that can be asked in lieu of addressing a mother question directly. This is done when the daughter nodes are simpler to answer. This is to be distinguished from the case when we ask a follow-up question that is more specific so as to obtain additional information. If, for instance, I asked the question q 1 What vehicle do you use for your commute? and get the response car I could ask the follow-up question q 2 What kind of car do you commute with?. This latter question q 2 is more specific in that it asks for finer-grained information, the type of the car. However, q 2 does not narrow q 1 because q 2 is obtained from q 1 by taking the cell corresponding to the answer car and subdividing it into several cells, one for each car type. 22 22 In fact, in this case q 1 narrows q 2 . This is a borderline case. While the answers to q 1 are coarser than those to q 2 , asking q 1 with the goal of answering q 2 is not useful. I decided to keep the definition of S-trees as simple as possible. There are conversational principles that would keep us from using q 1 as a way of getting at the answers of q 2 . However, we could easily expand the definition of an S-tree to prevent this case by requiring, in addition to the other constraints on S-trees, that each daughter question q of a mother question Q have the property that q ∩Q ≠Q. This would prevent us from including trivial daughter nodes, i.e. ones the answering of which does not bring us closer to finding the answer to the parent question. I am grateful to Jan Anderssen for calling my attention to this type of example.

Main result: a formal proposal for überhaupt
We are now ready to propose a meaning for überhaupt. When überhaupt can be used felicitously and what it means are two notions that are intertwined. I will use the letter 23 q to denote either a question or a declarative sentence. Let q \ ü denote the same utterance but with überhaupt removed. This is always a legitimate utterance of German, though it has different distributional properties in discourse. Say the set of utterances so far in the conversation is U . Then Stra(U ) is the set of all compatible strategy trees. The tree "favored" by each interlocutor i after the set of utterances U is denoted S(i, U ). It is an element of Stra(U ). The tree S(i, U) represents i's view of how the discourse is organized.

Main Claim
The meaning of überhaupt: After utterances U, interlocutor i may felicitously utter a sentence q which includes überhaupt if and only if q \ ü is an ascending move in S(i, U ) ∈ Stra(U ). Thus the presence of überhaupt in q signals that q \ ü is an ascending move in S(i, U ).
We know that there are exactly two types of ascending move in an S-tree, doubting moves and resolving moves, each of which can appear either in question or statement form. Thus there are the two doubting expressions ¬Q andQ, and the two resolving expressions n i=1 ans(q i ) and n i=1 ans(q i ). In Section 2, we gave examples suggesting that unfocused überhaupt corresponds to a doubting move, while focused überhaupt corresponds to a resolving move. This leads us to the following formulae for überhaupt.
Proposal What is the (one) drink you want?
What is the soda you want?
What is the juice you want?
What is the alcohol you want?
Assume, for simplicity, that wine, beer and vodka are the only types of alcohol that exist. Then the combination of the three daughter questions intersected with the presupposition of the parent, equals the parent question Q (What is the alcohol you want?). We may exit Q by asking Möchtest du überhaupt Alkohol? or by stating Ich möchte überhaupt keinen Alkohol.
When the presupposition of an utterance is spelled out, as in the example below (Is the alcohol you want wine? as opposed to just Do you want wine?), then an überhaupt question or statement could be uttered after just a single utterance. ¬p i make sense for either of the trees above. Thus the formalism explains why, when focused überhaupt is used, the speaker is taking no stance as to whether there was a presupposition. Thus, while a sentence using focused überhaupt may go against a previous presupposition, the fact that someone utters a sentence with focused überhaupt does not mean they are claiming there was a presupposition.
Finally, consider the tree below: Tree 25 What were things like in Rome?
Were the people in Rome nice?
Was the food good?
Was the weather nice?
. Tania Rojas-Esponda no reference to speaker alternation, and could be applied equally well to any organized monologue.
Moreover, even if we're analyzing a text without questions, we can extract the necessary structures using Wittgenstein's assumption (Wittgenstein & Anscombe 1953) that any statement p corresponds to raising the question of whether or not p is true, and then resolving it.
5 Syntactic and discourse constraints on überhaupt I believe the main issues surrounding überhaupt to be the following: (I) The construction of a syntactically well-formed überhaupt-sentence: How can überhaupt be incorporated into sentences, what governs its syntactic combinatorial behavior? (e.g. why can überhaupt appear with negation, with universal quantifiers and with scalar predicates?) (II) The distribution of syntactically well-formed überhaupt-sentences in discourse: Given a well-formed utterance that includes überhaupt, where can it be used felicitously in discourse?
(III) Unifying the uses of überhaupt: What ties together the different uses of überhaupt and what is their (common) meaning?
In this paper, I set out to address (II) and (III), taking the well-formed überhaupt-utterances as given and asking only where we can place these utterances in discourse. I regard (II) as the key to (III), getting the unifying characteristics and the common meaning of überhaupt. However, even though (II) and (III) together can be investigated in a logically self-contained manner, it remains an interesting question how (I) works. This is essentially what Anderssen investigates. His paper sets out, among other things, to explain why überhaupt can occur both with negation and with a universal quantifier.
I think Anderssen pointed to an important and intriguing issue by highlighting (I). However, unlike Anderssen, I believe that (II), not (I), is the key to getting at the meaning of überhaupt.
There might still be a way to bring together Anderssen's and my account fruitfully. A wild speculation would be that maybe überhaupt started out with a narrower set of uses describable by widening (let's call them 'widening uses' for short). These widening uses explain most of the syntactic co-occurrence 1:32 behavior of überhaupt (what determiners and quantifiers it can occur with) quite well. überhaupt may then later have evolved to work as a more general question-under-discussion-navigating-device, 24 for which widening is just a special case (as explained in Section 7). However, the syntactic restrictions on where überhaupt may occur within a sentence may have stayed the same, explaining why the narrower set of widening cases seems more suitable when explaining the syntactic behavior of überhaupt. Thus, it would be exciting to pursue a historical investigation of the development of überhaupt, as this may shed light on how (I), (II) and (III) fit together.
In the remainder of this section, I would like to discuss a number of additional examples illustrating constraints on überhaupt. These constraints can be discourse constraints or syntactic constraints or sometimes both. 25 In the following example, interlocutor A is asking a series of questions of the form Are you from X?. A possible parent question would be Where are you from?. Even though (C26.iv) answers this parent question, überhaupt is not felicitous in this sentence. This is because (C26.iv) is a relevant reply to (C26.iii) and thus fails to satisfy one of the requirements for an ascending move. Therefore Main Claim correctly predicts that überhaupt cannot be used in this utterance. Are you from Washington? (iv) B: Ja, ich komme (#überhaupt/ überhaupt F ) aus Washington.
In conversation (C27.iv) below, überhaupt is ruled out for different reasons. While (C27.iv) is not a relevant reply to (C27.iii), thus satisfying one of the requirements of an ascending move, it does not explicitly settle all questions of the form Would you like X drink?. In particular, it does not settle the question of whether B wants sherry or not. Thus (C27.iv) is not a full 24 Thanks to Jan Anderssen for this term. 25 I would like to thank Jan Anderssen and an anonymous reviewer for these insightful examples, which helped in improving the presentation of my formalism. 1:33 answer to the parent question and therefore is not an ascending move. Hence überhaupt is not felicitous here, as predicted by Main Claim . Interestingly, überhaupt must be located where the exhaustification takes place. Thus (C29.iv) in the following conversation is infelicitous. This simply reflects that discourse is updated dynamically (see Roberts 2004or Bittner 2003, and it is to be expected that a discourse particle is sensitive to this (see Roßdeutscher & von Stutterheim 2006). As mentioned above, either syntactic or semantic restrictions, or both, can be at play. I explained how discourse requirements preclude (C26.iv), (C27.iv) and (C29.iv) from being felicitous in their respective contexts. In addition, these utterances do not sound well-formed, at least to this native speaker. That is, the restrictions on the placement of überhaupt within an utterance are involved as well (concern I). So (I) and (II) can work together in ruling out some uses of überhaupt. By contrast, the infelicitous uses of überhaupt given in 2.2.2 are due to conditions on (II) alone. Lastly, the following is an example that is ruled out just for syntactic reasons, the concern of (I): Utterance (C30.iv) resolves the parent question, so it should allow the use of focused überhaupt if we go just by the felicity restrictions. What goes wrong here, however, is that the sentence (C30.iv) is not syntactically well-formed in the first place.
An advantage of keeping (I) and (II) separate is that it may make it easier to adapt the analysis presented here to other languages. There are particles in a number of languages that exhibit all, or sometimes a subset, of the discourse uses of überhaupt (Migron 2005a,b, Greenberg & Kharizman 2012. But I suspect that in some cases these particles will have different syntactic restrictions at the level of a single sentence. One case in point is the Russian particle voobshche, which can be used in all the discourse situations described in Section 2.1 (Kuznecov 1998, Migron 2005a. However, in contrast to überhaupt, the particle voobshche has more syntactic freedom in terms of where it can appear within a sentence. For example, I have been told by native informants that voobshche could be used in (C30) without disturbing well-formedness. This motivates why (I) and (II) should be treated as separate and orthogonal issues.