Context Structure for Dialogues

Context Structure for Dialogues Kim B. Bruce Pomona College Donka F. Farkas UC Santa Cruz August 8, 2007 1 Introduction The foundation of current work in formal pragmatics is Stalnaker (1978), where it is suggested that discourse unfolds against an ever changing background made up of a set of propositions already accepted by the discourse participants (the common ground). The essential effect of assertions on the context in which they are made, according to Stalnaker, is to add the content of the assertion to the common ground, provided, Stalnaker adds, that there are no objections from the other participants in the conversation. Assertion, then, is a proposal to change the context by adding the propositional content of the asserted sentence to the common ground of the conversation. 1 The proposal nature of assertive conversational moves, in our view, has not been given the weight it deserves. Our main goal in this paper is to remedy this shortcoming. Section 2 is devoted to developing an expanded context structure that allows us to capture the proposal nature of assertions and makes room for conversational moves that accept or reject them, and Section 3 gives the notation to be used in the rest of the paper. We then explore the consequences of this structure with respect to assertions and polar questions (Sections 4-6). In Section 7 we turn to the consequences of our proposals for capturing similarities and differences between moves that react to assertions and moves that react to polar questions and in Section 8 we examine some consequences for the treatment of presuppositions. Section 9 provides an extended example of the use of context states, and Section 10 concludes by summing up the main results of the paper and by looking ahead at research directions it opens. In the rest of this section we pose the questions the paper is meant to provide answers for. 1.1 Assertions and Questions Our empirical focus is to capture similarities and differences between a series of conversational moves. First, let us consider the move of making an assertion (uttering a declarative sentence with falling intonation) and that of asking an ordinary polar question by means of uttering a polar interrogative sentence. On the similarity side, note that both types of moves allow the same reactions, as exemplified in (1) and (2): (1) A: Sam is home. B: Yes/Yeah, he s home./no, he isn t home. 1 A large amount of work since Stalnaker (1978) has been devoted to the need to refine the notion of context so as to get to the level of discourse referents. The issues we are dealing with here are not relevant to these concerns and therefore they will be ignored below. 1

(2) A: Is Sam home? B: Yes/Yeah, he s home./no, he isn t home. On the differences side, note that a no answer is more dramatic in (1) than in (2). While B may be accused of being a bit laconic and rude in (3), in (4) there seems to be something amiss with both A and B (assuming A s utterances are made with falling intonation): (3) A: Is Sam home now? B: No, he isn t. A: Has Lee started dinner yet? B: No, he hasn t. A: Is there food in the fridge? B: No, there isn t. (4) A: Sam is home now. B: No, he isn t. A: Lee has started dinner already. B: No, he hasn t. A: There is food in the fridge. B: No, there isn t. The difference between negative reactions to assertions and negative reactions to questions affects their formal properties as well. Thus, there are negative answer forms that are appropriate when responding to a question but not when reacting to an assertion. In Romanian, the particle ba is possible when contradicting a positive assertion but not in a negative answer to a positive question: (5) A: Horea e acasă? Is Horea home? B: Nu, nu e./*ba nu, nu e. No, he isn t. (6) A: Horea e acasă. Horea is home. B: Nu, nu e./ba nu, nu e. No, he isn t. Note also that, typically, assertions do not necessarily require a reaction (though they are consistent with one), while also typically, questions do require an audience reaction. 2 The formal treatment of assertions and polar questions should help capture both the overlap and the differences in possible addressee reactions. Next, note that various types of assertions have radically different effects on the conversation in which they occur. The utterances of A and B in (1) above are both assertions and yet their conversational effects are sharply different. A s move is initiative in that it proposes the addition of a new proposition to the common ground. B s move, on the other hand, is reactive: it is offered to signal that B accepts A s proposal. The particles yes and no in these examples, we suggest, signal precisely this reactive nature of the move. Thus, it would be inappropriate for A to start the conversation with the utterances in (7): (7) a. Yes, Sam is at home. b. No, Sam is not at home. There are two inter-related questions that have been raised so far: (i) how to account for the similarities and differences between ordinary assertions and ordinary polar questions when it comes to addressee reactions; 2 Atypical cases are assertions flagged by a special interrogative tag that requires addressee reaction, and questions specially marked making an answer optional, or rhetorical questions that do not have to be answered. 2

(ii) how to account for the similarities and differences between initiating and reactive assertions. With respect to the first issue, under the standard view, the C(ontext)C(hange)P(otential) of assertions is the addition of their propositional content to the common ground (or to their author s commitment set, in Gunlogson 2001). As a result, assertions are taken to eliminate from the context set the worlds that are not in their denotation. The CCP of questions, on the other hand, is the partitioning of the context set into cells that correspond to full answers to the question. Under these assumptions, there is no reason to expect significant similarities in reactions to these two move types. With respect to the second issue, under standard assumptions B s assertion in (1) is uninformative in the affirmative case, and inconsistent in the negative case. And yet, in the former instance B s move serves a conversational purpose that differs from mere repetition. In the latter instance, although B s move places the conversation in an inconsistent state, B herself is most likely innocent of the charge of inconsistency (unless she previously committed to the proposition that Sam is at home now). These simple examples point to the necessity of refining our theoretical tools. Below we propose an articulated context structure which allows a finer-grained characterization of the contextual effects of assertions and questions. 1.2 Special discourse states, special discourse moves There are two further types of issues that motivate our proposals. One is the necessity of distinguishing between discourse states that can serve as natural end states of a conversation and discourse states that can not. Thus, the state of a discourse immediately after a question has been asked, but before the conversation participants have had a chance to react to the question would not be a natural end state. On the other hand, a context state that results after all the issues that have been raised up to that point have been settled or at least addressed to everybody s satisfaction, is one where the conversation can gracefully end. The notion of conversational table that we introduce below is crucial in capturing this difference. With respect to special moves, note that until now we have talked of questions and initiating assertions raising issues and reactive moves that address the issue. Such reactive moves may lead to settling the issue to everybody s satisfaction and thereby increasing the common ground. There are, however, more exotic conversational moves that we have to be able to model. Participants in a conversation may disagree on an issue and then decide to get out of the impasse that the disagreement creates by agreeing to disagree. Similarly, a question can be removed from the discourse table not only by settling it but also by the participants agreeing not to pursue it further. It is desirable to make room for such conversational moves, and at the same time capture in what way they differ from moves that resolve a question or accept an assertion. In the next section we propose an expanded context structure and then explore its consequences with respect to the issues mentioned above. 2 An expanded context structure 2.1 Context components We follow here Stalnaker (1978) in taking the common ground to be an essential component of context structure. The common ground of a context state K, cg K, is the set of propositions that all the participants in the conversation have publicly accepted as being true of the world in which the conversation takes place, w K. The propositions in the common ground are joint public commitments. We also follow here Stalnaker s original insight, as developed in Gunlogson (2001), in assuming that the propositions that a participant publicly commits to during a conversation are entered on his or her commit- 3

ment list. In Gunlogson, the common ground is decomposed, as it were, into each participant s commitment list, with the common ground itself being defined as an ancillary notion made up of the union of the participants lists. For her then, the main effect of assertions is to publicize their author s commitment to the asserted proposition. For Stalnaker on the other hand, the main effect of assertions is to reach a context state where the asserted proposition is added to the common ground. We wish to capture both effects. We therefore separate the common ground as a special context component, while maintaining participant commitment lists as separate elements as well. A participant s commitment list is made up of public commitments that have not (yet) become joint commitments. The common ground is made up of those propositions that have reached joint commitment status. If all that mattered was publicizing our commitments, there would be nothing strange about participants publicizing a series of mutually inconsistent commitments as long as each participant s own commitments are consistent, as in (4) above. We therefore maintain that part of the effect of an assertion is to register the proposal of having the asserted proposition added to the common ground without in fact adding it yet to the current common ground. In order to capture the proposal nature of assertions (and that of other conversational moves), our contexts contain a special space we call the Table, where matters under discussion are entered. In this, we follow Büring (2003), Roberts (1996) and Ginzburg (1996), among others, who emphasize the role the question under discussion (QUD) plays in discourse. The Table is the part of the context structure that registers what is at issue. We assume that every conversational move that places an item on the Table brings with it a canonical way of removing that item from the Table. A way of removing an item from the Table counts as canonical if and only if it is a step that eventually leads to addition of information to the common ground. This, in effect, captures the Stalnakerian view that the wish to increase the common ground is one of the main engines that drives discourse. We treat as canonical reactions conversational moves whose input contexts have a nonempty Table and whose effect directs the conversation towards context states where the Table is emptied in a way that leads to an increase in the common ground. Placing an item on the Table then projects a set of future context states arrived at by performing the operations involved in the canonical removal of the relevant item from the Table. We represent them here as a set of future common grounds. Assertions project future common grounds where the asserted proposition is true. Questions project a set of priviledged future common grounds that result after possible answers to the question have been added to the common ground. We use a special context space, called the p(rojected) s(et) to record the projected common grounds that result from canonical removals of the items currently on the conversational Table. When the Table is empty, the ps contains the current common ground as its only element. When the Table is not empty, the common grounds in ps are computed based on what counts as a canonical removal of the items from the Table. 3 Because canonical removals of items from the Table result in additions to the common ground, projected common grounds are supersets of the current common ground. In particular, all sets in the ps of a context structure will be supersets of that structure s common ground. We represent the context K of a conversation between two participants A and B in diagrams of the form below: set. 3 Our ps is different from the projection set of Gunlogson (2001), which contains all non-empty subsets of the current context 4

A Table B DC A S DC B Common Ground cg Projected Set ps Figure 1 In Figure 1, DC A and DC B are the sets of propositions A and B have individually publicly committed to and which are not yet joint commitments. The cg is the set of propositions that are joint commitments of the participants. The set of public commitments of a participant A is DC A cg. If a participant is sincere, the union of her commitment list and the common ground is a subset (hopefully a proper subset) of the propositions she actually holds as true of w K. 4 The contents of the Table, S, are statements and questions still under discussion, while ps is a set of common grounds that reflects canonical ways of settling the issues on the Table. Given that commitments in a conversation are propositions taken by the relevant participants to be true of w K, for each participant X, the sets DC X and DC X cg have to be consistent. 2.2 Justification and uses of the expanded context structure Before presenting the details of our proposals concerning assertions and polar questions in this new structure we briefly justify here its non-standard components. The separation of public commitments by participants is crucial in treating an assertion as a proposal to add the asserted proposition to the common ground. After A has asserted a sentence S with propositional content p, A is committed to p, but, until B has signaled acceptance, p is not yet in the common ground. Separating participants discourse commitments from the common ground is crucial in accounting for disagreements in coherent discourses. It is desirable, we suggest, to allow discourses to survive contradictions by allowing moves whereby participants agree to disagree. Assume A and B have agreed to disagree on p. The state of the discourse in this case is as in Figure 2, where s is the set of propositions in the common ground: A Table B p p Common Ground s Projected Set ps = {s} Figure 2 Here then A is committed to p, B to p but neither proposition is in the common ground. 5 The context structure we assume allows us to define consistency at multiple levels, as in (8): (8) a. A context state K is globally consistent if and only if the propositions in cg K are consistent. b. A context state K is consistent relative to a participant X if and only if DC X cg K is consistent. There are two sources of conversational crisis in this view. The first obtains when a discourse state K is 4 An important and difficult issue with which we will not deal here is what happens if participants are wrong, i.e., what happens if either commitment lists or the common ground contain propositions that are in fact false in w K. 5 The move of agreeing to disagree is defined formally in Section 4.2.2. 5

inconsistent at any of the two levels mentioned above. The second obtains when all common grounds in the projected set are inconsistent. In such a situation, one cannot empty the Table in a canonical way without reaching a globally inconsistent context state. Note that a context state can be consistent at both levels without the commitment lists of the participants being mutually consistent. In Figure 2 this is the case if s is consistent and if both p and p are consistent with s. Such a discourse then is not in crisis. In order for A and B to remain consistent, A s future commitments will have to be consistent with p, and B s future commitments will have to be consistent with p. Discourses therefore may stay globally consistent and have consistent participants even after public disagreements. This, we think, is a welcome result. Keeping commitment lists separate from the common ground allows us to capture what we take is a common conversational goal, namely increasing the store of information the conversational community has about the world in which the conversation takes place. For us, just as for Stalnaker, this amounts to increasing the common ground. As mentioned above, the Table plays a central role in capturing the proposal nature of ordinary assertions, and thus in making room for moves that accept or reject such assertions. It also plays a crucial role in characterizing the conversational move of agreeing to disagree as well as in our account of various subtypes of assertions. We assume here that what is entered on the Table is a syntactic object. This assumption is needed because in order to develop a grammar for cross-turn conversation and ellipsis it is crucial to be able to refer to the grammatical form (and not just the propositional content) of immediately previous utterances. For instance, we will see in Section 6.3 that in Romanian ba da can only be uttered in reaction to an immediately previous utterance that places a negative sentence on the Table, while ba nu can only be uttered in reaction to a previous move that places a positive sentence on the Table: (9) A: Horea bea bere. Horea drinks beer. B: Ba nu, (nu bea)./*ba da, nu bea. No, he doesn t. (10) A: Horea nu bea bere. Horea does not drink beer. B: Ba da, (bea)./*ba nu, bea. Yes, he does. If we only had access to the propositional content of A s utterance here we would have to assume that negative and positive propositions are distinguished, an assumption that one does not necessarily wish to make. Items on the Table form a stack. 6 When the Table is not empty, the immediate goal of the conversation is to empty it, and therefore conversational moves that affect the Table must be relevant to whatever item is on the top of the stack. The stack discipline allows us to capture the tight connection between initiating conversational moves, such as an initiating assertion or question, and responding moves that address the issue an immediately preceding utterance has placed on the Table, a matter we come back to in Section 4.2 below. Finally, having the Table as a separate context structure component allows us to differentiate between context states that can be end states, i.e., states which can serve as natural endpoints of a discourse, and states that are not. The items on the Table are at issue. Therefore, as long as the Table is not empty the conversation is not in a stable state. When the Table is empty, there is no open issue and the only projected set is necessarily identical to the current common ground. Stable context states and possible end states are 6 See Kaufmann (2000) for a similar proposal. Other authors (see Ginzburg (1995)) prefer a partial order, but we have no need for the extra flexibility of redefining a partial order to reorder statements in the simple examples handled here. 6

defined below: (11) Stable context state A context state K is stable if and only if its Table is empty. (12) Possible end state A context state K is a possible end state of a conversation if and only if it is stable. We take it that a default conversational goal is to reach a stable output state whose common ground is larger than the common ground of the input state. As mentioned above, each conversational move that places an item on the Table steers the conversation towards a future state where the item is removed in such a way as to increase the common ground. We record these future states at the level of changes they bring to the common ground in the form of a set of common grounds we call the projected set. The projected set allows us to capture the intended changes to the common ground associated with a particular move. Recording these changes at the level of the projected set rather than the level of the current common ground allows us to capture the anticipatory nature of certain moves and at the same time makes room for conversational moves that react to them. The projected set assumes a central role in our treatment of conversational moves because it is at this level that the manipulative nature of these moves is captured. We need it crucially in characterizing what A s proposal is when she makes her assertion, as well as what the questioner directs the conversation toward when he asks his question. There are, of course, many other possible choices in designing a context structure. For the matters we discuss here further additions such as the agendas of participants or public information about their private doxastic states are not necessary. As already mentioned, since we are not concerned with referential or anaphoric issues we also leave aside the question of registering discourse referents. We also note here that the information we do represent is redundant. In particular, the projected set can always be calculated from the common ground and the items on the Table stack. We nevertheless include these components in our representation for the sake of clarity and for emphasizing similarities between what we propose and earlier approaches. Maintaining the projected set as a separate component allows us to capture the fact that in our proposal the CCP of assertions and questions is essentially the same as traditionally assumed. The difference is located at the level of the context component affected. The projected set is also useful in allowing a simple characterization of the crisis the conversation reaches when a contradiction is placed on the Table. Finally, note that we are talking here in terms of removing items from the Table. Alternatively, if needed, one could opt for never removing items from the conversational Table but rather, of distinguishing between active and non-active items on the Table. Our choice is not crucial for the matters we discuss below. 3 Notation for manipulating context structures Before turning to discussing assertions and polar questions, we define here the notation we use throughout the rest of the paper. The table T in a context structure is represented as a stack. The following notation is used to represent traditional stack operations on T : 1. push(s, T ) represents the new stack obtained by adding sentence S to the top of the stack T. 2. pop(t ) represents the stack obtained by popping off the top element of T. 3. top(t ) represents the top element of the stack T. 7

4. remove(s, T ) represents the stack obtained by removing the top-most occurrence of S from stack T. If S does not occur in T, then T is returned. Let ps = {cg 1,..., cg n } be a collection of sets of propositions (e.g., possible common grounds) and let p be a proposition. Then define ps p = {cg 1 {p},..., cg n {p}}. The operation P S(cg, T ) rebuilds the projected set ps from the current common ground cg and the contents of the Table T by iteratively performing the ps changes associated with each item on the Table. This operation is needed after a non-canonical conversational move is made, such as retracting an assertion, agreeing to disagree or withdrawing a question. The move is defined at the beginning of Section 6. 4 Assertion 4.1 The C(ontext) C(hange) P(otential) of an assertion Stalnaker s insight on assertion is that when a speaker A asserts a sentence S with propositional content p she proposes to add p to the common ground of the input context. In making her assertion, A has not only proposed the addition of p to the input common ground but has also publicly committed herself to p. We take these effects to be characteristic of default assertions. Once we discuss all subtypes, however, it will turn out that the essential change is registering author commitment. Whether the Table, and therefore the ps are affected as well, and if so, how, depends on the details of the input context state. In what follows we will first characterize default assertions and turn to subspecies in Sections 5 and 6. Let us assume then that A asserts S with propositional content p against an input context K 1, whose common ground cg is s 1. For simplicity s sake, let us also assume that the discourse commitment lists of the two participants, A and B, are empty, and so is the Table. The projected set of K 1 then includes only s 1. The representation of K 1 is as in Figure 3: A Table B Common Ground s 1 Projected Set ps 1 = {s 1 } Figure 3: K 1 In our terms, the effects of A s move of asserting S on K 1 are to add p to A s discourse commitment list, to place S on the Table, and to project a future common ground where p is accepted, i.e., a state whose common ground is built from the current common ground by adding p to it. It will be important to what follows for the Table to record whether a sentence has been entered as an assertion or a question. We assume that both declarative and interrogative sentences have a syntactic marker, D and I, encoding their declarative and interrogative status respectively. Crucial for what follows is that the sentence radical these markers attach to be visible in the discourse and therefore we treat them as features on sentences. The context state that results after A has asserted S[D] relative to the input context K 1 is given Figure 4: A Table B p S[D] Common Ground s 2 = s 1 Projected Set ps 2 = {s 1 {p}} 8

Figure 4 K 2 : A asserted S[D] relative to K 1 The common ground of the output context K 2 is the same as that of the input context, K 1. The change involves the author s discourse commitment list, the Table, and, consequently, the ps. Further conversational moves must now attend to S[D] and eventually remove it from the Table in order to reach a stable state. If at the point of asserting S[D] the Table is not empty, the input projected set, ps i, contains a (possibly singleton) set of projected common grounds. The impact of asserting S[D] in such a case is to add p to all the common grounds in ps i. If the addition of p to a common ground in ps i results in an inconsistent set, the set is removed from the output projected set, ps o. Thus, ps o is made up of the result of adding p to all the sets in ps i and then removing those sets that are inconsistent. If the resulting projected set, ps o, is the empty set, the conversation is in crisis, because then the canonical removal of S[D] from the Table results in an inconsistent common ground. We follow Krifka (2001) (see also Ginzburg (forthcoming)) in assuming that there are speech act operators that take sentences as arguments, and which are functions from input context states to output context states. 7 We define the assertion operator A as a function from an input context K i to an output context K o, defined in (13), where S[D] is a declarative sentence with propositional content p, a is that discourse participant who is the author of the assertion, DC a,o and DC a,i are a s input and output context discourse commitments, and T o, T i are the output and input Table respectively: (13) A(S[D], a, K i ) = K o such that (i) DC a,o = DC a,i {p} (ii) T o = push(s[d], T i ) (iii) ps o = ps i p Those elements of K o that are unaffected by the move have not been listed here. We will see below that all types of assertions involve the context change potential in (i). The changes in (ii) and (iii) accompany all assertions as well except for those that simply accept a previous assertion. Varieties of assertions are defined below by imposing further constraints on the input or output context states. 4.2 Reacting to an assertion After A has asserted S[D], S[D] is at the top of the stack on the Table. Consequently, the conversation is not in a stable state. The canonical future, reflected in the ps, is one where the assertion is accepted, i.e., its propositional content is added to the common ground. This is a canonical continuation because it leads to an increase in the information of the conversational community without resulting in inconsistency at any level. Accepting the assertion is, in fact, the only way of canonically removing the asserted sentence from the Table given that the author of the assertion is committed to its propositional content. Because of this commitment, the context state that results after an assertion is biased in favor of that assertion in Gunlogson s terms. A non-canonical way of reacting to an assertion is partial or total denial, which does lead to crisis. We will discuss briefly both canonical and non-canonical reactions. An ordinary assertion then commits its author to the propositional content of the asserted sentence and raises an issue (places an item on the Table) while at the same time directing the conversation towards a unique resolution of that issue, namely acceptance of the assertion. 7 Given the draft status of Ginzburg (forthcoming) we do not undertake a full scale comparison of our approach and his. 9

4.2.1 Accepting an assertion B s acceptance of A s assertion has the effect of adding p to B s discourse commitment list. If, as a result of this addition, p becomes a joint commitment in the conversation, i.e., it is now present on the commitment lists of all participants, the following changes are triggered: S[D] is popped off the Table stack. p is added to the common ground of the conversation. p is removed from the commitment lists of all participants. 8 The changes just listed occur whenever all the participants in the conversation reach agreement on the issue that is at the top of the stack on the Table. Given the need to model multi-participant discourses it is convenient to introduce an auxiliary operation, M, that occurs after a move M whose outcome results in agreement relative to the resolution of the issue on top of the stack on the Table. (14) Common ground increasing operation M If M contains a change of the form DC X,o = DC X,i {p}, and as a result p is now present on the commitment lists of all participants in the conversation in K o, add the following changes to M: 1.Pop off of the top of the Table all occurrences of items that have p as an element of their denotation. 2.cg o = cg i {p}, 3.DC X,o = DC X,o -{p} for all participants o. The common ground increasing operation M turns a commitment that is shared by all participants in the discourse into an element of the common ground and cleans items sharing that propositional content from the Table, indicating that it is no longer a focus of discussion. In a move to reduce redundancy, it also deletes all instances of p from all participants discourse commitments lists. All the elements of the input ps to M already contain p because every move that adds a proposition to a discourse commitment list also adds it to every element of the ps. After M p will be an element of the output cg as well as an element of all sets in the output ps. Returning to our example, if S[D] was the only item on the Table, B s acceptance of A s assertion triggers M, which results in an empty Table. Now the projected set contains only the new common ground, and the conversation is in a stable state. Assuming that the input context state is K 2, the output context state after B accepts A s assertion and after M applies is K 3 in Figure 5. A Table B Common Ground s 3 = s 2 {p} Projected Set ps 3 = ps 2 = {s 3 } Figure 5 K 3 : B has accepted A s assertion Acceptance of an assertion can be signaled by silence, by the particles yes, yeah, or ok, sure, or by saying right / correct. We suggest that these are acceptance particles signaling an acceptance move. Acceptance 8 This is a housekeeping move meant to eliminate redundancy. Recall that the participants in a conversation are taken to be publicly committed to the propositions in the common ground in addition to the propositions in their discourse commitment list. 10

of an assertion can also be performed by repeating the asserted sentence in a more or less truncated form preceded by yes or yeah. 9 We treat such utterances as a special subtype of assertion signaling acceptance in Section 6. The fact that the move of accepting an assertion does not have to be overtly signaled is expected given the biasing nature of assertions that results in acceptance being the only canonical way of removing the asserted sentence from the Table. We define an operator for accepting an assertion, AA, as a function from an input context K i to an output context K o, defined in (15), where S[D] is a declarative sentence with propositional content p that has been asserted by some discourse participant a in a previous move and is currently at the top of the Table as well as on a s commitment list in K i. Let b be the participant accepting the assertion. (15) Assertion Acceptance (AA) a. Input context conditions: (i) top(t i ) = S[D] with propositional content p (ii) p in DC a,i, where a is a participant other than b. b. Change: AA(b, K i ) = K o where DC b = DC a {p} The fact that p is an element of some discourse commitment list, required in condition (ii) above follows from S[D] being on the Table. We require the author of the acceptance to be a different participant from the author of the initial assertion because assertion acceptance by the author of the assertion is a redundant move in all respects. If b s move results in p becoming a joint commitment in the conversation, M is triggered whereby S[D] is popped off the Table, p is moved into the common ground and at the same time removed from the participants commitment lists. Thus, if a and b are the only participants in the conversation, b s acceptance of an assertion S[D] with propositional content p results in an output context state where: cg o = cg i {p} DC a,o = DC a,i {p} T o = pop(t i ), where S[D] = top(t i ) The operation AA is defined only for input contexts resulting after a sentence S[D] has been asserted. This is captured in the two input context conditions in (a) above. The effect of AA is to add the propositional content of the previously asserted sentence to the discourse commitment list of the author of the move. 4.2.2 Contradicting an assertion Addressees don t always acquiesce. A possible reaction to an assertion is to deny it totally or partially: (16) A: Mary ordered chicken yesterday. B: No, she didn t. C: No, it was beef B s reaction is a flat or total denial, while C s is a partial denial. We follow van der Sandt & Maier (2003) in taking denials to be special conversational moves. In our terms, a discourse move by a participant X is a 9 There are interesting differences between various ways of signaling acceptance which we will not go into here. 11

denial if and only if X asserts S[D] with propositional content p relative to a context state K i such that the top of the Table in K i contains S [D] with propositional content q, and q and p are inconsistent. We define total denial in (17), where S[D] denotes p and S [D] denotes p, and where a is the author of the denial move: (17) Total denial a. Input context condition: S [D] = top(t i ) b. Change: T D(S[D], a, K i ) = K o such that (i) DC a,o = DC a,i {p} (ii) T o = push(s[d], T i ) (iii) ps o = ps i p A total denial is an assertion: the changes it performs, listed under (b), make it an ordinary assertion: it registers author commitment and it adds a sentence to the Table. What is special about it are the conditions it imposes on its input context which entail that the output context, K o, is in crisis. In contrast with assertion acceptance, we do not have here a condition requiring the author of the denial to be a different participant from the author of the assertion being denied. This is so because conversation participants may change their minds with respect to an assertion they just made. Unlike in the case of assertion acceptance, such a move is non-redundant. It does place the conversation in doulbe crisis, however, since now the discourse commitment of the author of the denial are inconsistent. Such self-denials, we assume, are accompanied by removal of the denied proposition from the author s commitment list. Partial denials, as noted by van der Sandt and Maier, must have access to the totality of the message conveyed by the assertion they react to: its entailments, presuppositions, as well as conventional and conversational implicatures. This means that assertions have to add more than just their propositional content on the Table but we will not go into these complications here. Note also that in the case of a partial denial, the author signals acceptance of all the parts of the previous assertion that are not explicitly denied in his move. This explains why B s second assertion below leads to self-contradiction: (18) A: Mary ordered chicken yesterday at the restaurant. B: No, it was beef. A: It was chicken. I heard her order it. B: #Mary didn t even go out yesterday. We do not deal here with partial denials or with the complications an account of them would lead us to introduce. We only note that even partial denials result in an empty projected set. Total denials (or contradictions) involve rejecting the previous assertion wholesale. A conversation made up of two such consecutive moves is in crisis because the ps of the conversational state that results after B s denial is inconsistent. The starkest form of denial is asserting the opposite of the asserted sentence on the top of the input Table stack. Such a situation is given in Figure 6, where s i is the common ground of the discourse state that served as input to A s assertion and ps 4 is inconsistent: 12

A Table B p S[D] p S[D] Common Ground s 4 = s 1 Projected Set ps 4 = Figure 6 K 4 : Contradiction on the Table Denials, just as acceptance moves, are reactive, i.e., they presuppose an immediately preceding conversational move, namely an assertion. They differ from acceptance, however, in that they place the conversation in crisis. The proposition the author commits to when denying a previous assertion cannot become a joint commitment without retraction on the part of another participant, and therefore the assertion cannot be removed from the Table in a canonical way. How is a stable state to be reached once there is a contradiction on the Table? There are two possibilities: (i) one of the participants retracts their assertion; (ii) the participants agree to disagree. The former move involves removing an assertion from the Table and from the commitment list of the author, thus making room for accepting the opposite proposition, which can now be added to the common ground without leading to inconsistency. If A retracts her assertion in Figure 4, the changes involved are given below: p is removed from A s commitment list. S[D] is removed from the table. The ps is recalculated based on what remains on the Table and the current common ground. We define the operator for retracting an assertion, RA, as a function from an input context K i to an output context K o, defined in (19), where S[D] is a declarative sentence with propositional content p, and a is the discourse participant who originally made the assertion S[D] and is now withdrawing it: (19) Retracting an assertion (RA) RA(S, a, K i ) = K o such that (i) DC a,o = DC a,i {p} (ii) T o = remove(s[d], T i ) (iii) ps o = P S(cg o, T o ) 10 A possible complication is that retracting an assertion may also lead to removing other propositions that p provided support for, such as propositions that were added as entailments of p or which were accepted on the basis of having accepted p. We do not deal with these complications here. Retraction moves are most often implicit. The author of the retraction usually signals it by asserting a sentence that signals the acceptance of the denial and the retraction of the previous assertion: (20) A: Sam has left town this morning. B: Oh, no, he hasn t. I ran into him five minutes ago in the coffee shop. A: He must have changed his plans then./he hasn t left then after all./something must have come up then. 10 The PS operator for reconstructing the projected set from the common ground and Table is defined in Section 6. 13

Dealing with such mechanisms that appear to soften retractions is an interesting issue that we have to leave open at present. The move of agreeing to disagree involves removing both S[D] and S [D] from the table without removing either p or q from the relevant commitment lists. Each participant remains publicly committed to the propositional content of whatever they asserted, but neither p nor q are added to the common ground. The state of the conversation after the move of agreeing to disagree has been carried out in our example is as in Figure 7: A Table B p p Common Ground s 5 = s 4 Projected Set ps 5 = {s 5 } Figure 7 K 5 : A and B have agreed to disagree relative to K 4 The common ground in K 5 is identical to the common ground in K i, the context state that served as input to the move in which A asserted S[D]. Under the assumption that that common ground was consistent and that it was consistent with both p and p, the context state K 5 is consistent at every level and therefore the conversation is no longer in crisis even though the commitment lists of the two participants in K 5 are mutually inconsistent. Separating commitment lists from the common ground is crucial in capturing the fact that after a move of agreeing to disagree the conversation is not in crisis. In future context states, however, in order for A to stay consistent, her public commitments have to be consistent with p, while in order for B to be consistent, his public commitments have to be consistent with p. Informally, the move of agreeing to disagree results in the following changes: S[D] and S [D] are removed from the Table. The ps is recalculated from the Table and common ground. We define the operator for agreeing to disagree, AD, as a function from an input context K i to an output context K o, defined in (21), where S[D] and S [D] have inconsistent propositional contents, and T o, T i are the output and input Table respectively: (21) Agreeing to disagree AD(K i ) = K o such that a. Input context conditions (i) S[D] and S [D] are on the top of the Table. (ii) p is in DC A and q is in DC B, where A and B are different participants. b. Change (i) T o = remove(s[d], remove(s [D], T i )) (ii) ps o = P S(cg, T o ) The move of agreeing to disagree is carried out in actual conversations by a participant proposing it by uttering an imperative such as Let s agree to disagree/let s not pursue this further and the other participant agreeing to comply with the imperative. The imperative in this case is special in that compliance with it involves a change in the context state of the conversation. 14

4.3 Conclusion In this section we have put to use the context structure proposed in Section 2 in order to capture the proposal nature of assertions. The enlarged context structure makes room for capturing reactions to assertions (both acceptance and denial) and allows us to characterize the non-canonical conversational moves of agreeing to disagree and of retracting an assertion. The old insights concerning the CCP of assertive moves are preserved, but distributed among the various context components we are working with. The conversational moves are characterized by the change in the input context they bring about as well as by possible conditions they impose on various components of the input state. We are thus able to characterize what is common to the two assertive moves we have discussed so far, default assertions and denials: they add their propositional content on the author s commitment list, place the asserted sentence on the Table, and affect the projected set by adding the propositional content of the asserted sentence to all members of the input ps. Denials are special in that their propositional content is inconsistent with the propositional content of a sentence asserted in the immediately previous move and therefore they place the context in crisis. 5 Polar questions In this section we consider the CCP of polar interrogative sentences illustrated in (22): (22) Is it raining? We refer to a question speech act performed by a polar interrogative as a polar question. Polar interrogatives are closest in form, and, as we will see, in CCP, to the corresponding declarative sentence. Our aim is to capture the similarities and differences between making an assertion and asking a polar question. We focus here on polar interrogatives but some aspects of the analysis generalize directly to constituent questions as well, and those will be formulated in general terms. The special issues posed by constituent questions remain outside the scope of this paper. 5.1 The CCP of interrogative sentences We assume, following standard wisdom, that the denotation of an interrogative sentence is the set of its possible complete answers. Assuming then that the propositional content of It is raining is p, the propositional content of (22) is {p, p}. Syntactically, polar questions are made up of a question operator and a sentence radical S denoting a proposition p. We represent such sentences as S[I] and assume that S and its denotation are available for discourse manipulation. The denotation of the polar interrogative sentence S[I] is {p, p}. The effect of A asking the question in (22) in a context K is to place the issue of whether it is raining on the Table without registering any absolute commitment relative to p or p. In Gunlogson s terms, posing a polar question S[I] (or any other sort of question) does not lead to an absolute bias in favor of the denotation of S. 11 In our terms, asking a polar question S[I] results in adding S[I] to the top of the stack of the Table without the concomitant addition of p to the author s commitment list. Placing a question on the Table steers the conversation towards a state in which the question is settled. For us, this means that the effect of asking a question on the ps is to take each projected common ground in it and replace it by new ones, each obtained by adding a proposition in the denotation of the question to the projected common ground. This is equivalent to the Groenendijk and Stokhof CCP of questions obtained 11 We do not, of course, exclude cases where the context was already biased, which, arguably, is what happens in rhetorical questions. 15

by partitioning context sets. In the case of a polar question S[I], each common ground cg in the input ps is replaced by two sets, cg {p} and cg { p}. Polar interrogative sentences can be positive or negative. In English, negation can be internal, as in (23-a) or external, as in (23-b). (23) a. Is it not raining? b. Isn t it raining? The common wisdom on negative questions of the type in (23-a) is that they involve some sort of bias in favor of the proposition in the scope of the negative operator, i.e., in this case, in favor of it being, in fact, raining. There are languages, such as Hungarian or Romanian, where there is no parallel syntactic difference. The only way of asking the corresponding negative question is as in (24): (24) a. Nu plouă? (R) Not rains? b. Nem esik? (H) Not rains? Biasing effects are associated in these languages with various intonation contours or the use of special particles. The issue of question bias is much debated in the literature (see, for instance, van Rooij & Safarova (2003), Romero & Han (2004) and references therein). We cannot add anything substantive to this debate. We only note that the bias a negative polar question brings about is never quite as radical as that of an assertion. Neither a positive nor a negative answer to a polar question brings the conversation to a crisis, the way a contradictory reaction to an assertion does. In our view this is mirrored by the fact that assertions project acceptance only, while polar questions, even if biasing, project a future for each answer. Our approach is consistent with van Rooij & Safarova (2003), where it is argued that the bias associated with a negative question is a pragmatic phenomenon. A way to register questioner bias towards one or the other of the possible ways of settling a polar question would be to encode a preference for one or the other cell in the partition. In our terms, this would mean registering a preference for certain future common grounds in the output ps over others. This preference can be rooted either in the author s own bias in favor of one of the alternatives or in the author s guess as to the preference of the addressee toward one or the other of the answers. This type of bias, then, has to do with whether the author thinks one or the other of the two conversational futures projected by the question is more likely to become the actual future or is a preferable alternative. It is different in kind from the type of bias assertions introduce. In our view, just as in Gunlogson (2001), this difference is rooted in the fact that assertions register author commitment while questions do not. We first present the effect of asking a polar question on a context state informally and then turn to defining a question speech act operator Q. Assuming that both the Table and the participants discourse commitment lists are empty, the context state after asking a polar interrogative sentence S[I] is as in Figure 8: A Table B S[I] Common Ground s 8 Projected Set ps 8 = {s 8 {p}, s 8 { p}} 16

Figure 8 K 8 : S[I] was asked relative to some input context K i In the concrete case of a polar question such as Is it raining?, we assume that the denotation of S is p and the denotation of S[I] is {p, p}. A crucial difference between questions and assertions is that in the case of the former, the commitment sets remain unchanged, while in the case of the latter, this is not the case. The nature of the change on the ps is the same for both questions and assertions: the projected common grounds in the input ps are replaced by sets obtained by adding each of the denotations of the item placed on the Table. The difference resides in the fact that the denotation of an interrogative sentence is a set of propositions while that of a declarative sentence is a single proposition. 12 The overall effect of asking a polar interrogative S[I] on a context state K is given below. Add S[I] to the top of the stack on the Table. Replace each projected common ground in the input ps by two sets, one in which p is added to the projected common ground and one in which p is added to it. Eliminate inconsistent sets. More formally now, we define the polar question operator PQ as in (25), where S[I] is a polar interrogative sentence, and where P is the set of propositions {p, p} that form the denotation of the interrogative sentence: (25) Polar question operator (PQ) PQ(S[I], K i ) = K o such that (i) T o = push(s[i], T i ) (ii) ps o = {s {p} p P, s ps i } {s s is inconsistent} = ( p P ps i p) {s s is inconsistent}. Constituent questions, we assume, have the same effect on the projected set as polar questions: ps o is the result of augmenting each s in ps i with each proposition in the denotation of the question. The major difference is that the sentence radical in their case is not a closed sentence. Returning to polar questions and assertions, the two sentences It is raining and Is it raining? share a sentence radical and differ in the [D] and [I] feature associated with it. Questions (whether polar or constituent questions) are like ordinary assertions in that they place the conversation in an unstable state by placing an item on the Table. The resulting state in the case of questions is different, however, because the ps now includes separate projections for each of the possible answers to the question. We call such a context state inquisitive. The default way in which an inquisitive context is returned to a stable state is by the addressee providing a complete answer to the question and the questioner accepting it. There are, however, special questions that indicate that the author does not, in fact, expect the addressee to settle the issue immediately. In Romanian, for instance, the particle oare in a question signals that the author does not necessarily expect an answer: (26) Oare Petru a sosit deja? oare Peter has arrived already Has Peter arrived already? In our framework questions marked by oare widen the ps by including not only projected common grounds in which the question is settled but also a copy of the elements of the input ps thereby indicating that not 12 Our discourse structures do not keep track of who placed a particular item on the Table because nothing we discuss here is sensitive to this aspect. Adding such a mechanism would not cause any problems we are aware of. 17

answering the question is one of the canonical discourse futures. These questions therefore are special in that for them removal from the Table without change in the current cg is a canonical option. Oare in Romanian is among the morphemes associated with free choice. This connection is not surprising under the suggested account since adding oare to a question widens the domain of alternative projected sets and free choice is commonly associated with a widened domain of alternatives. We now consider some of the consequences of this approach to (polar) questions. First, note that having the uttered sentence entered on the Table allows us to differentiate between positive and negative polar questions while at the same time assigning them the same denotation. A negative polar question such as: (27) Is it not raining? has the same effect on the ps as its positive counterpart. It differs from it, however, in that it places a negative rather than a positive sentence on the Table. Pragmatic differences between positive and negative polar questions as well as differences in the way they can be answered are connected in our view to the difference in the contribution the two types of questions make to the Table without having to give up the uniform account of the denotation of questions. 13 Let us now turn to comparing an assertion using a declarative sentence S[D] and a polar question using the polar interrogative S[I], exemplified below. (28) a. It is raining. b. Is it raining? The two speech acts are similar in that they both raise an issue, namely the issue of whether it is raining, and direct the conversation towards a state where the issue is resolved. The fact that the two utterances raise the same issue is captured in our view by the fact that they add the same sentence radical to the Table, namely it is raining. The fact that they direct the conversation toward a state in which the issue is resolved is captured by the fact that once each of these utterances has been processed, each projected common ground in ps in the output context state is such that either it contains the proposition that it is raining or its negation. In both cases the propositional content of the sentence being processed is decided relative to each projected common ground in the output ps. We define when a proposition p is decided relative to a common ground cg and a conversational state in (29): (29) a. A proposition p is decided relative to a common ground cg if and only if either cg implies p or it implies its negation. b. If a proposition p is decided relative to the common ground cg K of a conversational state K, p is decided in K. c. If the denotation of a sentence is decided relative to a common ground cg, the sentence is decided relative to cg. Given this definition, an interrogative sentence is decided relative to a common ground cg if and only if each proposition in its denotation is decided relative to cg. If an assertion is informative and if a question is non-redundant, their denotation is not (yet) decided relative to the cg in their input context state. We follow Gunlogson (2001) in placing the crucial difference between assertions and questions at the level of their effect on the discourse commitments of participants. Assertions of all types commit their author to the propositional content of the asserted sentence, while questions, of all types, do not add any of 13 As mentioned above, we leave open the important issue of the difference between inner negation questions such as (27) and outer negation questions, such as Isn t it raining? 18

the propositions in their denotation to the commitment list of their author. 14 5.2 Responding to a polar question The canonical way of removing a polar question from the Table is to settle it. If a participant asserts an answer, the context changes in the ways in which ordinary assertions affect the input context state, namely, the asserted answer is added to the Table, the propositional content is added to each common ground in the projected set, and the propositional content of the asserted sentence is added to the author s commitment list. Let us assume that A has asked a polar question S[I] with propositional content {p, p} putting the context in the state given in Figure 8. If B now responds by asserting answer S[D], with propositional content p, the context is changed to the state in Figure 9, where S[D] is now on top of the stack on the Table, p is on B s commitment list, and the projected set has been modified by adding p to each set in ps and then removing all context sets in the input projected set that contained p (because they are now inconsistent): A Table B S[I] p S[D] Common Ground s 9 = s 8 Projected Set ps 9 = {s 8 {p}} Figure 9 K 9 : B has answered the question raised in K 8 B s answer affects the Table, the projected set, and B s commitment list. Crucially, reacting to a question differs from reacting to an assertion because in the latter case the input context is categorically biased in favor of the denotation of the sentence on top of the Table stack. Since the question did not commit its author to either proposition in the denotation of the question, the asserted answer will not be a joint commitment. Thus, if A asks S[I] and B answers by asserting S[D], the propositional content of the sentence radical S is in DC b,o but is not in DC a,o and therefore the auxiliary operation M cannot apply. Note also that answering a question negatively does not lead to conversational crisis while reacting negatively to an assertion does. The similarity between B s asserting S[D] out of the blue and his asserting S[D] against the input context in K 8, which is inquisitive with respect to p, comes from the assertive CCP of the two moves: in both cases S[D] is added to the Table, the propositional content of S is added to the author s discourse commitment list, and the output ps is obtained by adding p to each set in the input ps. The difference between asserting out of the blue and asserting as a reaction to a question concerns the input context state. In the case of an answer, the input context state is inquisitive. As we will see below, this difference matters to the form the asserted sentence can take. 5.3 Accepting the response to a question If A accepts B s response S[D] to the question whether S, which is the canonical reaction, p is added to A s commitment list exactly as in the case in which an ordinary assertion is accepted, using the AA operator. Note that in ordinary cases this is the most canonical reaction of all since in canonical cases the questioner 14 Questions do, however, commit their author to whatever presuppositions are associated with the interrogative sentence. See section 8 for further discussion. 19

is assumed not to know the answer to her question while the addressee is assumed to know it. If accepting the answer results in p becoming a joint commitment, M applies as before with the only difference that now both S[I] and S[D] are removed from the stack. The resulting state of the context is as in Figure 10: A Table B Common Ground s 10 = s 9 p Projected Set ps 10 = {s 10 } Figure 10: A has accepted B s answer Had B answered A s question by asserting the negation of S, S, and had A accepted that answer, the context change moves would have been the same except that the addition to the common ground had been p. But in that case too, both S[I], the sentence the questioner placed on the Table, and S[D], the negative answer, are removed from the Table once the answer has been accepted. Sometimes the answer to a question is provided by answers to a series of subquestions or via the accumulated effect of assertions made in response to the question. A question is fully answered in a context state K if and only if it is decided relative to cg K, the common ground of K. We make here the simplifying assumption that a question may be removed from the Table when its propositional content is decided in the common ground. This means that rhetorical questions may be removed without an explicit answer, which we think is the right result. Getting to a more realistic account of when questions count as answered and how and when they are removed from the conversational table would require a separate paper. The moves that followed A s question in our example were canonical in that they eventually led to enlarging the common ground by settling the issue the question raised. In the view we have given here then, making an assertion and asking a polar question are parallel in that the issue of the acceptance or rejection of a proposition (the denotation of the sentence radical) is raised. The assertion raises the issue and simultaneously offers a way of settling it, namely accepting the proposition expressed by the sentence placed on the Table. The polar question raises the issue but leaves its resolution open (even in the case of biased questions that favor more or less strongly one resolution over the other). The CCP of a sequence of conversational moves made up of asking S[I], answering it positively and accepting the answer is, by the end of the sequence, the same as if someone had asserted S[D] and the interlocutors had accepted the assertion. Just as agreeing to disagree is a non-canonical way of removing an assertion from the table, one can agree not to pursue a question if there is no agreement on an answer, thereby removing it from the Table without having resolved it. Such a move is non-canonical because the question is popped off the stack without any concomitant enrichment of the common ground. Once the question is removed, the projection set must be recalculated. Such moves may be prompted by participants declaring their inability to provide information that would lead to answering the question. Agreeing not to pursue a question is similar to retracting an assertion except that the commitment list of participants is not affected. The changes involved are given below: Q is removed from the table. The projected set is recalculated from the Table and common ground. We define the operator for retracting a question, RQ, as a function from an input context K i to an output context K o, defined in (30), where Q is a question (polar or not), and T o, T i are the output and input Table respectively: 20