What Ifs

We develop a dynamic account of what if questions on which they re-pose questions inside local contexts introduced by their if -clauses subject to the felicity constraint that the resulting context is inquisitive. While this analysis is directly motivated by cases where a what if questioner challenges another speaker’s attempt to answer a current question under discussion (QUD) by seeming to re-ask this question over a more restricted contextual domain, it can also explain the flexibility of what if since other uses trigger accommodation with new QUDs to ensure that the post-suppositional inquisitivity condition is met. While QUD accommodation is a complex phenomenon that isn’t specific to just what if constructions, the pragmatic flexibility of what if furnishes a nice range of examples for investigating such repair. In the latter part of the paper, we focus on practical what if questions which trigger accommodation with QUDs that subserve the real-world domain goals of the speakers. We offer a systematic working theory of this accommodation within a formal model of discourse that involves goal stacks populated with both questions and decision problems tethered together by relevance. The larger contribution of this paper is to add to the understanding of how discourse felicity and update conditions at the level of speech acts can be encoded in natural languages.


Introduction
Research at the semantics-pragmatics interface has progressed greatly in recent years by developing formal accounts of the discourse effect of different speech acts (see 1929, Stalnaker 1968, Lewis 1973, Adams 1975, Heim 1983, Veltman 1985, Kratzer 1986, Edgington 1986, 1995, Jackson 1987; among many others).
That said, what if s raise issues that have been much less studied in both of these literatures: for question acts, their non-canonical form raises issues about what the limits might be for the category of linguistic questions, and for conditionals, what if s force us to confront questions about how conditionality relates to discourse structure.
To account for the senses in which what if s are question-like and conditional-like, we analyze them in a dynamic model of discourse that tracks the current assumptions of the conversational participants along with the questions they are currently busy discussing (Questions Under Discussion (QUDs); Roberts 1996, 2012b, Ginzburg 1996, van Kuppevelt 1996, Büring 2003 and the salient real-world domain goals that these questions subserve (as in Roberts 2004Roberts , 2012aRoberts , 2015Roberts , 2018. Our rough proposal is that what if questions serve to re-pose QUDs inside the local subordinate contexts introduced by their if -clauses (where the speakers are assuming these ifclauses hold) subject to the felicity condition that the resulting context is inquisitive (Groenendijk 1999, Rawlins 2010. This post-suppositional inquisitivity requirement is key to explaining the full range of interpretations, as we argue that many what if s trigger accommodation with new QUDs to ensure that the context reaches an inquisitive state, and this repair mechanism is in many cases constrained by the domain goals of the speakers. While in this paper we have space and time to deal with only one case of discourse conditionals, the pieces of the proposal for what if s provide a starting point for analyzing the whole family, and, more generally, contribute to our understanding of what kinds of discourse constraints at the semantics-pragmatics interface can be encoded into natural language morphology.

The Many Functions of What If
We will have more to say by way of introduction in a moment. But let us first survey some of the functional heterogeneity of what if. We have already seen some typical "hypothetical" uses of what if in (1) Such hypothetical what if questions are asked discourse-initial in order to start a fresh line of speculation about some new (often non-factual) topic; this initial topic can be narrowed by follow-up questions as in (5) and (6). The questioner is interested in what things are or would be like in the scenario introduced by the if -clause. However, what if has multiple other uses besides the discourse-new hypothetical. For instance, an "elaborative" what if question can be used following an assertion or some other informational contribution to ask about the consequences of a situation previously described under the assumption contributed by its if -clause. These elaborative what if s can occur both same-speaker and cross-speaker: Such elaborative what if s are like discourse-new hypothetical uses but require an informational antecedent that sets the stage, and so are often more limited in scope.
Next up, a what if question can be asked following an assertion to challenge or resist the speaker's attempt to update the discourse context with the proposition expressed (Rawlins' 2010 "conversational backoff", Bledin & Rawlins' 2016 "resistance moves"). In the following examples, the resister thinks that the resistee might be overlooking some relevant possibility that bears on her proposal, and the resister wants to hear more about this before making a call on acceptance or rejection: It is instructive to compare (12) with (9). The Uh oh in (9) signals that B has accepted A's claim and is now curious about what will happen if both Joanna and Alfonso attend. In contrast, when B challenges A in (12), B has not yet accepted A's claim and is effectively asking if Alfonso will still come to the party if Joanna is there. Offers, commands, requests, invitations, etc., as well as biased and even some non-rhetorical questions can also be challenged in this way: 5 (14) The boy came right over and boldly proposed that, since they were both there at the same time every week, they could start sharing a paper and save a tree. "What if we both want the same section?" Pip said with some hostility. These "challenging" or "resisting" what if s are cross-speaker response moves that stall an existing stream of discourse. By bringing up new possibilities or issues, they can serve as enticements for the challenged speaker to change her mind. Finally, there are also "suggestive" uses. Besides their exploratory and challenging functions, what ifs can be used to offer resolutions to salient questions under discussion (QUDs). These suggestive what if s come in both practical and theoretical flavors. After a planning question, a hearer can propose a course of action: 6 5 Thanks to Cleo Condoravdi (p.c.) for bringing the data involving imperatives to our attention. 6 Though we focus on planning what if s following explicit questions in this paper, such what if s can also follow assertions and other non-interrogative speech acts: (i) A: I was going to bake a cake but I haven't got any eggs.
B Though we present some reasons for preferring this second minimal version of our proposal, the CQ and SQ analyses are quite closely related and much of what we say in the remainder of the paper doesn't depend on the choice between them, so we ultimately leave both the CQ and SQ options on the table. In §4.6, however, we reject a third option according to which what if s are not even, properly speaking, questions-there is no requirement that they render the context inquisitive-but are rather suppose statements in disguise. This purely suppositional story goes too far: we argue that what if questions must have both suppositional and questioning aspects, where these aspects can be spelled out by treating these constructions as either conditional or suppositional questions. Now, whether one goes the CQ or SQ route, one can easily account for the challenging uses of what if in cases like (28) that these analyses were designed to handle. However, as we discuss in §5, other uses of what if pose a prima facie problem for either analysis. First, hypothetical and some elaborative what if s are problematic because these questions can be asked when the local overtly triggered QUD is closed or there is no obviously open QUD at all to be re-asked or re-posed in the local context created by the what if update. So it is not immediately clear how the questioning comes about. 10 Second, even when what if s directly respond to a prior explicit question, they often do not seem to be re-asking or re-posing this question over a restricted domain. For instance, in the suggestive exchange (21), repeated as (29) below, B does not seem to be asking the rather odd question of whom they should invite to speak on the assumption that they will invite Professor Plum. 11 Rather, he seems to be inquiring about what would happen if they invite Professor Plum in respects that matter for their common goal of having an interesting, well-attended colloquium.
(29) A: Who should we invite to speak at the next colloquium?
B: What if we invite Professor Plum?
Here, too, a flat-footed application of the conditional or suppositional question account delivers a bad result. This is where the post-suppositional inquisitivity requirement present in both the CQ and SQ accounts comes into play. We argue that in many, if not all, of the problematic cases where there is no QUD available for re-posing or the what if questioner seems to pose a totally different question than the current discourse topic, hearers will respond to the what if question by accommodating with a new QUD to ensure that the inquisitivity constraint is satisfied-for instance, a hearer might respond to one of the hypothetical what if s in (1)-(3) by implicitly raising the "Big Question" (Roberts 1996) or some suitable coarsening of it (though more on this in §5). The remainder of the paper is devoted to clarifying this repair mechanism. It is worth emphasizing at the onset that QUD accommodation is a complex phonemenon that isn't specific to what if questions, even if it is necessary for an account of what if s. So, in a sense, this latter part of the paper isn't just about what if s. However, the what if construction provides a nice gateway into the problem of QUD accommodation because its pragmatic flexibility furnishes a broad range of examples for investigating this kind of repair, and we often have reasonably clear intuitions in these examples about which question is being accommodated.
It is difficult to be very precise about how QUD accommodation works in general-like accommodation in other domains, it is a messy business-but we think that a nice story can be told in many cases where the implicitly introduced QUD subserves the practical goals and interests of the speakers. To provide a more systematic working theory of the repair in such cases, we extend our formal model of discourse in §6 by introducing goal stacks loaded up with both question denotations (encoding the speakers' discourse goals of resolving QUDs) and decision problems (encoding their domain goals like arranging a successful talk, acquiring a newspaper, and so forth; van Rooy 2003b, Davis 2009, Franke & de Jager 2010, Kaufmann 2012, Malamud 2012, Cariani, Kaufmann & Kaufmann 2013. With this decision-theoretic structure in place, we can formulate a new "Subservience" constraint between the 11 The challenging use in (18) raises a similar difficulty. 9 questions and decision problems on a goal stack that captures how the questions that speakers ask in practical contexts depend on their domain goals and plans. In §7, we apply Subservience to an example where a what if question is used to challenge a discourse-new command-which must trigger QUD accommodation on the assumption that such commands do not introduce or respond to QUDs-and we show how this relevance constraint narrows the range of accommodation options. In §8, we then revisit the colloquium example (29) and show how Subservience helps to constrain the repair in this example as well. We conclude in §9 with some open problems.

The Structure of What If
Before presenting the core of our analysis, let us first introduce some additional data bearing on the structure of what if questions. 12  And, of course, this is in stark contrast to all sorts of other constituent questions that support the full range of wh-items.
The idiosyncrasy of what in what if is also suggested by data showing that it cannot undergo normal wh-modification (these tests are due to Baker 1968Baker , 1970; see also Gawron 2001 andRawlins 2008 In addition, whereas what if s can be modified by speaker-oriented adverbs, they cannot combine with slack regulators (Lasersohn 1999) or be modified by other lower classes (Cinque 1999, Ernst 2002 Rawlins's (2008) dynamic semantics for conditional questions more generally (see Heim 1982, Veltman 1996, Beaver 2001 for some dynamic semantics classics; see Hulstijn 1997, Velissaratou 2000, Ciardelli, Groenendijk & Roelofsen 2013 for related work on CQs). Where c is a discourse context, In broad outline, updating c with the conditional interrogative if ϕ, ψ? involves the following two steps: • Temporarily assuming the antecedent ϕ and thereby entering a new local context c + Assume(ϕ) in which the set of possibilities under consideration is restricted to those in which this antecedent holds (cf. suppositional accounts in Ramsey 1929, Adams 1965, Mackie 1973, Heim 1983, Edgington 1986. 19 • Asking the question expressed by the consequent ψ? in this derived local context.
According to Rawlins, what if questions work in much the same way, except that the question posed inside the hypothetical context is a QUD supplied anaphorically by context: For example, on the conditional question analysis of A's what if in (52), this question serves first to assume that Joanna is coming, triggering a local hypothetical context where worlds in which Joanna is not coming are temporarily off the table, and then to re-ask the QUD (Is Alfonso coming?) over this restricted domain. This accords with intuition: since this is basically just the update corresponding to (53), the CQ analysis delivers a nice result.

Formal Discourse Model
To flesh out both the CQ analysis and the SQ analysis we offer later on, we develop a broadly Stalnakerian model of discourse (Stalnaker 1978(Stalnaker , 2002(Stalnaker , 2014. In §6, we upgrade our model with decision-theoretic structure, but for the time being we want to remain in more familiar territory and work with a representation of context in the style of Roberts (1996) and Farkas & Bruce (2010). Letting W be a non-empty set of possible worlds, we first define our discourse contexts as follows: A c is a stack of propositions (the assertion stack) 20 d. Q c is a stack of sets of propositions (the topic stack) For Stalnaker, the context set cs c includes the possibilities compatible with what is being taken for granted or presupposed for the purposes of the conversation (in its latest incarnation in Stalnaker 2014, cs c is spelled out using epistemic logic in terms of the higher-order notion of common acceptance, though see Lederman (2018b,a) for arguments against using higher-order "common" attitudes to explain coordination in discourse). Here we take cs c to represent fairly stable information in the common ground (not what is being temporarily assumed)-for present purposes, you can think of cs c as modeling what is publicly believed by the discourse participants. Instead of building speakers' suppositions directly into the context set itself, we represent them separately using another parameter: the assumption slot a c (Rawlins 2010 calls this the "view"). This slot serves as a temporary window onto part of the context set-the participants' current view is restricted to the worlds inside cs c ∩ a c (the default setting is a c = W when no assumptions are in force). 21 In this paper, we use the assumption slot to model the local contexts generated in the evaluation of various indicative conditional constructions. But this parameter could also be used to model the context change potential of other conditional constructions and related expressions that trigger hypothetical contexts (like suppose sentences). 22 The remaining components of a context c are the assertion stack A c and topic stack Q c (cf. Farkas & Bruce's 2010 "tables", which generalize the QUD stacks in Roberts 1996). These stacks keep a short history of assertions made by conversational participants, and a short history of questions that are currently under consideration 20 We assume some familiarity with stacks; see for example Kaufmann (2000) and Isaacs & Rawlins (2008) for similar uses. We use the following (standard) notation: push(x, s) is the stack obtained by adding x to the top of stack s, pop(s) is the stack obtained by removing the top element of s, and top(s) designates the top element. 21 See Isaacs & Rawlins (2008) for an alternative way of implementing assumptions using stacks of context sets ("macro contexts"). We could have equally well worked in their framework where making an assumption adds a new context set incorporating the assumed content to the top of the macro context. 22 Although some of our earlier examples like (1) and (2) involve counterfactual suppositions, formally modeling these counterfactual what if s would require us to introduce even more discourse structure (similarity orderings, structural equations, or whatnot). So we model only ordinary 'factual' discourse here. That said, while we don't develop a theory of counterfactual what if s in all its glory, in footnotes we do suggest ways to adjust our model to handle them. respectively. The assertion stack A c captures how assertions are proposals to update the context set cs c with their content. 23 In our formal system, asserting a proposition places it in this purgatory where it must wait for either acceptance (i.e., incorporation into the context set) or rejection by the audience. 24 Meanwhile, asking a question places it on the topic stack Q c . This second stack is loaded up with questions awaiting resolution, each represented as a set of propositions (the questions on the topic stack encode the discourse participants' "strategies of inquiry" as in Roberts 1996).
Without going into details, we assume that question denotations are compositionally constructed as alternative sets in the style of Hamblin (1973) and Kratzer & Shimoyama (2002). Polar interrogatives denote singleton sets (Roberts 1996, Biezma & Rawlins 2012: Alternative questions are the union of the disjuncts: Is it raining↑ or snowing↓? = {λ w s .raining in w, λ w s .snowing in w} Constituent questions are constructed pointwise based on the domain of the wh-item: What is the weather like? = {λ w s .raining in w, λ w s .sunny in w, ...} Moreover, we adopt the following notions of answerhood based on those in Roberts (1996): Given a question Q that is not yet settled in context c because one or more of its members is not yet evaluated in c (i.e., there is some A ∈ Q such that cs c ∩ a c ⊆ A and cs c ∩ a c ⊆ W − A): a. P partially answers Q in c iff for some alternative A ∈ Q that is not yet evaluated in c, P contextually entails either A or W − A. b. P completely answers Q in c iff for each alternative A ∈ Q, P contextually entails either A or W − A. (where P contextually entails P in c iff P ∩ cs c ∩ a c ⊆ P ) 23 Stalnaker (1978) recognizes the proposal nature of assertion but he deemphasizes it, so many working in the Stalnakerian tradition simply assume that assertions automatically update the discourse context unless rejected. 24 For ease of exposition, we assume that assertions are always proposals to update with their propositional content and ignore cases like epistemic modalized claims where it is unclear that assertors are even expressing propositions (Yalcin 2011). See Bledin & Rawlins (2016) for more discussion and for alternative models of the assertion stack that can handle such cases. Like Roberts, we require that complete answers to questions higher up on the topic stack Q c partially answer questions lower down. This will follow from the dynamics of questioning in our model.
The question at the top of the stack, top(Q c ), is under immediate discussion and is interpreted through the lens of the current context (we refer to this topmost question as the "current QUD" or sometimes just "QUD" for short). Note that any question Q induces an equivalence relation between possible worlds (or subject matter, in the sense of Lewis 1988a,b) where w and v are equivalent iff these worlds are members of the same propositions in Q (a singleton alternative set denoted by a polar interrogative generates a bipartite equivalence relation): 25 We assume here that speakers always want complete answers to their questions, so we identify the QUD in context with the set of equivalence classes induced by top(Q c ) over the context set cs c visible within a c (cf. Groenendijk & Stokhof 1984, Groenendijk 1999; that is, the QUD in context is the quotient set over the current domain of live options cs c ∩ a c determined by the equivalence relation ∼ top(Q c ) : 26 Where c is a context, For example, suppose that W = {w 1 , w 2 , w 3 , w 4 } where Carlos is having a birthday party in w 1 and w 2 , Maggie is having a party in w 1 and w 3 , and nobody is having a 25 While (as both an anonymous reviewer and Josh Dever p.c. suggest) it might be technically simpler to work with a partition from the beginning, we have taken the current approach as a compact but general implementation of Roberts (1996) that allows for a clean integration of alternative sets and contextual domain restriction, without the assumption that the alternative sets generated by the compositional semantics are partitions (Ciardelli, Groenendijk & Roelofsen 2013, Biezma & Rawlins 2012. Even if top(Q c ) were a partition already, we would still need a mechanism (on our approach) for doing contextual domain restriction. 26 We do not worry about mention-some readings here. This is not to say that these readings are unimportant; mention-some wh-questions are, for instance, one of the motivations for the recent development of inquisitive semantics (Ciardelli, Groenendijk & Roelofsen 2013, 2018. While we think that the main ideas of this paper can be recast in an inquisitive semantics framework, we do not have space to go into details here. 27 The quotient set notation S/ ∼ shouldn't be confused with the set difference notation S \ T . The former is the set of all equivalence classes in S with respect to the equivalence relation ∼ while the latter is the set of all elements in S not in T . party in w 4 . Suppose also that cs c ∩a c = W and top(Q c ) = Who is having a party? = {{w 1 , w 2 }, {w 1 , w 3 }}. Then QUD c = W / ∼ top(Q c ) = {{w 1 }, {w 2 }, {w 3 }, {w 4 }} and the immediate discourse goal of the participants is to locate the world inside one of the cells in this partition (or to at least establish that the world does not lie inside this or that cell). Note that the QUD in context can change as either the context set cs c or assumption slot a c changes, even if the topic stack Q c remains unchanged. For instance, if the speakers assume that Carlos is having a party, then the QUD in context shifts to Turning to the dynamic component of our model, let us first consider an assertion + acceptance sequence. 28 Our assertive update is fairly straightforward: it simply adds the proposition asserted to the assertion stack. An assertion is relevant just in case the proposition added to the stack partially answers top(Q c ) by excluding at least one of the alternatives in QUD c . Admittedly, this requirement from Roberts (1996) is overly restrictive, as assertions that only shift a speaker's credences over QUD c without ruling out a cell (Büring 2003, Simons et al. 2010 or that serve only to bring alternatives in QUD c to one's attention (Franke & de Jager 2010) can also be relevant. But we will not pursue a weaker relevance requirement here.
While relevance is necessary for appropriate assertion, it is certainly not sufficient; presumably, one will also want a sincerity condition (Searle 1969) or a stronger epistemic requirement like Williamson's (1996Williamson's ( , 2000 rule that we assert only what we know. 29 But since we don't distinguish between public and private information in our model, we do not make this other dimension of felicity explicit. What happens when an assertion is accepted? As in the original Stalnakerian theory, the asserted content is added to the context set. In our current framework, however, this update takes place only within the window of the current view a c . Because what a speaker asserts will often depend on the assumptions currently in play-that is, assertions are often conditional on a c -we take accepted assertions to update only the visible field of the discourse cs c ∩ a c , not the full context set cs c . To formalize this, we use the following operation (cf. "support" in Kaufmann 2000, "percolation" in Isaacs & Rawlins 2008: After an update with ϕ in c, the worlds remaining are those in the visible light gray region cs c ∩ a c ∩ ϕ together with those in the darker region cs c − a c (these latter worlds cannot be eliminated, as they are not even in view). Equivalently, the update serves to kick all and only the not-ϕ-worlds out of the live field of the discourse cs c ∩ a c .
Acceptance can now be defined in terms of this domain-restricted update. When an assertion is accepted the context is updated with the top element of the assertion stack, which is then removed from the stack: Felicity condition: appropriate in c only if cs c a c top(A c ) = / 0.
It is worth noting that in the limiting case where cs c ⊆ a c , the update (62) amounts to regular intersection: cs c a c ϕ = cs c ∩ ϕ . So the conversational sequence c + Assert(ϕ) + Accept behaves like an ordinary Stalnakerian assertion move. 30 Moving on to questions, our questioning update is at its core much like the earlier assertive update: it simply places a new question onto the topic stack. But, as before, this added question must be relevant to the current state of the discourse. If there was already a question on the stack, then a complete answer to the new question must partially answer the question that was previously on top (Roberts 1996). 31 Whether there was previously a question on the stack or not, the new question must also render the context inquisitive (in the sense of (60)). This blocks questions that are already settled in the context relative to the suppositions in force. Formally, this is captured by the following update: 30 Because acceptance is the default response to assertion (as Farkas & Bruce 2010 put it, assertions "project" their acceptance), the acceptance step often happens silently/implicitly. But sometimes acceptance is explicitly marked by particles like Okay, Sure, etc., or signaled by nodding and other physical gestures. 31 We actually impose a stronger constraint where the added question must be a subquestion of the question previously on top in the sense that any complete answer to the old question is a complete answer to the new question.
Felicity conditions: appropriate in c only if a. if Q c = , then each cell in the partition QUD c is the union of cells from QUD c (i.e., for every P ∈ QUD c , there are P , P , .
The relevance and inquisitivity conditions are necessary but not jointly sufficient for appropriate questioning. Presumably, a questioner must not already accept one of the answers to her question-though this Searlean "preparatory condition" can be suspended in exam contexts and other non-standard situations-and she must also think that the addressees can potentially help to resolve the question. Let us put just a few more pieces into place. To deal with conditional constructions, we still need a dynamic update to capture the effects of if -clauses. In our formal system, these clauses intersectively update the assumption slot with their content and thereby restrict the speakers' window onto the context set (Kaufmann 2000, Isaacs & Rawlins 2008: In the other direction, the following utility update resets the assumption slot: 32 Though we are not trying to model counterfactual discourse in this paper, one might worry at this point about the prospects of extending our formal system in this direction given the definability condition for Assume. If intersecting with the proposition expressed by a counterfactual if -clause results in the empty set, then the system crashes. However, we see this as a potentially useful design feature rather than a bug. In his work on counterfactuals (CFs), von Fintel (2001) argues that CFs can be analyzed as strict conditionals that quantify over a "modal horizon" that can widen as discourse proceeds (Gillies 2007 also analyzes CFs along these lines). Something like this contextual parameter could help us deal with counterfactual what if s. Suppose, following von Fintel, that in addition to the context set cs c , each context c comes equipped with an "accessibility function" f c mapping worlds in W to sets of worlds, with the default setting f c (w) = {w}. We can then take the counterfactual domain to be f cs c = w∈cs c f c (w), which collects all of the worlds obtained by applying f c to each world w in cs c (we have as the default that f cs c = cs c ). Now the fix: let Assume be defined only if f cs c ∩ a c ∩ ϕ = / 0 and allow for repair when this condition is violated by expanding the domain f cs beyond the context set (see von Fintel for details on how this might be implemented using similarity orders). Note that the revised definability condition would be similar to Heim's compatibility presupposition for CFs reported by von Fintel (what Gillies calls an "entertainability presupposition"), but there is nothing distinctively counterfactual about it-as part of Assume, it would apply to indicative conditionals as well.
We also want some more ways to dial back the table. Right now, the only way to pop the assertion stack is through acceptance. However, not all assertions are accepted, so a complete system must also allow for Retraction and perhaps also Agreement to Disagree (see Bruce 2010 andBledin &Rawlins 2016 for implementations; we will not require these moves here). The topic stack can also be popped with the following utility update: We regard Clear and Dispel as discourse maintenance operations needed to keep track of the changing attitudes and goals of the participants. Unlike our other updates, they are not triggered by specific linguistic constructions. However, the need for a Clear or Dispel operation must still be inferred from what is being said. When speakers stop using modal subordination morphology (e.g., would; Roberts 1989) on their responses, this can signal that they have departed a local context via Clear. When speakers' assertions/acceptance leave the discourse in an uninquisitive state where |QUD c | = 1 (i.e., the QUD, if any, has been resolved with respect to the live possibilities cs c ∩ a c ), we take it that this will typically trigger an adjustment of the context by Clear, Dispel, or both, so that inquisitivity is restored.

Example
Before turning to the formal CQ analysis of what if questions, it will be instructive to walk through a quick example to get more of a feel for how our machinery works. So consider W = {w 1 , w 2 , w 3 , w 4 } where Alfonso is coming to the party in w 1 and w 2 only, Joanna is coming in w 1 and w 3 only, and nobody else matters.
A's "OK" response signals acceptance, and so the update {w 1 ,w 3 } {w 3 , w 4 } removes the world w 1 in which both Alfonso and Joanna are coming from the context set: At this point, QUD c 3 = {{w 3 }} so things are no longer inquisitive. But a Clear update takes us back to the main categorical context of the discourse, which is inquisitive: Note that without this Clear, A's follow-up polar interrogative (which adds another question to the topic stack) would be infelicitious: 's subsequent answer now adds the proposition that Joanna is coming to the assertion stack: In the end, the assumption slot is clear, the table is empty, and the context set has been reduced to the maximally informed context set {w 3 } containing only the world where Joanna is coming to the party but Alfonso is not.

What If s as Conditional Questions
We have just seen an example of conditional assertion. Schematically, this takes the following form: Updating with if ϕ, ψ amounts to first assuming ϕ and then asserting ψ over the temporarily restricted domain. Importantly, the assumption constrains the effect of the subsequent assertion should ψ move from the table to the context set via acceptance. 33 A conditional question update works in much the same way. The only difference is that the post-assumption move is now asking rather than asserting: There is still significant interaction between the subcomponents of this update: the initial assumption step delimits the QUD induced by the question ψ? expressed by the consequent, as the discourse participants will try to resolve this question relative only to the temporarily restricted domain.
As for what if questions, these can now be treated as conditional questions where the question component is re-asking the current QUD: 33 Note that c + Assume(ϕ) + Assert(ψ) + Accept + Clear is effectively Heim's (1983) classic dynamic entry for conditionals. On our theory, the latter Accept and Clear operations are not part of the semantic clause for if but are rather updates that occur if the speaker's (conditional) assertion using ψ is accepted in the local context in which ϕ is being assumed and the interlocutors then exit this local context.
In our earlier example from §4.1, repeated below as (72), A's what if question serves to first intersect the current assumption slot a c with the proposition Joanna is coming and then re-ask the QUD by re-adding Is Alfonso coming? to the topic stack (crucially, A's resistance move indicates that she isn't yet willing to accept B's answer by moving the proposition Alfonso is coming from the assertion stack into the context set, and so the inquisitivity condition is satisfied): Following A's challenge, the immediate discourse goal is to determine whether Alfonso is coming to the party given the assumption that Joanna will be there.

What If s as Suppositional Questions
The CQ analysis seems to work well for the what if "resistance move" in (72). It captures how A is resisting B's proposed answer to the QUD by re-posing this question over the restricted domain where the issue of whether Joanna is coming is settled in the affirmative-an issue that A thinks B might be overlooking. 34 However, one might worry that the CQ analysis delivers the right end result but still gets the mechanism wrong. After all, do we really need the questioning part of the conditional question? Note that in (73)

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What if s when the topic stack is non-empty). But one might want to ban or restrict such trivial updating in a more refined model of discourse. This is partly a technical theory-internal worry, but it also has an empirical component. Full CQ responses that literally repeat a previous question tend to sound quite odd, and typically require some alternate phrasing or subjunctive marking: Note that the connection to questions has not been completely severed. While the questioning update +QUD c ? in (71) is gone, we have retained the inquisitivity condition familiar from questioning in general, which now requires that the posterior context c be inquisitive: after updating with +Assume(ϕ), QUD c must partition cs c ∩ a c into multiple cells. This ensures that there is a live question to be resolved in the temporary local context created by the what if. 35 As on the CQ analysis, we can still say that what if s serve to re-pose (or transpose) questions under the suppositions contributed by their if -clauses. But whereas both suppositional and questioning updates were hardwired into the original CQ update (71), only the suppositional update now remains. On the more minimal SQ update (75), the sole conventional discourse effect of what if is to introduce an assumption, which serves-when what if is felicitous-to re-pose an already existing question over a more restricted contextual domain (technically there is no re-asking). Any new questioning must be a secondary pragmatic effect coming by way of the accompanying felicity condition requiring that the discourse context be inquisitive after the assumption slot has been narrowed (much more on this still to come). 36

A Non-inquisitive Suppose Story?
Because the CQ and SQ versions of our proposal are so closely related and the differences between them will not matter for the issues we want to take up in the remainder of the paper, we will not try to further motivate the SQ analysis over its CQ rival. If you aren't worried about redundancy and want to hold on to the idea that what if s explicitly ask questions, then feel free to stick with the CQ analysis.
We do, however, want to reject a third candidate analysis of what if constructions. Given that we have already suggested weakening the question-hood component of what if s in the proposed move from the CQ to the SQ analysis, one might want to do away with question-hood altogether and assimilate what if s to suppositional imperatives (Isaacs 2007). On the non-inquisitive story we have in mind, what if s have only the assumption step but no inquisitivity requirement (though they still have whatever felicity conditions govern supposing). 37 A proponent of such a suppose-style account might argue that it is only because we have been theorizing about challenging what if s that conditional-question-style analyses seem so attractive. Start theorizing from the perspective of suggestive uses and a non-inquisitive analysis seems equally if not more promising. Recall for instance the following example from §2 where what if is used to suggest a course of action in response to a planning question:  Farkas & Roelofsen (2017). 37 We are grateful to Josh Dever (p.c.) for pressing us to give this non-inquisitive suppose account more consideration.
In this exchange, the main point of B's what if response is not really to pose a question in the subordinate context where it is assumed that A stops worrying and drives, but rather to foreground the possibility of him driving. The what if response does this by intersecting the assumption slot with the proposition that A drives his Ferrari to the party, and this could equally be achieved using an ordinary suppose sentence. It doesn't seem to matter at all whether the resulting local context is inquisitive. Now, what if constructions certainly have empirical properties of suppositions, as we should expect on any of the three analyses offered so far. Their suppositional nature can be seen from their licensing of modal subordination morphology, like would, on subsequent responses, similar to regular modal subordination (Roberts 1989) and to modal subordination triggered by conditionals (Roberts 1989 In dialogues, what if s also act like questions in terms of turn-taking and responsehood. When one asks what if...? to a hearer in the course of ordinary discourse, there is the usual expectation that the hearer will answer. In contrast, regular suppositionintroducing moves, such as suppositional imperatives and modal utterances intended to trigger modal subordination, are often infelicitous, or at least very awkward, if used to try to get an answer. 39 Compare: Although A's lead-off utterances in (80) and (81) both introduce the same assumptionthat Napolean won at Waterloo-the hypothetical what if calls for an answer in a way that the suppose directive does not. This observation that what if s invite hearers to answer cross-cuts all four uses discussed earlier in this paper (though it should be noted that in monologues and written text, questions can be self-addressed, and at least hypothetical, elaborative, and suggestive what if s can be used in this way). Even in (76) We conclude that what if s have both suppositional and questioning components. While we leave it open whether the best way to capture this is by treating what if s as CQs or SQs, we insist on maintaining some connection to questioning and so reject the proposal to do away with the post-suppositional inquisitivity requirement altogether. In fact, when we turn to the full functional spectrum of what if questions, this requirement will play a crucial role in getting good results.

QUD Accommodation
Let us pause and take stock. Motivated primarily by challenging uses of what if that contextually restrict an explicit QUD from preceding discourse, we first presented a conditional question analysis of what if s that expands and improves on the earlier analysis of Rawlins (2010). We then offered a new suppositional question analysis with a weaker questioning component, but rejected a non-inquisitive suppose account that breaks the tie to questioning altogether. Both of our proposals can account for the challenging and elaborative "re-asking" uses that Rawlins (2010) dealt with in a similar but cleaner way, so if our explananda were only these re-asking cases, we would arguably be finished our project. The CQ and SQ analyses of what if nicely explain examples like (72) where the what if questioner is continuing a preexisting line of inquiry introduced by a prior question in discourse.
Things get trickier, however, when we turn to the many other cases of what if, such as hypothetical and suggestive uses, because it isn't clear that either the CQ or SQ account generalizes. Given that many hypothetical, suggestive, and even some challenging and elaborative what if s seem to involve non-trivial questioning that doesn't simply amount to transposing the current QUD into a more limited domain, we need to say more about these other cases. In the remainder of this section, we suggest the beginnings of a story for some examples involving QUD accommodation. In the rest of the paper, we then use this idea to bootstrap an analysis involving accommodation of and coordination on shared discourse and domain goals, where the former are analyzed as QUDs and the latter as decision problems ( (75) is violated. 40 The second kind of challenge for the CQ and SQ analyses comes from cases where there is an open QUD available for re-asking or re-posing, but the what if question seems to pose a different question in the local context generated by its if -clause. Many suggestive uses are like this, as we have already seen in (76) Informally, the challenge is that B does not seem to be asking who they will invite if they invite Professor Plum. Formally, the problem is that a direct application of either our current CQ or SQ update (not to mention the Rawlins 2010 analysis) crashes. The denotation of A's opening question is: (87) Who are we going to invite to the next colloquium? =    λ w s .we invite Professor Plum in w, λ w s .we invite Professor McGonagall in w, λ w s .we invite Professor Xavier in w, ...

  
If this topic determines the current QUD in context and we intersect the assumption slot with the proposition λ w s .(we invite Prof. Plum in w) (and then perhaps re-ask the current QUD), then the resulting context is uninquisitive (assuming that the context set excludes the possibility of inviting multiple speakers). So what should we say about all these troubling examples? Our basic solution is to appeal to QUD accommodation (Lewis 1979;see Cooper & Larsson 2010 for discussion of QUD accommodation in particular). 41 In both kinds of cases that create difficulties for the CQ and SQ analyses, we suggest that the what if questions can trigger a repair mechanism whereby hearers will push a new implicit question onto the topic stack Q c to ensure that the post-suppositional context is inquisitive, or that there is even a QUD available for re-asking in the QUD-less cases that pose a more immediate problem for the CQ story. The devil, of course, is in the details. The appeal to QUD accommodation raises the puzzle of which questions are accommodated in particular examples and why these questions are used to repair the discourse context rather than others. 42 Reflecting first on XKCD-type examples, a natural first suggestion is that hearers will accommodate with the "Big Question" (Roberts 1996) that asks what things are/would be like in every respect. Formally, the Big Question can be modeled with the power set P(W ), which induces the finest possible partitioning of cs c ∩ a c into singleton sets; that is, after the repair, QUD c = {{w} : w ∈ cs c ∩ a c }.
Note that accommodating with the Big Question would leave things very unconstrained: any assertion that eliminates any world in cs c ∩ a c is relevant to it. But 41 See also the China example (47)  before worrying too much about this, note that-as exemplified by the XKCD examples (83) and (84) issue of what things will be like in respects that bear on whether inviting Professor Plum best achieves the speakers' common goals for the talk series. Suppose that the speakers definitely want to invite Professor Plum if he will give a semantics talk, they do not want to invite him if he will give a phonology talk (as the previous two talks in the series were by phonologists), and it isn't yet common ground what kind of talk Plum would give. Then the accommodated question is presumably whether Professor Plum gives semantics or phonology talks and this issue is open even on the assumption that Plum is invited.
At this point, we are relying on QUD accommodation without much discussion of how such an accommodation process might be constrained, and much of the explanatory power of our account rests on understanding such constraints. To address this concern, we spend the remainder of this paper developing a more careful, detailed account of how QUDs can be constructed in practical, action-directed exchanges like (86) from the real-world domain goals and plans of the speakers. To be clear, QUD accommodation is a very complex phenomenon and we are certainly not 43 In fact, when it comes to examples like (83) and (84) where a what if question heads a sequence in which follow-up questions immediately whittle down the space of inquiry, one might reasonably wonder whether accommodation is even necessary at all. Plausibly, hearers do not need to accommodate in such question sequences because the speaker herself quickly corrects the context by asking more questions that render it inquisitive. 31 early access Bledin and Rawlins going to try to explain how this kind of repair works in all cases in which what if s trigger it-to repeat, we restrict our attention in what follows to a range of practical contexts. Nor are we even looking to give a comprehensive account of the repair in (86) and related practical cases; though the discourse moves in our formal system from §4 might suggest otherwise, it is far from clear that the complete dynamics of repair in such examples lends itself to a compact systematization in terms of general principles. Our more limited aim in what follows is just to put one particular dimension of the process of QUD accommodation into the formal spotlight by developing a working theory of how new questions introduced in discourse are often constrained by the underlying domain goals in play (like the goal of arranging a successful colloquium) and how this relevance constraint provides one mechanism by which speakers coordinate on QUDs in a range of contexts.
Because QUD accommodation isn't specific to what if constructions, the remainder of the paper has reach beyond what if. The theory that we develop can be carried over to more standard question constructions that trigger the same kind of repair mechanism. Consider the following example (from Josh Dever p.c.): (88) A: Who are we going to invite to speak at the next colloquium? B: Should we invite Professor Plum? C: He'll give a phonology talk. A: That would be our third one in a row.
Here, too, C's response seems to call for the conversational relevance of the question

Bringing in Decision Problems
Our proposal about QUD accommodation in practical contexts will take a bit of setting up. In this section, we first review how speakers' domain goals can be explicated using decision problems (DPs) and show how these DPs can be embedded into a broader "conversational scoreboard" (to borrow Lewis' 1979 metaphor) that extends our formal discourse model from §4. We then define a "Subservience" constraint between new questions and the active DPs in a context. The payoff comes only later in §7 and §8 when we take another look at our colloquium example (86) and also consider a challenging use of what if that similarly forces accommodation with a new QUD shaped by the speaker's goals and interests. With Subservience to wield, we will be in a better position to predict the repair in such cases. 32

Decision Problems
Up to now, we have focused on only a single kind of discourse goal: the goal of publicly resolving a QUD by adding one of its answers to the context set. But besides these goals of inquiry, interlocutors often have various non-discursive domain goals that they hope to achieve in the world, such as finding a newspaper, making it to a party on time, or arranging a successful speaker series. These discourse and domain goals are not independent-the domain goals or plans of a group of interlocutors will generally dictate the questions they take up because, as Roberts (2012b) recognizes, "we are, naturally, most likely to inquire first about those matters that directly concern the achievement of our domain goals" (p. 7).
To integrate domain goals into our model of context, let us now bring in some ideas from decision theory. It has become increasingly popular for linguists to appeal to decision problems when the natural language phenomenon that they are investigating is sensitive in some way to the real-world domain plans and interests of speakers (for some applications, see van Rooy 2003b on questions, Davis 2009 on Japanese discourse particles, Franke & de Jager 2010 on awareness, Kaufmann 2012 on imperatives, Malamud 2012 on plural definites, and Cariani, Kaufmann & Kaufmann 2013 on deliberative modals). DPs encode an agent's preferences-partly in the form of a utility function-over a set of relevant outcomes that can obtain if the agent acts in certain ways and certain states of the world prevail. Note that this is precisely the kind of underlying information that B's suggestive what if in (86) seems to target: if A and B decide to take the collective action of inviting Professor Plum to speak at their colloquium, then what will things be like in respects that they care about?
For present purposes, it will be helpful to work with a somewhat nonstandard, purely qualitative formulation of DPs: Orthogonality condition: DP is well-formed iff for each a ∈ A and s ∈ S, a ∩ s = / 0 (i.e., A and S are pairwise compatible or "orthogonal" in the sense of Lewis 1988a).
Unlike in typical multi-sortal presentations, we take both actions and states to be propositions (see Lewis 1981, Jeffrey 1983 for approaches along these lines). The 44 We do not require that the actions in A or states in S cover all of W . 33 action set A specifies the alternative options available to the decision maker, who can act at will to realize any of the propositions in the set. In contrast, which state in S obtains is assumed to be outside the agent's control. We take it that each action-state pair a, s determines a particular outcome that the decision maker cares about, like making it to a party on time, being late to a party, hosting a successful colloquium, and so forth. This outcome holds throughout the set of worlds a ∩ s which are nonempty so long as DP is well-formed. 45 Which action the agent performs will depend on the preferences she has over the achievable outcomes and on how she goes about making decisions. Her preferences are encoded in the utility function U, where U(a, s) ≤ U(a , s ) just in case the possible worlds in a ∩ s are at least as preferred as those in a ∩ s. 46 We work with only ordinal properties of an agent's preferences in this paper. While a utility function can also encode the relative strength or intensity of preferences, we do not require such additional information in what follows.
Together with acts, states, and utilities, DPs used to model decisions under risk (as opposed to those under strict uncertainty) also typically include a probability measure Pr over the state space S that represents the agent's degrees of belief or credences in these exogenous states. With both probabilities and cardinal utilities to draw on, rational decision makers can then be regarded as expected utility maximizers who will refrain from performing any action in A whose expected utility across the state space is less than some alternative action. However, we are looking to bring in neither probabilities nor the methods of statistical decision theory (unlike van Rooy 2003b,a, Franke & de Jager 2010, who put these to good use). So we will continue to qualitatively represent the shared public beliefs of a group of speakers using a Stalnakerian context set, a separate component of the discourse context. We also assume that speakers facing DPs in practical deliberative contexts want to get rid of any uncertainty they have about what to do by reaching a future discourse context where their problem is resolved in the sense that one of their potential actions has optimal consequences in every state that remains open (more on this in the next section).
Despite these differences, domain goals can still be modeled in a fairly typical fashion. For a stock example, suppose I am looking to purchase an Italian newspaper and I consider whether to walk to the station or to the palace to buy one (Groenendijk & Stokhof 1984, van Rooy 2003a. Assuming that all I care about is getting my 45 The orthogonality condition is meant to capture the requirement of Savage (1954) that the acts and states used in framing DPs be independent of each other. 46 For readers familiar with Condoravdi & Lauer (2012): the set of propositions {a∩s : a ∈ A, s ∈ S} with ordering a ∩ s ≤ a ∩ s iff U(a, s) ≤ U(a , s ) is a "preference structure" (we could have taken this ordering as basic).
hands on a paper and avoiding unnecessary movement, my goal can be represented as follows: λ w s .newspaper available only at station in w, λ w s .newspaper available only at palace in w, λ w s .newspaper available at both locations in w, λ w s .newspaper available at neither location in w          U(λ w s .go to station in w, λ w s .paper only at station in w) = 1 U(λ w s .go to palace in w, λ w s .paper only at station in w) = −1 U(λ w s .stay put in w, λ w s .paper only at station in w) = 0 etcetera.
If Italian newspapers are available at the train station, then I do well to go there. Same thing with the palace. But if I can get a paper at neither place, then I am better off staying where I am, as I am bound to lose a util no matter where I walk.

DPs in Context
To situate decision problems in a broader account of discourse structure, our next step is to generalize the topic stacks from §4 to goal stacks by letting DPs coexist alongside questions in these data structures (cf. the shift from "information structure" to "intentional structure" in Roberts 2004Roberts , 2012aRoberts , 2015Roberts , 2018; you can think of the theory-building in this section as an attempt to flesh out some aspects of Roberts' intentional structure in a decision-theoretic way): 47 A context c is a tuple cs c , a c , A c , G c with cs c , a c ⊆ W as before but where the table now includes a goal stack G c loaded up with both DPs and question denotations in addition to the assertion stack A c .
Each element on G c encodes a mutually recognized discourse or domain goal of one or more speakers, having been explicitly introduced or made salient in prior conversation. Living within G c is the (possibly empty) substack G Q c of questions awaiting resolution. This representation of the speakers' common interrogative goals is just the topic stack from our earlier tables relabeled. The remaining elements on G c − G Q c are DPs representing any public domain goals of the speakers in the discourse. The speakers will of course have all sorts of real-world goals and interests at the time they are talking, but the goals reified on the stack as DPs are those immediately relevant to the conversational exchange that is directed at resolving them. We reference this (possibly empty) substack of DPs with G DP c . As before, the goals on G c are interpreted through the lens of the current discourse context. The current QUD in context can still be defined as in (59) simply by replacing Q c with G Q c : Where c is a context, After our decision-theoretic upgrade, we can also define the analogous concept of a DP in context. This is generated by restricting the top element of the DP substack, top(G DP c ) (the "current DP"), to the live alternatives in the context. We intersect the elements of its action set and state space with cs c ∩ a c (holding on to nonempty propositions) and assign each of the resulting action-state pairs the same utility as the pair it was derived from ( Now recall the earlier relevance condition from Roberts discussed in §4, according to which the complete answers to questions higher up on the topic stack must partially answer questions lower down. We can now state a decision-theoretic analog of this: if the speakers' current goal is to resolve a DP on the goal stack, then the complete answers to any new question added to the stack should help to resolve it; that is, the speakers should ask questions as part of strategies for achieving their underlying domain goals, where the answers to their questions can ultimately help them decide what to do in the world. We take both of these relevance constraints to be operative in a discourse.

Subservience
When is a DP "resolved" exactly? And how can the answers to questions "help to resolve" DPs? To formalize the new notion of relevance that we are after, we follow van Rooy (2003b,a) and first assign each action of a DP the set of states in which it is optimal (i.e., where there is no alternative action that is strictly better): (95) Best action sets (BASes) Given a decision problem DP = A, S,U : a. The best action set for a ∈ A is a * = {s : U(a, s) ≥ U(b, s) for all b ∈ A} b. The best action set for DP is Q DP = { a * : a ∈ A} Notice our use of the notation "Q DP " in (95-b) because the BAS for DP is a Hamblinstyle alternative set corresponding to the issue What is the best thing to do?. In the Italian newspaper example (90), for instance, the best action set is Q (90) =    λ w s .newspaper available at station in w, λ w s .newspaper available at palace in w, λ w s .newspaper available at neither location in w    The top proposition (λ w s .walk to station in w) * is the union of the best action set for heading to the train station, the middle proposition (λ w s .walk to palace in w) * consists of those worlds in which heading to the palace is best, and the bottom proposition (λ w s .stay put in w) * consists of worlds in which you do well to stay where you are. Note that the first two alternatives overlap, as walking to either the station or palace hits the maximum utility if you can get an Italian newspaper at both locations.
We can define DP resolution in terms of BASes. Intuitively, a decision problem is resolved in a discourse context just in case one of the members of its action set restricted to the context has optimal consequences come what may. In this happy situation, there is no longer any uncertainty about how best to achieve the explicated domain goal: Next, we can say that a proposition P resolves a decision problem DP iff informationally updating with this proposition takes us to a context in which this problem is resolved (van Rooy 2003b): (97) Resolving DPs Given a decision problem DP that is not yet resolved in c: P resolves DP in c iff DP is resolved in cs c a c P, a c , A c , G c .
It might help to think of (97) as the decision-theoretic analog of complete answerhood for questions. The notion of "helping to resolve" a decision problem, which one might think of as the decision-theoretic analog of partial answerhood, is more complicated. Earlier in §4.2 when we introduced our Assert update (61), we noted how information that fails to eliminate any live options can still be relevant to a QUD by virtue of shifting probabilities over its alternatives or bringing new possibilities to a speaker's attention. Presumably, information can also help to resolve a DP in context in these ways. However, if we focus exclusively on the world-excluding impact of assertion, then we can define a relatively straightforward notion of partial resolution for decision problems. The basic idea is this: a proposition P helps to resolve a DP in context iff P rules out at least one member of either its contextually restricted action set or state space and in so doing brings the decision maker closer to a situation where her choice is clear. 49 But, importantly, not just any action or state will do. Regarding actions: P helps to resolve the DP only by eliminating a potentially optimal action a that is "in play" in the sense that there is no alternative action b that is optimal in all of the same states as a and in some additional ones besides; if there is such an alternative action b, then eliminating a needn't help to resolve the DP since b is strictly closer to being a resolving action anyway. Regarding states: P helps to resolve the DP only by excluding a "conflict state" where some action in play is optimal while some other action in play is not, thereby helping with the choice between these competing actions.
This proposal can be formalized as follows: (98) Acts in play & conflict states Given a decision problem DP = A, S,U : a. Action a ∈ A is in play iff there is no b ∈ A s.t. a * b * . 49 We implicitly assume that the new information P is wholly about either the action set or state space in the sense of Lewis (1988a). b. State s ∈ S is a conflict state iff there are actions a, b ∈ A in play such that s ∈ a * but s ∈ b * . (99) Helping to resolve DPs Given a decision problem DP that is not yet resolved in c, P helps to resolve DP in c iff one of the following holds: a. P ∩ a = / 0 for some action a of DP ⊗ (cs c ∩ a c ) in play, or b. P ∩ s = / 0 for some conflict state s of DP ⊗ (cs c ∩ a c ).
It follows from our definitions that information which resolves a DP also helps to resolve it, but the converse needn't be true.
We are now in position to articulate more precisely the DP-based relevance constraint from the end of §6.2. Working alongside the felicity conditions in our earlier Question update (64), we assume that the following felicity condition governs both conditional and unconditional question-introducing speech acts: If the speakers in c face a decision problem top(G DP c ) that is not yet resolved in c (i.e., DP c is unresolved) and a speech act is performed that results in a new question Q being pushed onto the goal stack, then this speech act is appropriate only if completely answering Q (in the sense of (57-b)) helps to resolve DP c (in the sense of (99)).
For example, if a speaker asks a conditional question using if ϕ, ψ? in c and thereby shifts the context to c + Assume(ϕ) + Question(ψ?), then the complete answers to ψ? in this posterior context must each help to resolve the current DP in the initial context c. Importantly, it does not suffice that the answers to ψ? help to resolve the current DP in the local context c + Assume(ϕ); indeed, as we will later see in §8, the current DP might already be resolved in this local context. We began §6.1 with the informal platitude that speakers in practical contexts will generally take up questions the answers to which can help them achieve their real-world domain goals and plans. Though such an idea is difficult-perhaps impossible-to formalize perfectly, we now have a first-draft working theory that implements it. In the next two sections, we put this theory to work. We consider a couple of cases where challenging and suggestive what if s trigger QUD accommodation and show how the Subservience condition (100) constrains the repair possibilities in these examples.

Case Study: Resisting Commands
We will revisit the troublesome colloquium example (86)  But how can we recover this result? Since we aren't looking to defend a full theory of imperatives here, feel free to think of A's directive as adding the property of opening the window to B's "To-Do List" (Portner 2004, cf. Roberts 2004, 2018, issuing a performative using a deontic modal (Kaufmann 2012), expressing an "effective preference" for the window to be open (Condoravdi & Lauer 2012), or performing some other function. We require only two things. First, that the denotatum of A's imperative (whatever it is) together with the conditions of authority and deference in the context of issuance make it such that B now faces a decision problem-he wants to satisfy A's wishes with respect to the window but thinks that A may have failed to consider the possibility that it is raining when issuing her command, so B is not sure whether opening the window is in fact the thing to do. And, second, that this decision problem (call it "DP B ") is added to the goal stack at some point before B's what if question is processed. 50 Bracketing off any other discourse effects of A's imperative, let us take up the example in the context c 0 = cs c 0 , W , , DP B , which has an empty assumption slot, empty assertion stack, and B's decision problem sitting on the goal stack ready for his what if question to exploit. Applying the what if update intersectively updates the assumption slot with the proposition that it is raining: = c 0 + Assume(It is raining) (+QUD c on CQ version) = cs c 0 , It is raining , , DP B The state space of B's decision problem restricted to c 0 + Assume(It is raining), represented by the light gray columns in the below decision matrix, partitions the raining-worlds in the context set into those in which A wants the window open (regardless of rain) and those in which she wants it closed: Assuming that A's command did not raise or respond to a QUD, B's challenging what if must also adjust the discourse context to ensure that there is a question available for re-asking (on the CQ analysis) or that the inquisitivity requirement is met (on both the CQ and SQ analyses). Which question gets added to the goal stack G c 1 ? Back in §5, we would have simply said "some coarsening of the Big Question" and left things pretty much at that. But we can now offer this more precise answer: "some question the complete answers to which each help resolve B's decision problem in c 0 (i.e., DP c 0 )". If we restrict attention to questions that aren't about B's behavior (i.e., whose answers cannot exclude an action of DP c 0 that is in play), then the accommodated question must be one whose answers exclude at least one of the light grey columns of the above decision matrix-these are both conflict states of DP c 0 as opening the window is preferred if it rains but A wants the window open, while keeping the window closed is preferred if it rains and A wants the window closed (and both actions are in play). One obvious candidate is the BAS of B's current decision problem restricted to the local context c 0 + Assume(It is raining): Q DP c 1 = λ w s .rain and A (really) wants the window open in w, λ w s .rain and A wants the window closed in w Note that various more fine-grained questions would also fit the bill. But it is presumably a rational requirement of practical questioning that one avoid unnecessary processing costs by not asking for extra irrelevant information to decide what to do (van Rooy 2003b). So we take it that A will accommodate with Q DP c 1 rather than some more specific question that also satisfies Subservience: This yields the intuitively right result. Suppose we had instead updated c 0 with the conditional interrogative (102): Suppose we are working our way through the agenda at a meeting when the chair broaches the subject of colloquium scheduling with the lead-off question in (103). How does this change the discourse context? Well, first and foremost, this adds the denotation Who will we invite? from (87) to the goal stack. The chair's question also has a secondary effect: because this question clearly subserves (and is mutually understood to subserve) our common domain goal of hosting a successful colloquium, the question raises the domain goal to salience.
Let us begin our analysis in the following context, where our goal of hosting a good talk is represented by the decision problem DP C , which rests below the chair's question on the goal stack: Now recall the trouble: though there is a question Who will we invite? sitting on top of the goal stack G c 1 , it does not partition the light gray invite-Plum row into multiple cells. So the inquisitivity condition of the what if update is violated and the context must be repaired. Fortunately, as in our previous case study in §7, the decision-theoretic structure in c 1 helps to narrow the repair options. Given that we are presently focusing on the possibility of inviting Professor Plum, it is natural to assume that we will accommodate with the coarsening of the state space of the current DP that concerns him-call this "Q Plum ": 51 51 Suppose, following Lewis (1988a), that Professor Plum determines a subject matter (i.e., a partition of a subset of W ) that groups together worlds that are exactly alike with respect to his state. More accurately, then, we can take Q Plum to be the finest common coarsening of this "Plum matter" and the partition of conflict states of the current DP. See Lewis (1988a,b), Yablo (2014) for more on the mereology of subject matters. c 2 = c 1 + Repair! = cs c 0 , We invite Plum , , Q Plum , Who will we invite? , DP C In our simplified setting: Q Plum = λ w s .Plum gives semantics talks in w, λ w s .Plum gives phonology talks in w So B is effectively asking the following conditional question: (104) If we invite Plum, will he give a semantics or phonology talk?
More generally, B is asking the following question: (105) If we invite Plum, then what will things be like Plum-wise that bear on whether we made the best choice?
C goes on to answer this question: c 3 = c 2 + Assert(Plum will give a phonology talk) = cs c 0 , We invite Plum , Plum will give a phonology talk , Q Plum , Who will we invite? , DP C After this assertion is accepted, Professor Plum is out of the running. There is a complication, though. 52 Note that if we apply our earlier domainrestricted informative update (62) when C's reply is accepted, then this removes the possible worlds from the context set in which we invite Professor Plum and he gives a semantics talk but leaves any world in which we fail to invite him untouched (in the decision matrix, the update eliminates only the left-middle cell, not the full left column). So, strictly speaking, C's answer does not help to resolve DP c 0 as required by Subservience. But, intuitively, it does help: though the presence of will in C's response indicates subordination under the supposition that we will invite Plum to speak, and the information that Plum will give a phonology talk is clearly conditional on our invitation, C is also providing us with the unconditional information that Plum gives phonology talks, as Plum's research area is independent of our invitation.
The more general problem is that our update (62) is overly restrictive. When speakers convey information that is recognizably independent of what is being currently assumed in a context, we want to be able to unrestrictedly update the context set with this unconditional content. To implement this, let us now assume that each discourse context c comes equipped with a binary relation ⊥ c between propositions where P⊥ c P iff P and P are independent. 53 For any asserted proposition ϕ , we can then define the set of its contextual entailments in c that are independent of a c (we require that W ∈ ϕ ⊥a c so this entailment set will be nonempty): (106) ϕ ⊥a c = {P : ϕ ∩ cs c ∩ a c ⊆ P & P⊥ c a c } The fix is now to redefine our informative update so that it eliminates not only the not-ϕ-worlds in cs c ∩ a c (as before) but also any worlds throughout the rest of the context set that are excluded by a member of ϕ ⊥a c : (107) Informative update v. 2 cs c a c ϕ = (cs c ∩ a c ∩ ϕ ) ∪ ((cs c − a c ) ∩ ϕ ⊥a c ).
a ϕ ϕ ⊥a c cs The intuitive idea is this. If an assertion is accepted, we should at least update cs c ∩ a c with its content ϕ . However, because the assertion might depend on what is currently being assumed, we do not in general want to update the full context set cs c with ϕ . In contrast, any P ∈ ϕ ⊥a c is independent of the current assumptions, so speakers can safely update the full context set cs c with P. At one extreme where (cs c − a c ) ⊆ ϕ ⊥a c , the new update (107) coincides with the older update (62). At the other extreme where ϕ = ϕ ⊥a c , the assumptions are disregarded and the new update amounts to regular intersection: cs c a c ϕ = cs c ∩ ϕ . 54 Assuming that We invite Plum ⊥ c 3 Plum gives phonology talks , C's assertion now has the desired effect: c 4 = c 3 + Accept + Dispel + Clear = cs c 0 ∩ Plum gives phonology talks , W , , Who will we invite? , DP C 53 This qualitative notion of independence might be spelled out as in Goebel (2017) using Veltman's (2005) "cognitive states" or by using the causal modeling apparatus of Pearl (2000). 54 The update (107) allows a treatment of biscuit conditionals (BCs; Austin 1956, Franke 2009, Francez 2015Goebel 2017;Biezma & Goebel ms.) and biscuit what ifs where 'normal' biscuit conditional antecedents are posed as what if questions that license non-subordinated answers (Franke's "intelligibility conditionals" do not tend to work): (i) There are biscuits on the sideboard if you want them.
(ii) A: What if I want some biscuits?
B: There are some on the sideboard.
Assuming that A wants biscuits ⊥ c There are biscuits on the sideboard , accepting (i) or B's reply in (ii) will incorporate the unconditional information that there are biscuits on the sideboard into the context set.
By ruling out the entire left column of the decision matrix, it leaves the best action set for inviting Plum empty.

Conclusion
We have analyzed what if s as conditional/suppositional questions: they have the suppositional semantics of regular conditionals combined (at least) with a postupdate requirement for an inquisitive context, found with questions in general. This, together with the integration of decision problems into discourse structure and complicated but general principles involving QUD accommodation, has allowed us make progress on our  (2010), often involve accommodation with new QUDs constructed from contextually salient DPs in order to meet the inquisitivity constraint.
The fact that what if questions are so flexible follows from two main factors on our proposal: (i) the inquisitivity condition in the semantic entry for what if, and (ii) the range of maneuverability that agents have in discourse when trying to infer what QUDs their interlocutors intend. We have suggested that in general this accommodation obeys a number of constraints revealed by what if questionsconstraints on what kinds of QUDs are available in speculative hypothetical cases, constraints on how decision problems and QUDs interact, and so forth. An important part of our analysis has been a more general account of relevance to goals in discourse beyond just QUDs: what if s provide an argument for a model of discourse that allows for this sort of deeply underspecified move.
There are a number of open problems. One future task is to extend our formal system to handle various other linguistic phenomena, like counterfactual what if s. There is also the pressing issue of how DPs become salient. Considering other morphology that interacts with DPs (Davis 2009, etc.)  These data raise a number of puzzles for the analysis of conditionals in general. If, as we have suggested, what if questions are consequent-less conditionals, then any theory of conditionals that deeply relies on the existence of some kind of consequent might need to be revisited. For example, if conditionality relies on a main-clause operator (a position defended by many researchers in linguistics beginning with Lewis 1975, Kratzer 1977, Heim 1982, including one of the authors of this paper in Rawlins 2008), what do we make of the fact that there is no evidence for such an operator in what if s? The success of this account is prima facie evidence for a suppositional account of if -clauses, and works against accounts of conditional compositionality that structurally require the if -clause to interact with an operator (e.g., the restrictor account as usually implemented). This is not to say that we want to reject the insights of such a theory, but rather we suggest that this treatment of what if s calls for a reconciliation between the kind of suppositional view we advocate here and the data motivating the restrictor account. This reconciliation, necessary though it may be, has only begun, and we suggest that broader investigation of discourse conditionals both in English and across languages provides an important new direction for research on conditionals.