Kwong-Cheong Wong
email: wongkwongcheong@gmail.com

Jonathan Ginzburg
email: yonatan.ginzburg@univ-paris-diderot.fr

Investigating Non-Sentential Utterances in a Spoken Chinese Corpus

Keywords: Dialogue, Non-sentential utterances, Corpus analysis, Spoken Chinese

This paper describes a preliminary investigation into Chinese non-sentential utterances (NSUs) in a corpus of spoken Mandarin. It presents, with examples, a corpusbased taxonomy of Chinese NSUs. This taxonomy builds on the one by Fernández and Ginzburg for English NSUs in the British National Corpus (BNC)

. Partly due to the distinctiveness of spoken Chinese, eight new classes are added and their reasons for addition are explained. The paper concludes with discussions for future work.

Introduction

Dialogue is full of fragmentary utterances which are incomplete sentences but nevertheless convey a complete sentential meaning in the given context. These utterances are called non-sentential utterances (NSUs) in the literature (see, e.g., [START_REF] Fernández | Non-sentential utterances: a corpus study[END_REF], [START_REF] Fernández | Non-Sentential Utterances in Dialogue: Classification, Resolution and Use[END_REF], [START_REF] Ginzburg | The Interactive Stance: Meaning for Conversation[END_REF] and [START_REF] Schlangen | A Coherence-Based Approach to the Interpretation of Non-Sentential Utterances in Dialogue[END_REF]). In recent years, these frequently occurring NSUs, whose meaning being highly context dependent, have drawn the attention of researchers, especially those in the field of computational dialogue systems, for the reason that these NSUs can provide us with various clues to the structure of context. In fact, based on an investigation into NSUs in the British National Corpus (BNC), Ginzburg has developed a theory of meaning for spoken interaction which is basically a theory of dialogue context [START_REF] Ginzburg | The Interactive Stance: Meaning for Conversation[END_REF]. Building on the work by Fernández and Ginzburg on English NSUs in the BNC [START_REF] Fernández | Non-sentential utterances: a corpus study[END_REF], this paper describes a preliminary investigation into Chinese NSUs in a corpus of spoken Mandarin. This paper is structured as follows: Section 2 introduces the NCCU Corpus of Spoken Mandarin, the corpus we used in this research. Section 3 presents, with examples from the corpus, our corpus-based taxonomy of Chinese NSUs and explains the reasons why several new classes which are not in the taxonomy by Fernández and Ginzburg for English NSUs in the BNC (called, henceforth, the FG taxonomy) are added. Section 4 concludes the paper with discussions for future work.

The NCCU Corpus of Spoken Mandarin

While there are many spoken English corpora which have been established for different research and educational purposes, there are few spoken Chinese corpora. The NCCU Corpus of Spoken Chinese, which is established and maintained by the National Chengchi University (NCCU), Taiwan, is a project of language documentation whereby open online access to data of three major dialects of Chinese, namely, spoken Mandarin, spoken Hakka, and spoken Southern Min, are provided for non-profitingmaking research [START_REF] Chui | The NCCU Corpus of spoken Chinese: Mandarin, Hakka, and Southern Min[END_REF]. In the research reported in this paper, we used only the Mandarin sub-corpus of this NCCU Corpus, which is called, henceforth, the NCCU Corpus of Spoken Mandarin (NCCUCSM) and is available on the web at http://140.119.172.200/. The spoken data of NCCUCSM are mostly in the form of spontaneous faceto-face conversations. The data we used for this research are the first 10 conversations which are available for free. These 10 conversations are among college students who knew each other, totaling about 3.5 hours. Table 1 below, from [START_REF] Chui | The NCCU Corpus of spoken Chinese: Mandarin, Hakka, and Southern Min[END_REF], is a list of these 10 conversations (named M001-M010) with the information about the participants (F = Female, M = Male) and the length of the texts (in minutes, in number of turns, and in number of Chinese characters).

A Corpus-based Taxonomy for Chinese NSUs

Our taxonomy of Chinese NSUs is built on the FG taxonomy of English NSUs [START_REF] Fernández | Non-sentential utterances: a corpus study[END_REF]. However, due to the fact that Chinese is distinctly different from English (see, e.g., [START_REF] Xiao | Corpus-based Contrastive Studies of English and Chinese[END_REF] and [START_REF] Pan | Contrastive Linguistics: History, Philosophy and Methodology[END_REF]), it is reasonable to expect that our taxonomy of Chinese NSUs contains modifications to the FG taxonomy of English NSUs. The issues here concern the definition of "predicate" in these two different languages. As defined in [START_REF] Ginzburg | The Interactive Stance: Meaning for Conversation[END_REF], non-sentential utterances (NSUs) are "utterances without an overt predicate" (p. 2). In contrast to English, in which only verb phrases can act as predicates, there are non-verb predicates, besides verb predicates, in Chinese. Specifically, a Chinese predicate can be either a verb phrase, a noun phrase, an adjective phrase or even a subject-predicate phrase itself (see, e.g., [START_REF] Pan | Contrastive Linguistics: History, Philosophy and Methodology[END_REF]). Consequently, a Chinese utterance containing no verb cannot, by itself, be a sufficient condition for being counted as an NSU, for it may have other kinds of predicates. For example, the following two Chinese utterances are not NSUs as each of them contains an overt (non-verbal) predicate.

A Chinese utterance with a nominal predicate:

我 二十二歲 Here we use "NCCUSM C:L1-L2" or just "C:L1-L2" to indicate that the example is taken from lines L1-L2 of Conversation M00C of NCCUSM. On the other hand, there is a much more critical issue concerning whether Chinese modals are verbs. This issue is still highly controversial, as Huang et al. [START_REF] Huang | The Syntax of Chinese[END_REF] point out, "While the basic distinction between nouns and verbs is universally recognized in modern literature on Chinese syntax, scholars differ, sometimes drastically, on other categories" (p. 9). In particular, there is still no consensus on whether Chinese modals are verbs. In the literature, there are three common views on the categorization of Chinese modals:

(1) Chinese modals are verbs; (2) Chinese modals are adverbs; and (3) Some Chinese modals are verbs, some are adverbs and some are both [START_REF] Song | A Study on the Negation of Modals in Modern Chinese[END_REF]. In this research, we follow [START_REF] Huang | The Syntax of Chinese[END_REF], [START_REF] Lin | Modals as verbs in Chinese: a GB perspective[END_REF], [START_REF] Xiang | A Syntactic Comparison of Chinese and English Auxiliary Verbs[END_REF], [START_REF] Mccawley | Justifying part-of-speech assignments in Mandarin Chinese[END_REF]

Questions

There are 5 classes of Questions.

(

) 4

Extension Moves

There are 3 classes of Extension Moves.

( [START_REF] Guida | Les phrases sans verbes[END_REF].

) 21
Another future work is to use this taxonomy as the basis for an interpretation module of a dialogue system building on, e.g., GoDiS [START_REF] Larson | GoDiS -An accommodating dialogue system[END_REF], [START_REF] Larsson | Issue-based Dialogue Management[END_REF].

  -old very young 'Twenty years old is very young.'[NCCUCSM 6: 453]

  in the FG taxonomy. The reason for this addition is that in Chinese, to answer a polar question positively is by way of repeating the verb in the question (or negatively by repeating the verb with a negation adverb). Since we treat Chinese modals as verbs, there are two kinds of verbs: lexical verbs and modal verbs. In case that the verb being repeated is a lexical verb, the answer is

	not an NSU because the lexical verb (or the verb plus the
	negation adverb) itself constitutes a complete (verbal)
	predicate. For example,
	A: 你 還 記得		喔
	ni3 hai2	ji4de2		o
	2SG still	remember	QST
	'Do you still remember?' B: 記得 啊
	ji4de2	a
	remember PRT
	'I remember.'		[NCCUCSM 5: 210-211]
	na4ge zhen1de hui4	shui4zhao2	ma
	that really	will	sleep:RESULT QST
	'Does it really make people sleepy? ' B: 會
	hui4
	right
	'Right.'			[NCCUCSM 6: 109-110]
	It therefore follows that not containing a verb can neither
	be a sufficient nor a necessary condition for a Chinese
	utterance to be classified as an NSU. Not sufficient,
	because in Chinese there are non-verbal predicates; not
	necessary, because an utterance containing a modal verb
	can also be classified as an NSU. In order to distinguish
	these verb-containing NSUs from those which contain no
	verb, two new classes, namely Verbal Affirmative Answer
	and Verbal Rejection are added to our taxonomy. Apart
	from these two, we have also added six more new classes,
	namely, Helpful Acknowledgement, Helpful Affirmative
	Answer, Correction, Noun Phrase, Re-Affirmation and
	Interjection. Empirical evidence, i.e., examples from the
	corpus, for these NSUs is provided in the following
	subsections. Table 2 below is our taxonomy of Chinese
	NSUs. There are totally 23 classes of NSUs which are
	grouped under four categories: (1) Acknowledgement, (2)
	Questions, (3) Answers, and (4) Extension Moves. In the
	following, we describe each NSU class and illustrate it
	with an example from NCCUSM.

and others in treating Chinese modals as verbs. Note that this is a critical decision as it leads to the addition of new NSU classes to our taxonomy which are not By contrast, in case that the verb being repeated is a modal verb, the answer is an NSU because the modal verb, with or without a negation adverb, is an incomplete (verbal) predicate. For example, A: 那個 真的 會 睡著 嗎

Table 2 A

 2 Taxonomy for Chinese NSUs in the NCCU Corpus of Spoken Mandarin (NCCUSM)

				B: 對 馬屁大會
				dui4 ma3pi4da4hui4
		NSU Class		yes fawn-conference
		A. Acknowledgement	'Yes. It is a fawning conference.'	[4: 190-191]
	1	Plain Acknowledgement
	2	Repeated Acknowledgement	(3) Helpful Acknowledgement is an acknowledgment
	3	Helpful Acknowledgement	providing additional information. E.g.,
	B. Questions Classification Ellipsis Sluice Noun Phrase Check Question Filler C. Answers Short Answer 10 Affirmative Answer 4 5 6 7 8 9 11 Repeated Affirmative Answer	A: 像 我們 那個 班級 才 幾 個 而已 xiang4 wo3men ne4ge ban1ji2 cai2 ji3 ge er2yi3 like 1PL that CL class just few CL only 'Like our class, there are only a few students.' B: 對 呀 才 三 個 ei dui4 ya cai2 san1 ge ei right PRT just three CL PRT 'Right. There're only three students.' [1: 114-116]
	12 Verbal Affirmative Answer
	13 Helpful Affirmative Answer
	14 Re-Affirmation
	15 Rejection
	16 Verbal Rejection
	17 Helpful Rejection
	18 Correction
	19 Interjection
	20 Propositional Modifier
		D. Extension Moves
	21 Factive Modifier
	22 Bare Modifier Phrase
	23 Conjunction + Fragment
	3.1 Acknowledgement
	There are 3 classes of Acknowledgement.
	(1) Plain Acknowledgement signals understanding or
	acceptance. E.g.,
	A: 沒辦法
	mei2ban4fa3
	NEG:way
	'Nothing we can do.' B: 對呀
	dui4 ya
	right PRT
	'Right.'	[1: 4-5]
	(2) Repeated Acknowledgement is an acknowledgement
	containing verbatim repetition of a previous utterance.
	E.g.,
	A: 所以 是 一個 馬屁 大會
	suo2yi3 shi4 yi2ge ma3pi4 da4hui4
	so	COP one	fawn-conference
	'So, it's a fawning conference.'

  Factive Modifier is an evaluative adjective. E.g., This paper describes a preliminary investigation into Chinese non-sentential utterances (NSUs) in the NCCU Corpus of Spoken Mandarin. In particular, it presents, with examples from the corpus, a corpus-based taxonomy of Chinese NSUs. Although this taxonomy is built on the FJ taxonomy -the taxonomy by Fernández and Ginzburg for English NSUs in the British National Corpus (BNC), several new classes are added. The reasons for the addition have been explained by appealing, in part, to the distinctiveness of spoken Chinese. In the future, we plan to test this taxonomy with other Chinese corpora and on other languages. A recent work in this direction is by Floriane Guida who tests the FG taxonomy on French

	and	tall	PRT
	'And he's tall.'	[2: 53-54]
	4. Conclusions and Future Work
			ran2hou4 NB ke2yi3 dai4 hui2 jia1
			then notebook can	take back home
			'They can take the laptop home with them.' B: 好好
			hao2hao3
			good
			'That's really good.'	[3: 308-309]
			(22) Bare Modifier Phrase is an adjunct that can modify a
			previous utterance in the context. E.g.,
			A: 我 現 在 別 人 大 聲 我 會 比 他 更 大聲 的 那 一 種
			wo3 xian4zai4 bie2ren2 da4sheng1 wo3 hui4 bi3
			ta1 geng4 da4sheng1	de na4 yi4 zhong3
			1SG now other-person loud 1SG will COMPARE
			3SG even loud	ASSC that one kind
			'Now, I am the kind of person who, when other person
			is loud, will be even louder.' B: 人生 磨練 之後
			ren2sheng1 mo2lian4 zhi1hou4
			life	train	after
			'After training in life.'	[6: 546-547]
			(23) Conjunction + Fragment consists of a connective
			which introduces a fragment. E.g.,
			A: 蠻 瘦 的
			man2 shou4 de
			quite thin	PRT
			'He's quite thin.' B: 而且 高高	的
			er2qie3 gao1gao1 de

A: 然後 NB 可以 帶 回 家


Robin Cooper
email: cooper@ling.gu.se

Jonathan Ginzburg
email: yonatan.ginzburg@univ-paris-diderot.fr

TTR for Natural Language Semantics

Semantic analysis of interaction and coordination in dialogue (SAICD), by

Introduction

Given the state of the art, a simple actual conversation such as (1) 2 still constitutes a significant challenge to formal grammar of just about any theoretical flavour.

(1) types of non-sentential utterances, partial comprehension, and self answering.

Making sense of all these phenomena in a systematic way is a challenge undertaken in the TTR-based dialogue framework KoS [START_REF] Ginzburg | The Interactive Stance: Meaning for Conversation[END_REF]. While we will not have the space to develop a detailed analysis of this example, by the end of the paper we will have sketched a toolbox on the basis of which disfluencies, non-sentential utterances, partial comprehension self, answering, and multilogue can be explicated. A key ingredient to this is a theory of the structure and evolution of dialogue game-boards(DGBs), the publicised component of the conversationalists' information states. This, in turn, presupposes both means of developing semantic and grammatical ontologies to explicate notions such as propositions, questions, and utterances.

There are, nonetheless, a number of well established paradigms for doing just that and the obvious question to ask is: why develop a distinct framework?

We will illustrate throughout the paper intrinsic problems for frameworks such as possible worlds semantics and typed-feature structure (TFS)-based approaches:

• Semantic ontology: Why not a possible worlds-based approach? There are well known problems for this strategy that revolve around its coarseness of grain. These are often ignored (folk assumption: '. . . the attitudes are difficult and primarily a philosophical problem . . . ') Whether or not this is true we point to the problems posed by negation which cannot be brushed off so easily.

• syntax-semantics interface: Why is a TFS-based approach to a syntaxsemantics interface, as in frameworks such as Head-driven Phrase Structure Grammar (HPSG) [START_REF] Sag | Syntactic Theory: A Formal Introduction[END_REF]) and in Sign-based Construction Grammar [START_REF] Michaelis | Sign-based construction grammar[END_REF]) insufficient? Here again, there are well known problems (lack of proper binding, functions) and these can be solved in standard λ-calculus based approaches. We will point to issues that are difficult to the latter such as clarification interaction.

Our claim is that TTR enables a uniform theory of grammar, semantics, and interaction that can tackle such problems, while allowing one to maintain past insights (emanating from Montague Semantics and Discourse Representation Theory) and also, we think, future directions (e.g. probabilistic semantics). This article is structured as follows: the basics of TTR are described in section 2. Subsequently, in sections 3-5 we use this to sketch fundamental notions of grammar, semantic ontology, and dialogical interaction. These are eventually illustrated in more detail in sections 6-8, which deal with metacommunicative interaction, negation, quantification, and, more briefly, non sentential utterances and disfluencies.

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2 A theory of types and situations

Type theory and perception

In classical model theoretic semantics [START_REF] Montague | The Proper Treatment of Quantification in Ordinary English[END_REF][START_REF] Montague | Formal Philosophy: Selected Papers of Richard Montague[END_REF] there is an underlying type theory which presents an ontology of basic classes of objects such as, in Montague's type theory, entities, truth values, possible worlds and total functions between these objects. Here we will make use of a rich type theory inspired by the work of [START_REF] Martin-Löf | Intuitionistic Type Theory[END_REF] and much subsequent work on this kind of type theory in computer science. For a recent example relating to natural language see [START_REF] Luo | Contextual Analysis of Word Meanings in Type-Theoretical Semantics[END_REF]. Ranta (this volume) gives important background on Martin-Löf's type theory.

In a rich type theory of the kind we are considering there are not only types for basic ontological categories but also types corresponding to categories of objects such as Tree or types of situations such as Hugging of a dog by a boy. A fundamental notion of this kind of type theory is that of a judgement that an object (or situation) a is of type T , in symbols, a : T . In our view judgements are involved in perception. In perceiving an object we assign it a type. The type corresponds to what [START_REF] Gibson | The Ecological Approach to Visual Perception[END_REF] (and following him in their theory of situation semantics, [START_REF] Barwise | Situations and Attitudes[END_REF] would call an invariance. In order to perceive objects as being of certain types, agents must be attuned to this invariance or type. We take this to mean that the type corresponds to a certain pattern of neural activation in the agent's brain. Thus the types to which a human is attuned may be quite different from those to which an insect is attuned. A bee landing on a tree does not, presumably, perceive the tree in terms of the same type Tree that we are attuned to.

TTR: Type theory with records

The particular type theory we will discuss here is TTR which is particular variant of Type Theory with Records. The most recent published reference which gives details is [START_REF] Cooper | Type theory and semantics in flux[END_REF]. An earlier treatment is given in Cooper (2005b), and Cooper (2005c) discusses its relation to various semantic theories.

Here we will give a less detailed formal treatment of the type theory than in the first two of these references. We start by characterizing a system of basic types as a pair consisting of a non-empty set of types, Type, and a function,

A, whose domain is Type and which assigns to each type in Type a (possibly empty) set which does not overlap with Type. We say that a is of type T (in Type), a : T , according to Type, A just in case a ∈ A(T ). In general we will think of basic types as corresponding to basic ontological categories. The basic type we will use in this section is Ind for individuals.

We will use complex types for types of situations, inspired by the notion of situation in [START_REF] Barwise | Situations and Attitudes[END_REF]. The simplest complex type of situation is constructed from a predicate together with some appropriate arguments to the predicate. Consider, for example, the type of situation where a boy called Page: 4 job: rc-jg-ttrsem-final macro: handbook.cls date/time: 3-Apr-2015/11:40

Bill (whom we will represent by b) hugs a dog called Dinah, (represented by d). The type of situation in which Bill hugs Dinah will be constructed from the predicate 'hug' together with the arguments b and d. This type is represented in symbols as hug (b,d). Here we are treating 'hug' as a predicate which has arity Ind, Ind , that is, it requires two individuals as arguments.

Sometimes we may allow predicates to have more than one arity, that is they may allow different configurations of arguments. In this case we say that the predicate is polymorphic. 3 Types like this which are constructed with predicates we will call ptypes. A system of types containing ptypes, that is, a system of complex types, will be an extension of a system of basic types BType, A , Type, BType, PType, A, F where PType is a set of ptypes constructed from a particular set of predicates and arities associated with them by combining them with all possible arguments of appropriate types according to the type system and F is a function whose domain is PType which assigns a (possibly empty) set of situations to each ptype. The set Type includes both BType and PType.

This gives us a system of types which will allow us types of situations where particular individuals are related to each other. However, we want to be able to characterize more general types of situation than this, for example, the type of situations where some boy hugs a dog, that is, the type of any "boy hugs dog" situation. There are a number of ways to characterize such more general types in type theory. In TTR we use record types. The type of situation where a boy hugs a dog could be the record type in (2).

(2)

      x : Ind c boy : boy(x) y : Ind c dog : dog(y) e : hug(x,y)      
This record type consists of five fields each of which consists of a label (such as 'x' or 'c dog ') and a type (such as Ind or 'dog(y)'). Each field is an ordered pair of a label and a type and a record type is a set of such fields each of which have a distinct label. We use labels like 'x' and 'y' for fields introducing individuals and labels like 'c pred ' for fields introducing types which are ptypes with the predicate pred representing constraints or conditions (hence 'c') on objects in other fields. We will often use the label 'e' for the type representing the main event, such as hugging.

A record of this type is a set of fields (i.e. order is unimportant) with labels and objects such that no two fields have the same label, there is a field with each of the labels in the record type and the object in the field is of the type in the corresponding field in the record type. Note that there can be more fields in the record with labels not mentioned in the record type. A record of the type in (2), that is, a witness for this type, will be one of the form in (3).

(3)

         x = a c boy = s 1 y = b c dog = s 2 e = s 3 . . .         
where: a : Ind

s 3 : hug(a, b)
If the type (2) is non-empty there will be a boy and a dog such that the boy hugs the dog. Thus (2) could be used to represent the content of a boy hugs a dog. That is, we use it to play the role of a proposition in other theories.

(Later we will introduce a more complex notion of proposition which builds on such types.) Let r be a record of the form (3). We will refer to the objects in the fields using the notation r. where is some label in the record. Thus r.x will be a, r.c boy will be s 1 and so on. We will allow records to be objects in fields. Thus we can have records within records as in (4).

(4)

      f =   f = ff = a gg = b g = c   g = h = g = a h = d      
We can extend the dot notation above to refer to paths in a record, that is sequences of labels which will lead from the top of a record down a value within the record. Let r be (4). Then we can use paths to denote various parts of the record as in (5).

(5)

a. r.f =   f = ff = a gg = b g = c   b. r.g.h = g = a h = d c. r.f.f.ff = a
Technically, we have cheated a little in the presentation of record types.

'boy(x)', 'dog(y)' and 'hug(x,y)' are not technically ptypes since 'x' and 'y' are labels, not individuals as required by the arities of these predicates. What we mean by this notation is the ptype we can construct by substituting whatever individuals occur in the 'x' and 'y' fields of the record we are checking to see

:Ind . hug(v 1 , v 2 ), x,y        
There is good reason to use this more complex notation when we deal with more complex record types which have record types embedded within them.

However, for the most part we will use the simpler notation as it is easier to read. Functions from objects to types, dependent types, will play an important

role in what we have to say below.

In record types we will frequently make use of manifest fields5 A manifest field =a:T is a convenient notation for :T a where T a is a singleton type whose only witness is a. Singleton types are introduced by the clauses in ( 7).

( 7)

a. If a : T then T a is a type. b. b : T a iff b = a 2.

Subtyping

The notion of subtype in TTR plays a central inferential role within the system. T 1 is a subtype of T 2 (T 1 T 2 ) just in case for all assignments to basic types it is the case that if a : T 1 then a : T 2 . For more discussion of this notion see [START_REF] Cooper | Type theory and semantics in flux[END_REF].

Function types

We introduce function types as in (8).

(8)

a. If T 1 and T 2 are types, then so are (T 1 → T 2 ) and (T This means that f is a total function from objects of type T 1 to objects of type T 2 . In (8c) f is required to be a constant function. A function is associated with a graph, that is, a set of ordered pairs, as in the classical set theoretical model of a function. As in set theory we let functions be identified by the graphs, that is, for functions

1 → c T 2 ) b. f : (T 1 → T 2 ) iff f is a function with domain {a | a : T 1 } and range included in {a | a : T 2 } c. f : (T 1 → c T 2 ) iff f : (T 1 → T 2 )
f 1 , f 2 , if graph(f 1 ) = graph(f 2 ) then f 1 = f 2 .
We also require that for each graph whose domain (i.e. left projection) is the set of witnesses of a type and whose range (i.e. right projection) is included in the set of witnesses of another type there is a function which has this graph.

This makes the existence of a function of type (T 1 → T 2 ) correspond to a universal quantification, "for everything of type T 1 there is something of type T 2 ". Finally we stipulate that types (T 1 → T 2 ) and T 1 are incompatible. That is, you cannot have something which belongs to a function type and the type which characterizes the domain of the function. As a consequence, functions cannot apply to themselves. This is one way of avoiding paradoxes which can arise when we allow functions to apply to themselves.

We introduce a notation for functions which is borrowed from the λcalculus as used by [START_REF] Montague | The Proper Treatment of Quantification in Ordinary English[END_REF]. We let functions be identified by sets What should be the result of applying this function to the record type in (10)?

(10) x : Ind c 1 : λv 1 :Ind (dog(v 1 )), x

Given normal assumptions about function application the result would be (11).

(11) c 0 :

x : Ind c 1 : λv 1 :Ind (dog(v 1 )), x (incorrect!) But this would be incorrect. In fact it is not a well-formed record type since 'x' is not a path in it. Instead the result should be (12).

(12) c 0 :

x : Ind c 1 : λv 1 :Ind (dog(v 1 )), c 0 .x (Note that 'c 0 .x' is not a path in this record type.)

Suppose that we wish to represent a type which requires that there is some dog such that it appears to be approaching (that is a de re reading). In the abbreviatory notation we might be tempted to write (18)

a. If T 1 and T 2 are types then so are (

T 1 ∧ T 2 ), (T 1 ∨ T 2 ) and ¬T b. a : (T 1 ∧ T 2 ) iff a : T 1 and a : T 2 c. a : (T 1 ∨ T 2 ) iff a : T 1 or a : T 2 d. a : ¬T 1 iff there is some type T 2 which is incompatible with T 1 such that a : T 2
T 1 is incompatible with T 2 just in case there is no assignment to basic types such that there is some a such that a : T 1 and a : T 2 . That is, it is impossible for any object to belong to both types. This is a non-classical treatment of negation which we will discuss in Section 7.1.

Occasionally we will need types which are possibly infinite joins of types in order to characterize certain function types. We will represent these using a subscripted . Thus if T 1 and T 2 are types, then ( 23) is a type.

(

) X T1 (X → T 2 ) 23
Witnessing conditions for (23) are defined by ( 24).

(24) f :

X T1 (X → T 2 ) iff f : (T → T 2 )
for some type T such that T T 1 .

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As we have record types in our system we will be able to form meets, joins and negations of these types just like any other. When we form the meet of two record types, T 1 ∧ T 2 there is always a record type T 3 which is equivalent to T 1 ∧ T 2 in the sense that no matter what we assign to our basic types anything which is of T 1 ∧ T 2 will be of type T 3 and vice versa. T 3 is defined using the merge operator ∧ . . Thus, T 1 ∧ . T 2 is the merge of the two types T 1 ,T 2 .

If at least one of the two types is not a record type it is identical with the meet T 1 ∧ T 2 . The basic idea of merge for record types is illustrated by the examples in (25).

( In addition to merge we also introduce asymmetric merge, T 1 ∧ . T 2 . This is defined like ordinary merge except that in the case where one of the types is not a record type T 1 ∧ . T 2 = T 2 . This notion (which is discussed in detail in Cooper, in prep) is related to that of priority unification [START_REF] Shieber | An Introduction to Unification-Based Approaches to Grammar[END_REF].

) a. f:T 1 ∧ . g:T 2 = f:T 1 g:T 2 b. f:T 1 ∧ . f:T 2 = f:T 1 ∧ . T 2 (For 25

Set and list types

We introduce set and list types as in ( 26). So far we have talked about situations or events in terms of ptypes or record types which have ptypes in some of their fields. This gives us a rather static view of events and does not give an analysis of the changes that take place as an event unfolds. A single type is rather like a snapshot of an event at one point in its development. In an important series of papers including [START_REF] Fernando | A finite-state approach to events in natural language semantics[END_REF][START_REF] Fernando | Situations as strings[END_REF][START_REF] Fernando | Finite-state descriptions for temporal semantics[END_REF][START_REF] Fernando | Situations in LTL as strings[END_REF]; ?), Tim Fernando has proposed that events should be analyzed in terms of strings of snapshots or observations. In TTR we adapt these ideas by introducing regular types: types of strings of objects corresponding to the kinds of strings you find in regular languages in formal language theory [START_REF] Hopcroft | Introduction to Automata Theory, Languages and Computation[END_REF][START_REF] Partee | Mathematical Methods in Linguistics[END_REF]. ( 29) is an account of the two main kinds of regular types that we will use here where a b represents the concatenation of two objects a and b.

(

) a. if T 1 , T 2 ∈ Type, then T 1 T 2 ∈ Type a : T 1 T 2 iff a = x y, x : T 1 and y : T 2 b. if T ∈ Type then T + ∈ Type. a : T + iff a = x 1 . . . x n , n > 0 and for i, 1 ≤ i ≤ n, x i : T T 1 T 2 29
is the type of strings where something of type T 1 is concatenated with something of type T 2 . T + is the type of non-empty strings of objects of type T . Suppose for example that we want to represent the type a game of fetch as a game played between a human, a, and a dog, b, involving a stick, c, in which the human picks up the stick, attracts the attention of the dog, and throws the stick, whereupon the dog runs after the stick and picks it up, returning it to the human, after which the cycle can start from the beginning. The type of this event would be (30).

Inference from partial observation of events

An important fact about our perception of events is that we can predict the type of the whole event when we have only perceived part of the event. Thus if we see a human and a dog playing with a stick and we see the human pick up the stick and attract the dog's attention we might well predict that the type of the whole event is one of playing fetch. We can represent this prediction by a function, as in (31).

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(31) λr:

          x : Ind c human : human(x) y : Ind c dog : dog(y) z : Ind c stick : stick(z) e : pick up(x,z) attract attention(x,y)          
( e : play fetch(r.x,r.y) )

Notice that this function is what we have called a dependent type, that is, it takes an object (in this case the observed situation) and returns a type (in this case the type of the predicted situation). Notice that this ability to predict types of situations on the basis of partial observations is not particular to humans. The dog realizes what is going on and probably starts to run before the human has actually thrown the stick. However, in the Section 3 we will suggest that humans build on this ability in their perception and analysis of speech events.

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Grammar in TTR

In Section 2 we suggested that an important capability that agents have is the prediction of the type of a complete event on the basis of a partial observation of an event. We suggested that functions from observed situations to predicted situation type (a kind of dependent type) can be used in modelling this, taking the example of the game of fetch. Very similar inferences are involved in the perception of linguistic events, though there are also some important differences. In the case of the game of fetch the predicted type is a type of situation which you could in principle perceive completely. In the example we gave you are inferring the nature of the event as it will develop later in time. The case of linguistic perception is rather more abstract. We are inferring types which may hold simultaneously with what we have observed and the predicted event types may be of events that are not directly perceivable. Thus we are able to perceive events belonging to phonological or phonetic types but from these we infer types relating to syntactic and semantic structure whose instances are not directly perceivable. It is this kind of reasoning about abstract objects which seems so important to human linguistic ability. Nevertheless the fundamental mechanism is the same: we are mapping from an observation to a type of something unobserved.

Grammar rules involve a prediction on the basis of a string of linguistic events. Thus they are functions of the form (32).

(32

) λs : T 1 . . . T n (T )
where the T i and T are sign types, which, as we will see below, are types which have both a directly perceivable and a non-directly perceivable component.

Thus grammar rules are functions from strings of linguistic events to a type of a single linguistic event. An example would be the observation of a string consisting of a noun-phrase event followed by a verb-phrase event and predicting that there is a sentence event, that is, what is normally written in linguistic formalisms as the phrase-structure rule S → NP VP.

Sign types correspond to the notion of sign in HPSG [START_REF] Sag | Syntactic Theory: A Formal Introduction[END_REF].

The type Sign could be thought of as (33).9

(33)

    s-event : SEvent synsem :   cat : Cat constits : {Sign} cont : Cont      
Here we use 'synsem' ("syntax and semantics") as a field corresponding to both syntactic and semantic information, although this, and also what follows below, could be adjusted to fit more closely with other versions of HPSG.

However, for technical reasons having to do with recursion (ultimately signs

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(34) r:Sign iff r:

    s-event : SEvent synsem :   cat : Cat constits : {Sign} cont : Cont      
We have introduced three new types here: SEvent, the type of speech events;

Cat, the type of categories and Cont, the type of semantic contents. We will take each of these in turn and return to the 'constits'-field (for "constituents") in synsem later.

A minimal solution for the type SEvent is ( 35).

       
where we have in addition fields for event location, speaker and audience. This corresponds more closely to the kind of information we normally associate with speech act theory [START_REF] Searle | Speech Acts: an Essay in the Philosophy of Language[END_REF]. However, this type may be too restrictive: more than one person may be in the audience; more than one speaker may collaborate on a single speech event, as is shown by work on split utterances [START_REF] Purver | Splitting the I s and Crossing the Yous: Context, Speech Acts and Grammar[END_REF]. For present purposes it will be sufficient to use the simpler type (35) for speech events.

We will take the type Phon to be the type of a non-empty string of phoneme utterances, that is Phoneme + . We could use phonetic symbols to represent types of individual phoneme utterances. For example u : h would mean that u is an utterance of the phoneme h (the phoneme being modelled as a TTR type). u : h aey would mean that u is an utterance of the phoneme string which we denote in orthography by 'hi'. It is not our intention to give a detailed account of phonology here and we will represent this string type using the orthography as hi. Note that hi is a subtype of Phon.

We define the type TimeInt, for time interval, to be (37). where Time is a basic type whose witnesses are time points and < is a predicate (here used in infix notation) which requires that its first argument is ordered before its second argument.

The 'constits'-field in synsem if for the set of constituents (including all constituents, not just daughters (immediate constituents)).

In Section 5 we will extend the definition of Sign to include a field for a dialogue game board.

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A theory of abstract entities

An ontology including abstract entities-including entities such as propositions, questions, and outcomes is a necessary ingredient for accounts of illocutionary acts such as assertion, querying, and commanding, as well as of attitude reports. Building on a conception articulated 30 years earlier by [START_REF] Austin | Truth[END_REF], [START_REF] Barwise | The Liar[END_REF] developed a theory of propositions in which a proposition is a structured object prop(s, σ), individuated in terms of a situation s and a situation type σ. Given the ':' relation between situations

and their types there is a a straightforward notion of truth and falsity:

(38)

a. prop(s, σ) is true iff s : σ (s is of type σ). b. prop(s, σ) is false iff s : σ (s is not of type σ).
A detailed such ontology extending the original situation semantics ontology was developed in [START_REF] Ginzburg | Interrogative Investigations: the form, meaning and use of English Interrogatives[END_REF]. This approach has subsequently been developed in TTR in works such as [START_REF] Ginzburg | Situation semantics and the ontology of natural language[END_REF][START_REF] Ginzburg | The Interactive Stance: Meaning for Conversation[END_REF]. We start by discussing how to add propositions into TTR.

For many purposes the type theory already developed has entities that could be identified with Austinian propositions, an identification frequently assumed in past work in type theory via the slogan propositions as types.

Cooper (2005b) develops the former in which a proposition p is taken to be a record type. A witness for this type is a situation. On this strategy, a witness is not directly included in the semantic representation. Indeed, record types are competitive in such a role: they are sufficiently fine-grained to distinguish identity statements that involve distinct constituents. (39a)

would correspond to the record type in (39c), whereas (39b) to the record type in (39d)). Moreover, in this set up substitutivity of co-referentials (39e)

and cross-linguistic equivalents ((39e), the Hebrew equivalent of (39a)) can be enforced:

(39) a. Enescu is identical with himself.

b. Poulenc is identical with himself. f. Enesku zehe leacmo.

A situational witness for the record type could also be deduced to explicate cases of event anaphora, as in (40); indeed, a similar strategy is invoked when in an analysis of nominal anaphora in Ginzburg ( 2012 Nonetheless, here we develop an explicitly Austinian approach, where the situational witness is directly included in the semantic representation. The original Austinian conception was that s is a situation deictically indicated by a speaker making an assertion 10 -teasing out the semantic difference between implicit and explicit witnesses is a difficult semantic task. The Austinian approach is important for negation (see section 7.1). Explicitly Austinian propositions can also play a role in characterizing the communicative process: in section 6 we will show that locutionary propositions individuated in terms of an utterance event u 0 as well as to its grammatical type T u0 allows one to simultaneously define update and clarification potential for utterances. In this case, there are potentially many instances of distinct locutionary propositions, which need to be differentiated on the basis of the utterance token-minimally any two utterances classified as being of the same type by the grammar.

Assuming we adopt an explicitly Austinian approach, then on the current account the type of propositions is the record type (41a). The correspondence with the situation semantics conception is quite direct. We can define truth conditions as in (41b).

(41)

a. Prop = def sit : Rec sit-type : RecType † b. A proposition p = sit = s 0 sit-type = ST 0 is true iff s 0 : ST 0
Here the type RecType † is a basic type which denotes the type of records types closed under meet, join and negation. That is, we require:

(1) if T :RecType, then

T :RecType † (2) if T 1 , T 2 :RecType † , then T 1 ∧ T 2 , T 1 ∨ T 2 , ¬T 1 :RecType † (3)

Questions

Given the existence of Austinian-like propositions and a theory of λ-abstraction given to us by existence of functional types, it is relatively straightforward to develop a theory of questions as propositional abstracts in TTR. Extensive motivation for the view of questions as propositional abstracts is provided in [START_REF] Ginzburg | Resolving questions, I[END_REF]; [START_REF] Ginzburg | Interrogative Investigations: the form, meaning and use of English Interrogatives[END_REF]-TTR contributes to this by providing an improved notion of simultaneous, restricted abstraction, as we will see shortly.

A (basic, non-compound) question will be a function from records into What of polar questions? [START_REF] Ginzburg | Interrogative Investigations: the form, meaning and use of English Interrogatives[END_REF] proposed that these are 0-ary abstracts, though the technical apparatus involved in explicating this notion in their framework based on non-well-founded set theory was quite complex. TTR, however, offers a simple way to explicate 0-ary abstraction.

If (44)

Question = def X RecType (X → Prop)
The type of polar questions, PolQuestion, is given in ( 45).

(45)

PolQuestion = def (Rec→ c Prop)
That is, polar questions are constant functions from situations (records) to propositions as discussed in [START_REF] Ginzburg | The Interactive Stance: Meaning for Conversation[END_REF].

Answerhood is one of the essential testing grounds for a theory of questions. Abstracts can be used to underspecify answerhood. This is important

given that NL requires a variety of answerhood notions, not merely exhaustive answerhood or notions straightforwardly definable from it. Moreover, as with questions, answerhood needs to be explicable within type theory. This is because answerhood figures as a constituent relation of the lexical entries of resolutive verbs12 and in rules regulating felicitous responses in dialogue management (see section 5.). For current purposes this means that we need to be able to define notions of answerhood as types.

There are a number of notions of answerhood that are of importance to dialogue. One relates to coherence: any speaker of a given language can recognize, independently of domain knowledge and of the goals underlying an interaction, that certain propositions are about or directly concern a given question. We will call this Aboutness. The simplest notion of answerhood we can define on the basis of an abstract is one we will call, following [START_REF] Ginzburg | Interrogative Investigations: the form, meaning and use of English Interrogatives[END_REF], simple answerhood. In order to this we will use the following notion:

A proposition p is an instantiation of a question q just in case there is some r in the domain of q such that q(r) = p

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(46) α is a simple answer to q iff α is an instantiation of q or the negation of an instantiation of q.

Given these definitions it is straightforward to show:

(47)

a. If q is an n-ary question of type (T → Prop) and α is a simple answer to q then there is some r : T such that α is q(r) or ¬q(r).

b. In particular, if q is the polar question λr:[](p) and α is a simple answer to q then α is either p or ¬p.

Simple answerhood covers a fair amount of ground. But it clearly underdetermines aboutness. On the polar front, it leaves out the whole gamut of answers to polar questions that are weaker than p or ¬p such as conditional answers 'If r, then p' (e.g. 48a) or weakly modalized answers 'probably/possibly/maybe/possibly not p' (e.g. ( 48b)). As far as wh-questions go, it leaves out quantificational answers (48c-g), as well as disjunctive answers.

These missing class of propositions, are pervasive in actual linguistic use: One straightforward way to enrich simple answerhood is to consider the relation that emerges by closing simple answerhood under disjunction. Ginzburg (1995); [START_REF] Ginzburg | Interrogative Investigations: the form, meaning and use of English Interrogatives[END_REF] show that aboutness as defined in (49) seems to encompass the various classes of propositions exemplified in (48).

(49) p is About q iff p entails a disjunction of simple answers to q.

Answerhood in the 'aboutness' sense is clearly distinct from a highly restricted notion of answerhood, that of being a proposition that resolves or constitutes exhaustive information about a question. This latter sense of answerhood, which is restricted to true propositions, has been explored in great detail in the formal semantics literature, since it is a key ingredient in explicating the behaviour of interrogatives embedded by resolutive predicates such as 'know', 'tell' and 'discover'. We will not discuss this here but refer the reader to [START_REF] Ginzburg | The Interactive Stance: Meaning for Conversation[END_REF].

Many queries are responded to with a query. A large proportion of these are clarification requests, to be discussed in section 6. But in addition to these, there are query responses whose content directly addresses the question posed, as exemplified in ( 50):

(50) There has been much work on relations among questions within the framework of Inferential Erotetic Logic (IEL) (see e.g. [START_REF] Wiśniewski | Questions and inferences[END_REF][START_REF] Wiśniewski | Erotetic search scenarios[END_REF] and

Wiśniewski (this volume)), yielding notions of q(uestion)-implication. From this a natural hypothesis can be made about such query responses, as in (51a). A related proposal, first articulated by [START_REF] Carlson | Dialogue Games, Synthese Language Library[END_REF], is that they are constrained by the semantic relations of dependence, or its converse influence.

(51) a. q 2 can be used to respond to q 1 if q 2 influences q 1 . b. q 2 influences q 1 iff any proposition p such that p Resolves q 2 , also satisfies p entails r such that r is About q 1 . Its intuitive rationale is this: discussion of q 2 will necessarily bring about the provision of information about q 1 . The actual characterization of query responses is complex, both empirically and theoretically. For a detailed account using TTR see [START_REF] Lupkowski | Question answers[END_REF].

Page This allows us to construct a type corresponding to a lexical entry for a greeting word such as 'hi', as in ( 54). Here we assume that the definition of the type Sign in Section 3 has been modified to include a field for a dialogue game board:

Sign = def     s-event : SEvent synsem : cat : Cat cont : cont dgb : DGBType    
This represents an extension of the Saussurean notion of sign where we not only take account of the signifier ('s-event') and the signified ('synsem') but also the context in which the signification takes place (here represented by 'dgb').

(54) Sign∧ .

                  s-
                       
Here, as before in our discussion of questions, r ds is the described situation as determined by the context. The use of '⇑' in the 'sit-type'-field is a convenient informal notation for paths occurring in a record type embedded within a larger record type but not lying on a path in that record type. It indicates that the path is to be found in the next higher record type. It clears up an ambiguity that arises because we are using the notation that does not make explicit the dependent types that are being used as discussed on p. 6.

How do we specify the effect of a conversational move? The basic units of change are mappings between dialogue gameboards that specify how one gameboard configuration can be modified into another on the basis of dialogue moves. We call a mapping between DGB types a conversational rule. The types specifying its domain and its range we dub, respectively, the pre(conditions)

and the effects, both of which are supertypes of the type DGBType. A conversational rule that enables us to explain the effect a greeting, the initial conversational move, has on the DGB is given in (55). It is a record type which contains two fields. The 'pre(condition)'-field is for a dialogue gameboard of a certain type and the 'effects'-field provides a type for the updated gameboard. The precondition in this example requires that both Moves and

    :RecType              
The form for update rules proposed here is thus (56) pre : T 1 effects=T 2 : RecType An agent who believes that they have a current state s of type T 1 , that is, whose hypothesis about the current state is that it belongs to type T such that T T 1 , can use s to anchor T 2 to obtain T 2 [s] and then use asymmetric merge to obtain a type for the new state:

T ∧ . T 2 [s].
The rule (57) says that given a question q and ASK(A,B,q) being the LatestMove, one can update QUD with q as QUD-maximal.

               
Next we introduce the rule QSPEC. QSPEC can be thought of as a 'relevance maxim': it characterizes the contextual background of reactive queries and assertions. (58) says that if q is QUD-maximal, then subsequent to this the next move is constrained to be q-specific [START_REF] Ginzburg | The Interactive Stance: Meaning for Conversation[END_REF], that is, either about q (a partial answer) or a question on which q depends. Moreover, this move can be held by either of the speech event participants. The constraint in (58) involves merging a constraint concerning the information about QUD and Moves with a constraint entitled TurnUnderSpec, which merely specifies that the speaker and addressee of the effects are distinct and drawn from the set consisting of the initial speaker and addressee:

(58) a. QSPEC

            pre : qud. = q, Q : poset(Question) effects : TurnUnderspec ∧ .      r : Prop ∨ Question R: IllocRel LatestMove = R(spkr,addr,r) : IllocProp c1 : About(r,q) ∨ Depend(q,r)                  b. TurnUnderspec =           
PrevAud = pre.spkr,pre.addr : Ind spkr : Ind c1 : member(spkr, PrevAud) addr : Ind c2: member(addr, PrevAud)

∧ addr = spkr           
QSPEC involves factoring out turn taking from the assumption that A asking q involves B answering it. In other words, the fact that A has asked q leaves underspecified who is to address q (first or at all). This is justified by self-answering data such as the initial example we considered in the intro- Integrating metacommunicative interaction into the DGB involves two additions to the picture we have had so far, one minor and one major. The minor addition, drawing on an early insight of Conversation Analysis (see the notion of side sequence, Schegloff ( 2007)), is that repair can involve 'putting aside'

an utterance for a while, a while during which the utterance is repaired. The 'pending'-field in the dialogue gameboard is used for this. Note that this field contains a list of locutionary propositions. Most work on (dialogue) context to date involves reasoning and representation solely on a semantic/logical level.

But if we wish to explicate metacommunicative interaction, then we cannot limit ourselves in this way.

If p:LocProp, the relationship between p.sit and p.sit-type can be utilized in providing an analysis of grounding/CRification conditions:

(65) a. Grounding: p is true: the utterance type fully classifies the utterance token.

b. CRification: p is false, either because p.sit-type is weak (e.g. incomplete word recognition) or because u is incompletely specified (e.g.

incomplete contextual resolution-problems with reference resolution or sense disambiguation).

In principle one could have a theory of CRification based on generating all available CRs an utterance could give rise to. But in practice, as the data in (64) showed us, there are simply too many to be associated in a 'precompiled' form with a given utterance type.

Instead, repetition and meaning-oriented CRs can be specified by means

                    
                    
where q is λr: cont:Cont ( e : mean(⇑pre.spkr,⇑pre.u,r.cont) )

Parameter Identification (66) underpins CRs such as (67b-67c) as followups to (67a). We can also deal with corrections, as in (67d), since they address the issue of what A meant by u. We have now explicated the basis for partial comprehension in dialogue, relating to the requirements from the initial example (1).

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7 Traditional semantic concerns in a dialogue perspective

In this section we will discuss two traditional concerns in semantics, negation and quantification, and show that we get a rather different view of them when we consider dialogue phenomena relating to them.

Negation

The classical view of negation is that it is a truth functional connective that maps true to false and false to true. In intuitionistic approaches as standardly used in type theory, negative propositions, ¬p, are regarded as the type of refutations of p. This leads intuitionistic logic to abandon the principle of bivalence, that propositions are either true or false. On the intuitionistic view it is possible that a proposition p neither has a proof nor a refutation. Thus such a proposition is neither true nor false.

In this section, which contains revised material from Cooper & Ginzburg (2011a,b), we will suggest an alternative view: that negation is used to pick out a negative situation type. It is crucial for this proposal to work that we are able to distinguish between positive and negative types in a way that is not possible on the standard approaches to "truth-value flipping" negation.

Consider the uses of no in the (made-up) dialogue in ( 68) and the glosses given after them in square brackets. The first use of no does not relate back to any previous linguistic utterance but rather to an event which is in progress. The parent has observed the first part of the event and predicted a likely conclusion (as in the example of the game of fetch discussed in Section 2). The parent wishes to prevent the completion of the event, that is, make sure that the predicted complete event type is not realized. We claim that the central part of the meaning of negation has to do with the non-realization of some positive situation type (represented by a negative situation type), rather than a switching of truth values as on the classical logical view of negation. We see this again in the second use of no in response to the parent's query whether the type child-wants-to-be-electrocuted is realized. The child's negative response asserts that the type is not realized.

The third utterance of no agrees with the previous assertion, namely this asserts agreement that the type is (or should be) empty. A naive application of the classical view of negation as a flipping of truth values might say that no always changes the truth-value of the previous assertion. This would make the wrong prediction here, making the parent disagree with the child. Our view that negation has to do with a negative situation type means that it will be used to disagree with a positive assertion and agree with a negative assertion, which seems to be how negation works in most, if not all, natural languages.

Another important fact about this dialogue is the choice of the parent's question. The positive question is appropriate whereas the negative question would be very strange, suggesting that the child should want to be electrocuted. The classical view of negation as truth value flip has led to a view that positive and negative questions are equivalent [START_REF] Hamblin | Questions in Montague English[END_REF][START_REF] Groenendijk | Questions[END_REF]. This derives from a view of the contents of questions as the sets of propositions corresponding to their answers. While positive and negative questions do seem to have the same possible answers it appears that the content of the question should involve something more than the set of answers. The distinction between positive and negative questions was noted for embedded questions by [START_REF] Hoepelmann | On questions[END_REF] who noted the examples in (69).

(69)

a. The child wonders whether 2 is even.

b. The child wonders whether 2 isn't even. (There is evidence that 2 is even)

Hoepelmann's observation is that the same kind of inference as we noticed with the negative version of the parent's question about electrocution. That is, there is a suggestion that there is reason to believe the positive, that the type is realized. This kind of inference is not limited to negative questions but seems to be associated with negation in general. [START_REF] Fillmore | Frames and the semantics of understanding[END_REF] notes the examples in (70). The examples marked with # sound strange because they are contrary to our expectations. We in general expect that people have teeth but not walnut shells, a nose but not several noses and several toes but not just a single toe.

Fillmore discusses this in terms of frames. We would discuss this in terms of resources we have available. We can, however, create the expectations by raising issues for discussion within the dialogue thus creating the necessary resources locally as in ( 71).

( In French the word oui is used to agree with a positive proposition and the word si is used to disagree with a negative proposition. Similar words exists in other languages such as German (ja/doch) and Swedish (ja/jo).

How do we know that the distinction between positive and negative propositions is a semantic distinction rather than a syntactic distinction depending on how the propositions are introduced? There are lots of ways of making a negative sentence, by using various negative words such as not, no, none, nothing. In French you have there are discontinuous constructions ne. . . pas/point/rien corresponding to "not/not at all/nothing". However, in these constructions the ne can be omitted. Thus both of the following are possible: je n'en sais rien/ j'en sais rien ("I know nothing about it"). In Swedish there are two words for not which are stylistic variants: inte, ej The generalization that allows us to recognize all these morphemes or constructions as "negations" is the semantic property they share: namely that they introduce negative propositions.

On the traditional truth-value flipping view of negation it is hard to make this semantic distinction. For example, in a possible worlds semantics a proposition is regarded as a set of possible worlds -the set of worlds in which the proposition is true. On this view the negation of a proposition is the complement of that set of worlds belonging to the proposition. There is no way of distinguishing between "positive" and "negative" sets of possible worlds. However, on a type theoretic approach the distinction can be made in a straightforward manner.

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The account of negation we give here is slightly different to that given in Cooper & Ginzburg (2011a,b) and as a consequence the definitions are slightly more elegant and intuitive. We introduce negative types by the clause (74).

(74) If T is a type then ¬T is a type

Because types are intensional we can say that ¬T is distinct not only from

T but also from any other type, without worrying that there might be an equivalent type that has the same witnesses. Thus simply by introducing a negative operation on types (represented by ¬) we distinguish negative types from positive ones. We can also introduce types of negative types. For example, we can introduce a type RecType ¬ such that T :RecType ¬ iff T = ¬T and T :RecType. We can then define a type RecType ¬ † whose witnesses are the closure of the set of negated record types under negation (in a similar manner to our definition of RecordType † on p. 18).

We can characterize witnesses for negative types by: a : ¬T iff there is some T such that a : T and T precludes T . We say that T precludes T iff either (75a) or (75b) holds.

(75) a. T = ¬T b. or, T, T are non-negative and there is no a such that a : T and a : T for any models assigning witnesses to basic types and ptypes It follows from these two definitions that (1) a : ¬¬T iff a : T and that (2) a : T ∨ ¬T is not necessary (a may not be of type T and there may not be any type which precludes T either). Thus this negation is a hybrid of classical and intuitionistic negation in that (1) normally holds for classical negation but not intuitionistic whereas (2), that is failure of the law of the excluded middle, normally holds for intuitionistic negation but not classical negation.

Nothing in these definitions accounts for the fact that a : ¬T seems to require an expectation that a : T . One way to do this is to refine the clause that defines witnesses for negative types:a : ¬T iff there is some T such that a : T and T precludes T and there is some expectation that a : T . There is some question in our minds of whether this addition should be included here or in some theory of when agents are likely to make judgements. What does it mean for there to "be some expectation"? We would like to relate this to the kind of functions we used to predict completions of events and which we also used for grammar rules, that is to dependent types. [START_REF] Breitholtz | Clarification requests as enthymeme elicitors, in Aspects of Semantics and Pragmatics of Dialogue[END_REF]; [START_REF] Breitholtz | Enthymemes as rhetorical resources[END_REF] The type of positive Austinian propositions can be defined as (77). They use this to motivate a particular view of the semantics of quantified noun-phrases which is based on witness sets rather than families of sets as in the classical treatment. [START_REF] Cooper | Generalized quantifiers and clarification content[END_REF]Cooper ( , 2013) ) argues for combining a more classical treatment with their approach. We summarize the argument here.

In terms of TTR, a type corresponding to a "quantified proposition" can be regarded as ( 79). : q(restr,scope)

 

The third field represents a quantificational ptype of the form q(restriction,scope)

an example of which would be (80).

(80) every(λr: x:Ind ( c:dog(r.x) ), λr: x:Ind ( c:run(r.x) ))

That is, 'every' is a predicate which holds between two properties, the property of being a dog and the property of running. As an example, suppose we want to represent the record type which is the content of an utterance of A thief broke in here last night. For convenience we represent the property of being a thief as thief and the property corresponding to broke in here last night as bihln. Then the content of the sentence can be (81). That is, the set of objects that have the property. We say that there is a constraint on models such that the type q(P 1 , P 2 ) is non-empty iff the relation q *

holds between [↓ P 1 ] and [↓ P 2 ], where q * is the relation between sets corresponding to the quantifier in classical generalized quantifier theory. Examples are given in ( 83).

(83) a. some(P 1 ,P 2 ) is non-empty (that is, "true") just in case

[↓ P 1 ]∩[↓ P 2 ] = ∅ b. every(P 1 ,P 2 ) is non-empty just in case [↓ P 1 ] ⊆ [↓ P 2 ]. c. many(P 1 ,P 2 ) is non-empty just in case | [↓ P 1 ] ∩ [↓ P 2 ] | > n,
where n counts as many.

The alternative analysis of generalized quantifiers that [START_REF] Purver | Clarifying noun phrase semantics[END_REF]; [START_REF] Ginzburg | Quantfication, the reprise content hypothesis, and type theory[END_REF]; [START_REF] Ginzburg | The Interactive Stance: Meaning for Conversation[END_REF] propose is based on the notion of witness set from [START_REF] Barwise | Generalized quantifiers and natural language[END_REF]. Here we will relate this notion to the notion of a witness for a type, that is something which is of that type. We have not yet said exactly what it is that is of a quantifier ptype q(P 1 , P 2 ). One solution to this is to say that it is a witness set for the quantifier, that is (84).13

(84) a : q(P 1 , P 2 ) iff q * holds between [↓ P 1 ] and [↓ P 2 ] and a =

[↓ P 1 ]∩[↓ P 2 ]
This definition relies on the fact that all natural language quantifier relations are conservative [START_REF] Peters | Quantifiers in Language and Logics[END_REF], a notion which we can define as in ( 85).

(85) a quantifier q is conservative means q * holds between [↓ P 1 ] and [↓ P 2 ] just in case q * holds between [↓ P 1 ] and [↓ P 1 ] ∩ [↓ P 2 ] (every person runs iff every person is a person who runs)

Armed with this we can define the type of (potential) witness sets for a quantifier relation q and a property P , q † (P ), that is, witness sets in the sense of Barwise and Cooper as in (86).

(86) a : q † (P ) iff a ⊆ [↓ P ] and there is some set X such that q * holds between [↓ P ] and X Using these tools we present a modified version of Ginzburg and Purver's proposed analysis of most students left in (87), where the 'q-params'-field specifies quantifier parameters and the 'cont'-field specifies the content of the utterance.

(87) q-params : w:most † (student) cont : c q =q-params.w:most(student,left)

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In [START_REF] Cooper | Generalized quantifiers and clarification content[END_REF] we presented the two analyses as in competition with each other, but we now think that there is advantage to be gained by putting the two together. Our way of combining the two analyses predicts two readings for the noun-phrase most students, a referential reading which makes the witness set be a q-parameter in Purver and Ginzburg's analysis and a non-referential reading in which the witness set is incorporated in the content of the NP. These are given in ( 88).

(

) a. referential           q-params: restr i =student:Ppty w i :most † (q-params.restr i ) cont= λP :Ppty (   scope=P :Ppty c most =⇑q-params.w i :most(⇑q-params.restr i , scope)   ):Quant           b. non-referential           q-params:Rec cont= λP :Ppty (     restr i =student:Ppty w i :most † (restr i ) scope=P :Ppty c most =w i :most(restr i ,scope)     ):Quant           88
Given these types, what can a clarification address? Our claim is that the clarification must address something for which there is a path in the record type. In addition there appears to be a syntactic constraint that clarifications tend to be a "major constituent", that is a noun-phrase or a sentence, rather than a determiner or a noun. In a referential reading there are three paths available: 'q-params.restr i ', 'q-params.w i ' and 'cont'. The first of these, the restriction, is dispreferred for syntactic reasons since it is normally expressed by a noun. This leaves the witness and the whole NP content as possible clarifications. However, from the data it appears that the whole content can be expressed focussing either on the restriction or the quantifier relation. For non-referential readings only the whole content path is available.

In (89) we give one example of each kind of clarification from the data that Purver and Ginzburg adduce.

( The basic strategy adopted in KoS to analyze non sentential utterances (NSUs) is to specify construction types where the combinatorial operations integrate the (surface) denotata of the fragments with elements of the DGB.

We have provided one example of this earlier in our lexical entry for 'hi', (54).

Another simple example is given in (90), a lexical entry for the word 'yes'.

(90) Sign∧ .

   

s-event : phon : yes q max : PolQuestion

synsem : cat=adv ic : Cat cont=q max (r ds ) : Prop    
Here q max is a maximal element of dgb.qud which is of the type PolQuestion, exemplified in (43). Since q max is of the type PolQuestion, it is a constant function whose domain is the class of all records and its range is a proposition p. Hence the content of this function applied to any record is p. Thus, 'yes' gets as its content the proposition p, intuitively affirming the issue 'whether p' currently under discussion. See [START_REF] Fernández | Non-Sentential Utterances in Dialogue: Classification, Resolution and Use[END_REF]; [START_REF] Ginzburg | The Interactive Stance: Meaning for Conversation[END_REF] for a detailed account of this and a wide range of other more complex NSU types.

Disfluencies

Disfluencies are ubiquitous and observable in all but the briefest conversational interaction. Disfluencies have been studied by researchers in Conversational Analysis (e.g., [START_REF] Schegloff | The preference for selfcorrection in the organization of repair in conversation[END_REF]), in great detail by psycholinguists (e.g., [START_REF] Levelt | Monitoring and self-repair in speech[END_REF]; [START_REF] Brennan | How listeners compensate for disfluencies in spontaneous speech[END_REF]; [START_REF] Clark | Using uh and um in spontaneous speech[END_REF]), and by computational linguists working on speech applications (e.g., [START_REF] Shriberg | Preliminaries to a theory of speech disfluencies[END_REF]). To date, they have mostly been excluded from semantic analysis, primarily because they have been assumed to constitute low level 'noise', without semantic import. In fact, disfluencies participate in semantic and pragmatic processes such as anaphora, conversational implicature, and discourse particles, as illustrated in (91). In all three cases, the semantic process is dependent on the reparandum (the phrase to be repaired) as the antecedent.

Hesitations, another manifestation of disfluency, provide a particularly natural example of self-addressed queries, queries where the intended responder is the original querier: Since they can occur at just about any location in a given utterance and their effect is local, disfluencies provide strong motivation for an incremental semantics, that is, a semantics calculated on a word-by-word, left to right fashion (see e.g. [START_REF] Steedman | The Syntactic Process, Linguistic Inquiry Monographs[END_REF]; [START_REF] Kempson | Dynamic Syntax: The Flow of Language Understanding[END_REF], and et al (this volume)). Moreover, they require the content construction process to be nonmonotonic, since initial decisions can be overriden as a result of self-repair. In this paper we have presented a theory which encompasses both the analysis of dialogue structure and the traditional concerns of formal semantics. Our main claim is that the two should not be separated. We have used a rich type theory (TTR -type theory with records) in order to achieve this. The main advantages of TTR is that it presents a theory of types which are structured in a similar way to feature structures as employed in feature-based approaches to grammar while at the same time being a type theory including a theory of functions corresponding to the λ-calculus which can be used for a highly intensional theory of semantic interpretation. This type theory can be used to formulate both compositional semantics and the theory of dialogue structure embodied by KoS [START_REF] Ginzburg | The Interactive Stance: Meaning for Conversation[END_REF]. Among other things we have shown how these tools can be used to create a theory of events (both non-linguistic and linguistic) and thereby create a theory of grammar grounded in the perception of speech events. We have shown how these tools enable us to give an account of the kind of abstract entities needed for semantic analysis, such as propositions and questions. We have also shown how the same tools can be used to given an account of dialogue gameboards and dialogic interaction.

We have exemplified that with respect to variety of phenomena one needs to tackle in order to provide even a rudimentary analysis of an extract from an actual British National Corpus, example (1), which we presented at the beginning of the paper. While we cannot claim to have handled all the details of this example we have nevertheless presented a theory which begins to provide some of the pieces of the puzzle. In particular: non sentential utterances are analyzed using a dialogue game-board driven context exemplified in sections , 2011, 2012;[START_REF] Dobnik | Spatial descriptions in type theory with records[END_REF]; grounding semantics in terms of classifiers used for perception [START_REF] Larsson | Formal semantics for perceptual classification[END_REF]; probabilistic semantics [START_REF] Cooper | A probabilistic rich type theory for semantic interpretation[END_REF]; and language acquisition [START_REF] Larsson | Towards a formal view of corrective feedback[END_REF][START_REF] Ginzburg | The earliest utterances in dialogue: towards a formal theory of parent/child talk in interaction[END_REF].

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  and there is some a : T 1 such that if b : T 2 then f (b) = a Page: 7 job: rc-jg-ttrsem-final macro: handbook.cls date/time: 3-Apr-2015/11:40

  of ordered pairs as in the classical set theoretic construction of functions. Let O[v] be the notation for some object of our type theory which uses the variable v and let T be a type. Then the function λv : T . O[v] is to be the function identified by { v, O[v] | v : T }. For example, the function λv:Ind . run(v) is identified by the set of ordered pairs { v, run(v) | v : Ind }. Note that if f is the function λv:Ind . run(v) and a:Ind then f (a) (the result of applying f to a) is 'run(a)'. Our definition of function-argument application guarantees what is called β-equivalence in the λ-calculus. We allow both function types and dependent record types and we allow dependent record types to be arguments to functions. We have to be careful when considering what the result of applying a function to a dependent record type should be. Consider the simple example in (9).(9) λv 0 :RecType ( c 0 :v 0 )

  T is a type then {T } and [T ] are types b. A : {T } just in case A is a set and for any a ∈ A, a : T c. L : [T ] just in case L is a list and any member, a, of L is such that a : T We will also introduce a type Poset(T ) which can be regarded as (27) a : po(R, S) iff a = R, S and R is a partial order on S, that is, R is a set of pairs of members of S (coded as records with 'left' and 'right' fields as above) and R is reflexive or irreflexive, antisymmetric and transitive. If a : T , P : Poset(T ) and a ∈ P.set, then a ⊕ P : Poset(T ) where a ⊕ P is the record r:Poset(T ) such that the clauses in (28) hold. | x ∈ P .set} c. r.c po = r.rel, r.set 2.7 The string theory of events

  (a,c) attract attention(a,b) throw(a,c) run after(b,c) pick up(b,c) return(b,c,a)) +



  TimeInt utt at : uttered at(phon, s-time)Here we have introduced the types Phon, phonology, and TimeInt, time interval, which we will further specify below. A more detailed type for SEvent might be (36). sp,phon,au,e-time,e-loc)

  -jg-ttrsem-final macro: handbook.cls date/time: 3-Apr-2015/11:40

  c. c : Identical(enesco, enesco) d. c : Identical(poulenc, poulenc) e. He is identical with himself.

  : Jo and Mo got married yesterday. It was a wonderful occasion. Page: 17 job: rc-jg-ttrsem-final macro: handbook.cls date/time: 3-Apr-2015/11:40 b. A: Jo's arriving next week. B: No, that's happening in about a month.

  propositions. As such, questions are automatically part of the type theoretic ontology. Let us start by considering some very simple examples of interrogatives and their TTR representations. (42) exemplifies the denotations (contents) we can assign to a unary and a binary wh-interrogative. We use r ds here to represent the record that models the described situation in the context. The meaning of the interrogative would be a function defined on contexts which provide the described situation and which return as contents the functions given in (42). The unary question ranges over instantiations by persons of the proposition "x runs in situation r ds ". The binary question ranges over pairs of persons x and things y that instantiate the proposition "x touches y in situation r ds ": sit-type = c:touch(r.x,r.y) )

  we think of a unary abstract as involving a domain type with one field for an individual and a binary abstract as one whose domain type contains two such fields etc, then by analogy the domain type of a 0-ary type would simply be the empty record type [] (that is, the type Rec of records). 11 This makes a 0-ary abstract a constant function from the universe of all records . sit-type = c : run(bo) )The fact that questions individually are part of the type theoretic world is not the end of the story. For various linguistic tasks (e.g. specifying the selectional requirements of verbs like 'ask', 'wonder', and 'investigate'), and for various dialogical tasks (e.g. the formulation of various conversational rules) one needs to appeal to a type Question (see the chapter on questions, Wiśniewski (this volume).). This means that we need to have a characterization of this type within TTR. One such characterization is given in Ginzburg (2012); a more recent and, arguably, more constructive proposal can be found in Ginzburg et al. (2014a). Here we offer a somewhat simpler characterization.The domain of a question (polar or wh) is always characterized by a subtype of RecType. Thus we define the type Question by (44).

  : Can I have some ice-cream then? Dorothy: you can do if there is any. (BNC, KBW) b. Anon: Are you voting for Tory? Denise: I might. (BNC, KBU, slightly modified) c. Dorothy: What did grandma have to catch? Christopher: A bus. (BNC, KBW, slightly modified) d. Rhiannon: How much tape have you used up? Chris: About half of one side. (BNC, KBM) e. Dorothy: What do you want on this? Andrew: I would like some yogurt please. (BNC, KBW, slightly modified) f. Elinor: Where are you going to hide it? Tim: Somewhere you can't have it.(BNC, KBW) g. Christopher: Where is the box? Dorothy: Near the window. (BNC, KBW)

  ; the sole effect is to push the proposition associated with hi onto the list in the 'moves'-field.

  q=ques⊕pre.qud : poset(Question) :RecType

  duction (1), as well as (59a,b), where the querier can or indeed needs to keep the turn, as well as multi-party cases such as (59c) where the turn is : When is, when is Easter? March, April? (BNC, KC2) b. Brian: you could encourage, what's his name? Neil. (BNC, KSR) c. A: Who should we invite? B: Perhaps Noam. C: Martinu. D: Bedrich.Explicating the possibility of self-answering is one of the requirements for dealing with our initial example (1). most recent move has been understood to the satisfaction of the conversationalists, has come to be known as grounding, following extensive empirical work by Herb Clark and his collaborators[START_REF] Clark | Contributing to discourse[END_REF];[START_REF] Clark | Referring as a collaborative process[END_REF];[START_REF] Clark | Using Language[END_REF]). One concrete task for a theory of dialogue is to account for the potential for and meaning of acknowledgement phrases, as in (60), either once the the utterance is completed, as in (60a), or concurrently with the utterance as in (60b): (60) a. Tommy: So Dalmally I should safely say was my first schooling. Even though I was about eight and a half. Anon 1: Mm. Now your father was the the stocker at Tormore is that right ? (British National Corpus (BNC, K7D) b. A: Move the train . . . B: Aha A:. . . from Avon . . . B: Right A:. . . to Danville. (Adapted from the Trains corpus, Allen et al. (1995)) An additional task is to characterize the range of (potential) presuppositions emerging in the aftermath of an utterance, whose subject matter concerns both content and form. This is exemplified in the constructed examples in Mark send you a love letter? b. B: No, though it's interesting that you refer to Mark/my brother/our friend c. B: No, though it's interesting that you mention sending d. B: No, though it's interesting that you ask a question containing seven words.

  of a uniform class of conversational rules, dubbed Clarification Context Update Rules (CCURs) in Ginzburg (2012). Each CCUR specifies an accommodated MaxQUD built up from a sub-utterance u1 of the target utterance, the maximal element of Pending (MaxPending). Common to all CCURs is a license to follow up MaxPending with an utterance which is co-propositional with MaxQUD. (66) is a simplified formulation of one CCUR, Parameter identification, which allows B to raise the issue about A's sub-utterance u: what did A mean by u?:

(

  67) a. A: Is Bo here? b. B: Who do you mean 'Bo' ? c. B: Bo? (= Who is 'Bo' ?) d. B: You mean Jo.

  Child approaches socket with nail Parent: No. ["Don't put the nail in the socket."] Do(#n't) you want to be electrocuted? Child: No. ["I don't want to be electrocuted."] Parent: No. ["You don't want to be electrocuted."]

  father doesn't have any teeth b. #Her husband doesn't have any walnut shells c. Your drawing of the teacher has no nose/#noses d. The statue's left foot has no #toe/toes

  RecTypeThus we have a clear way of distinguishing negative and positive propositions.

7. 2

 2 Generalized quantifiers[START_REF] Purver | Clarifying noun phrase semantics[END_REF];[START_REF] Ginzburg | Quantfication, the reprise content hypothesis, and type theory[END_REF];[START_REF] Ginzburg | The Interactive Stance: Meaning for Conversation[END_REF] introduce the Reprise Content Hypothesis (RCH) given in (78). (78) a. RCH (weak) A fragment reprise question queries a part of the standard semantic content of the fragment being reprised. b. RCH (strong) A fragment reprise question queries exactly the standard semantic content of the fragment being reprised.

  this proposal back to classical generalized quantifier theory, as represented in[START_REF] Barwise | Generalized quantifiers and natural language[END_REF]. Let the extension of a type T , [ˇT ], be the set {a | a : T }, the set of witnesses for the type. Let the P-extension of a property P , [↓ P ], be the set in (82). {a | ∃r[r : x:Ind ∧ r.x = a ∧ [ˇP (r)] = ∅]}

  was + { well } he was ] fired. (Example from Heeman & Allen (1999)) b. A: Because I, [ [ [ any, + anyone, ] + any friend, ] + anyone ] I give my number to is welcome to call me (Example from the Switchboard corpus, Godfrey et al. (1992)) (implicature: 'It's not just her friends that are welcome to call her when A gives them her number') c. From yellow down to brown-NO-that's red. (

  : Well it's (pause) it's (pause) er (pause) what's his name? Bernard Matthews' turkey roast. (BNC, KBJ) b. Steve: They're pretty . . . um, how can I describe the Finns? They're quite an unusual crowd actually.

(

  Ginzburg et al. (2014b)) sketch how, given an incremental dialogue semantics, accommodating disfluencies is a straightforward extension of the account discussed in section 6 for clarification interaction: the monitoring and update/clarification cycle is modified to happen at the end of each word utterance event, and in case of the need for repair, a repair question gets accommodated into QUD. Self-corrections are handled by a slight generalisation of the rule (66), which just as with the rule QSPEC, underspecifies turn ownership. Hesitations are handled by a CCUR that triggers the accommodation of a question about the identity of the next utterance. Overt examples for such accommodation is exemplified in (92).

5 and 8. 1 .

 1 Disfluencies are handled using conversation rules of a similar form to Clarification Requests and, more generally, to general conversational rules.The possibility of answering one's own question is a consequence of factoring turn taking away from illocutionary specification, as in the conversational rule QSPEC. Misunderstanding is accommodated by (i) associating different dialogue gameboards with the conversational participants, and (ii) characterizing the grounding and clarification conditions of utterances using locutionary propositions (propositions constructed from utterance types/tokens). Multilogue involves scaling up of two-person conversational rules to include communal grounding and acceptance, and multi-agent turn taking. (See[START_REF] Ginzburg | Scaling up to multilogue: some benchmarks and principles[END_REF]; Ginzburg (2012))Beyond the treatment of real conversational interaction, we have looked at a couple of traditional concerns of formal semantics from a dialogical perspective: negation and generalized quantification. Some other areas which are currently being examined using these tools, but which we have not discussed in this article are: quotation(Ginzburg & Cooper, 2014)-where we argue for the use of utterance types and locutionary propositions as denotations for quotative constructions; the semantics for spatial descriptions and its relationship to robot perception and learning (Dobnik Page: 41 job: rc-jg-ttrsem-final macro: handbook.cls date/time: 3-Apr-2015/11:40 et al.

  whether it belongs to the type. Thus the ptypes will be different depending on which record you are checking. The official notation for this record type makes this more explicit by introducing functions from individuals to ptypes and pairing them with a list of path names indicating where in the record one should look for the arguments to the functions, as in (6). 4

			x	: Ind
	(6)	      	c boy : λv:Ind . boy(v), x y : Ind c dog : λv:Ind . dog(v), y e : λv 1 :Ind λv 2
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  We suggest that this is more radical because it ultimately leads to seemingly radical conclusions of an intrinsic semantic indeterminacy: in such a situation the public context is no longer identical for the interlocutorsthe original speaker can carry on, blissfully unaware that a problem exists, utilizing a 'grounded context', whereas if the original addressee takes over the context is shifted to one which underwrites a clarification request. This

	potential context-splitting is illustrated in (62), originally discussed in (Gin-
	zburg (1997)). The data in (62) illustrates that the contextual possibilities for
	resolving the fragment 'Bo?' are distinct for the original speaker A and the
	original addressee B. Whereas there is one common possibility, the short an-
	swer reading, only B has the two clarification request readings, whereas only
	A has a self-correction reading, albeit one that probably requires an further
	elaboration:
	(62)
	a. A: Who does Bo admire? B: Bo?
		Reading 1 (short answer): Does Bo admire Bo?
		Reading 2 (clausal confirmation): Are you asking who BO (of all
		people) admires?;
		Reading 2 (intended content ): Who do you mean 'Bo' ?)
	b. A: Who does Bo admire? Bo?
		Reading 1 (short answer): Does Bo admire Bo?
		Reading 2 (self correction): Did I say 'Bo' ?
	Clarification requests can take many forms, as illustrated in (63):
	(63)
	a. A: Did Bo leave?
	(64)
	a. Who rearranged the plug behind the table?
	b. Who? / rearranged?/ the plug? / behind? / the table?
	c. A: Is that the shark? B: The? B: Well OK, a. (based on an example
		in the film Jaws.)
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e. B: No, though it's interesting that the final two words you just uttered start with 'l' Developing a semantic theory that can fully accommodate the challenges of grounding is far from straightforward. A more radical challenge, nonetheless, is to explicate what goes on when an addressee cannot ground her interlocutor's utterance. b. Wot: B: Eh? / What? / Pardon? c. Explicit (exp) : B: What did you say? / Did you say 'Bo' /What do you mean 'leave' ? d. Literal reprise (lit): B: Did BO leave? / Did Bo LEAVE? e. Wh-substituted Reprise (sub): B: Did WHO leave? / Did Bo WHAT? f. Reprise sluice (slu): B: Who? / What? / Where? g. Reprise Fragments (RF): B: Bo? / Leave? h. Gap: B: Did Bo . . . ? i. Filler (fil): A: Did Bo . . . B: Win? (Table

I from Purver (2006))

Now, as (64a) indicates, a priori ANY sub-utterance is clarifiable, including function words like 'the', as in (64c). While the potential for repetitionoriented clarification interaction clearly applies to all utterances and their parts, it is an open question whether this is true for semantically/pragmatically oriented CRification. For empirical studies on this see

[START_REF] Healey | Experimenting with clarification in dialogue[END_REF]

;

[START_REF] Purver | Answering clarification questions[END_REF][START_REF] Purver | Lexical categories and clarificational potential, revised version under review[END_REF]

.

  The type of negative Austinian propositions can be defined as (76).

	(76)	sit sit-type : RecType ¬ † : Rec
		use dependent types to implement Aristotelian
	enthymemes that is defeasible inference patterns. Such enthymemes could be
	either general or local context-specific resources that we have available to
	create expectations.
	Finally, let us see how the techniques we have developed here could be
	combined with Austinian propositions.
	Page: 34	job: rc-jg-ttrsem-final	macro: handbook.cls	date/time: 3-Apr-2015/11:40

  Chorlton, mhm, he examined me, erm, he, he said now they were on about a slide unclear on my heart. Mhm, he couldn't find it.

	Unknown: And er they X-rayed me, and took a urine sample, took a 8 Grammar in dialogue
		blood sample. Er, the doctor
	Unknown: Chorlton? 8.1 Non Sentential Utterances
		Unknown: BNC file KPY, sentences 1005-1008
	b. Content clarification with restriction focus
		Terry: Richard hit the ball on the car.
		Nick: What car?
		Terry: The car that was going past.
			BNC file KR2, sentences 862-864
	c. Content clarification with quantifier relation focus
		Anon 2: Was it nice there?
		Anon 1: Oh yes, lovely.
		Anon 2: Mm.
		Anon 1: It had twenty rooms in it.
		Anon 2: Twenty rooms?
		Anon 1: Yes.
		Anon 2: How many people worked there?
			BNC file K6U, sentences 1493-1499
	)
	a. Witness clarification
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Our conclusion is that a combination of the classical approach to generalized quantifiers combined with a modification of the approach suggested by Purver and Ginzburg, adding a field for the witness, provides correct predictions about clarifications. This means that the strong version of the reprise clarification hypothesis is consistent with our analysis, allbeit now with a more complex interpretation of the clarification request than Purver and Ginzburg proposed. The interpretation proposed here involves a combination of the classical approach to generalized quantifiers and the witness approach suggested by Purver and Ginzburg. The clarification itself, however, can address different parts of the content of the clarification request.

This introduces one kind of polymorphism into the system. We will also introduce some polymorphism in the typing. Page: 5 job: rc-jg-ttrsem-final macro: handbook.cls date/time: 3-Apr-2015/11:40

Here we use the λ-notation for functions which is discussed in Section 2.4.

This notion was introduced in Coquand et al. (2004).

This convention of representing the path from the top of the record type to the "local" field by the final label on the path is new since[START_REF] Cooper | Type theory and semantics in flux[END_REF].

This record corresponds to the interpretation of it appears that a dog is approaching.

This notation is new since Cooper (2012).

For more detailed discussion of the grammar discussed here and below see[START_REF] Cooper | Type theory and semantics in flux[END_REF].

This is the type all records satisfy, since it places no contraints on them. Page: 19 job: rc-jg-ttrsem-final macro: handbook.cls date/time: 3-Apr-2015/11:40

For more detailed discussion seeGinzburg & Sag (2000, Chapter 3, section 3.2; Chapter 8, section 8.3.).

This appears to go against the intuition that we have introduced before that ptypes are types of situations. Ultimately we might wish to say that a witness for a quantifier type is a situation containing such a witness set, but we will not pursue this here.

This work was supported in part by Vetenskapsrådet project 2009-1569, the Lab(oratory of )Ex(cellence)-EFL (ANR/CGI), and by the Disfluency, Exclamations, and Laughter in Dialogue (DUEL) project within the projets franco-allemand en sciences humaines et sociales funded by the ANR and the DFG. We are grateful for comments to the participants in three courses we