Inquisitive Semantics and Implicature

Transcription

1 Utrecht Univerity Bachelor Thei (7.5 EC) Inquiitive Semantic and Implicature Author: Minke Brandenburg Student no Supervior: Dr. R. Iemhoff Augut 2014

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3 Content 1 Introduction 1 2 Inquiitive Semantic Sentence repreentation Support and meaning Claifying entence Formulae in inquiitive emantic Knowledge repreentation Inquiitive logic Entailment and the dijunction property Dijunctive negative variant InqL and other logic Converational Implicature Grice maxim Inquiitive Implicature Dicuion Inquiitive emantic application in reearch Inquiitive emantic in AI Concluion 35 iii

4 iv CONTENTS

5 Chapter 1 Introduction Language ha alway been a ource of interet for cientit. The human ability to produce and undertand peech and writing i part of what differentiate u from other animal pecie, and therefore help define what it i to be human. Linguitic and artificial intelligence reearcher try to model language to better undertand it mechanic, and to be able to reproduce it in a computer. Their goal i to have a computer undertanding and producing linguitic expreion that are inditinguihable from thoe produced by human. Reearch on emantic ha provided ome inight into the way language could be modelled. Many way exit to model all kind of entence in a way that i analyzable uing logic, but many of thoe model are baed on only the informative apect of language. In thee model, entence are informative tatement, ideal to model factual information like that given in a textbook. An example of uch a entence would be Mr. Smith live in London. Thi entence could then be modeled a omething like liveinlondon(m r.smith). We could then check whether Mr. Smith wa in the et of entitie living in London, and thereby would know whether the entence i true or fale. One iue that i rarely addreed i one of quetion: how to properly model a quetion? Some linguit ugget modeling a imple quetion Where doe Mr. Smith live? like the following three tatement: I do not know where Mr. Smith live, I want to know where Mr. Smith live, and I believe you know where Mr. Smith live. (ee Hamblin (1958)). While thi method might provide a olution at leat to ome extent, thi eem like a very unnatural way of thinking about quetion. Ciardelli and Roelofen (2011) and Groenendijk and Roelofen (2009) decribe an- 1

6 2 CHAPTER 1. INTRODUCTION other poible olution: a emantic more uited to natural converation, and the logic accompanying it. Inquiitive emantic wa developed a an alternative to claical emantic, in which the interactive exchange of information and quetion that are real-world converation could be modelled. Converation play a big role in human ociety, and here we will only be able to conider converation with the purpoe of haring information. Goal like convincing omeone of omething untrue or meaningle chit-chat will have to wait Thi thei will be an introduction to inquiitive emantic. In addition, we will look at how pragmatic can be handled in inquiitive emantic. We will conider thi by looking at the work of Grice (1975) and an adaption of hi maxim for thi emantic by Groenendijk and Roelofen (2009). To tart we will give a introduction to the definition of meaning and truth in inquiitive emantic. We will how how ome important theorem from claical logic hold up in inquiitive logic. In the following chapter we will conider the way converation i modelled uing inquiitive emantic. In chapter four we will then look at Grice converational maxim and their tranlation to inquiitive emantic. Thi will then be followed by more analyi on inquiitive emantic poibilitie and application, both in cience in general and in the field of artificial intelligence.

7 Chapter 2 Inquiitive Semantic 2.1 Sentence repreentation In thi chapter we will look at the logic behind inquiitive emantic. After ome example we will then continue with an explanation of how knowledge i repreented in the emantic. Mot of thi chapter i baed on the work of Ciardelli and Roelofen (2011) and Groenendijk and Roelofen (2009) Support and meaning Inquiitive emantic ha a bai of indice and tate. An index i a ubet of the propoition letter P. Sometime indice are called world, however, we will ue the term indice. A tate i a et of indice. In thi emantic, the language, and the entence in it, are built up from by propoition letter and the connective,,,!,? and, where: p = p!p = p?p = p p Becaue we want the emantic to be able to deal with both tatement and quetion, the concept of meaning i lightly different from that in claical emantic. We ay omething i true in a tate if the tate upport it. Support by a tate i defined a follow: 3

8 4 CHAPTER 2. INQUISITIVE SEMANTICS Definition 1. Support (for tate and index w) = p w : p w p q p and q p q p or q p q t : t p implie t q Indice, or ingle-index tate, behave claically. That mean that for any index w and any formula φ: {w} φ w φ (2.1) Becaue of upport being defined thi way, meaning will alo be different compared to claical emantic. A propoition φ will be conidered true in a tate if upport φ. With thi definition, we can define multiple other term that will come in ueful later on. Definition 2. For a formula φ: 1. A poibility for φ i a maximal tate upporting φ. A tate i a poibility for φ if it upport φ and it i not a ubet of another tate upporting φ. 2. A propoition of φ i the et of poibilitie for φ. 3. The truth-et of φ i the et of indice where φ i true, or the maximal tate where every index eparately upport φ. The propoition for φ can be written a [φ], while the correponding truth-et i written a φ. Example are given in figure 2.1 to clarify the difference between poibilitie, propoition, and truth-et. In thi figure (and many of the one following) the mall circle containing number or ymbol are depicting indice. The colored line urrounding them how the tate. Sometime the line may urround all indice, in which cae it how the maximal tate of all indice. We will ue the term illutrated in the figure to further define the different kind of entence in inquiitive emantic.

9 2.1. SENTENCE REPRESENTATION 5 1, 1 0, 1 1, 1 0, 1 1, 1 0, 1 1, 0 0, 0 1, 0 0, 0 1, 0 0, 0 (a) The poibilitie for p q (b) The propoition p q (c) Truth-et of p q Figure 2.1: Difference between poibility, propoition and truth-et of a formula Claifying entence To help u talk about the different kind of entence in thi emantic, we will introduce ome new definition. Firt we will define what a quetion i, and what an aertion i. To thi end, we will alo define inquiitivene and informativity. Definition 3. Inquiitive, informative A entence φ i inquiitive if the correponding propoition [φ] contain more than one poibility. A entence φ i informative if it propoe to remove at leat one index from it tate. Definition 4. Quetion & aertion A entence i a quetion if it i not informative. A entence i an aertion if it i not inquiitive. Of coure, a propoition can be both informative and inquiitive, for example if it propoe to eliminate one index, and to divide the remaining indice into group. Becaue of the definition of quetion, it i poible for a quetion to not propoe more than one poibility, for example if the quetion i meant to make ure everyone agree with the common ground at a certain time. Similarly, it i poible for an aertion to not actually give new information. We will look at what thi mean for converational implicature later. We will firt how ome example of inquiitive and informative entence.

10 6 CHAPTER 2. INQUISITIVE SEMANTICS Example 1. a) I Ada at home or at work? (Inquiitive (+ informative)) b) Ada i at work. (Informative) work market work market work home home chool (b) Inquiitive home chool (a) informative and inquiitive (c) Informative Figure 2.2: Example of inquiitive and informative entence In figure 2.2a we ee a knowledge tate containing three three poibilitie, two of which are marked in red. When uing thi knowledge tate, the entence in (a) i both inquiitive (ince there are till two poibilitie that are poible anwer to the quetion) and informative (ince two indice are eliminated, becaue they are not contained in any poibility for (a)). Figure 2.2b how the purely inquiitive entence. In thi cae, no indice are eliminated, but it i clear that the entence outline multiple poibilitie. Figure 2.2c how the propoal of entence (b): only one poibility i left, and all indice not contained in that poibility will be dicarded. In mot cae it will uffice to ay that if a propoition contain a dijunction operator, it will be a quetion. If it doe not contain a dijunction, it i an aertion. It i poible, however, to find example in which thi in t the cae. For thoe cae more pecific definition will be ufficient. To give the reader a better idea of what part of inquiitive logic and claical logic are imilar and different, we will look at ome theorem and formulae and the way they are interpreted uing inquiitive emantic.

11 2.2. FORMULAE IN INQUISITIVE SEMANTICS Formulae in inquiitive emantic A we have een earlier, there i a difference in how truth i defined between inquiitive and claical emantic. To how what exactly the poibilitie of inquiitive emantic are, I will here give ome example of poible tatement in inquiitive emantic. Thee example are of interet becaue they are true in claical logic, while thi i not a trivial in inquiitive logic. Firt we will how how the claical dijunction can be modelled in inquiitive emantic. We here define claical dijunction a w : (p w q w). So thi i the poibility in for which every index contain either p or q. For the following proof, recall that!φ i horthand for φ Lemma 1. Claical dijunction (p q) can be modelled in inquiitive emantic a!(p q) Proof.!(p q) (p q) (2.2) (p q) (2.3) t : t (p q) implie t (2.4) t : t (p q) (2.5) t : t (p q) (2.6) t : ( r t : r (p q) implie r ) (2.7) t : ( r t : r (p q)) (2.8) t : r t : r (p q) (2.9) t : r t : r p or r q) (2.10) (2.11) The final line of the proof ay that in every ubet t of, there mut be at leat one ubet r(thi can be a ingle index) for which it i true that r contain either p or q or both. Becaue thi i true for every ubet, it mut alo be true for every index in, if every index contain either p or q. So, the claical dijunction in inquiitive emantic can be modelled a the aertion!(p q). Above proof i only hown uing atom. Thi i becaue inquiitive emantic i not cloed under ubtitution. Therefore, it i hard to prove omething uing φ, ψ intead

12 8 CHAPTER 2. INQUISITIVE SEMANTICS of p, q. We will now how that inquiitive emantic i not cloed under uniform ubtitution. We can t jut replace a ubformula with another formula and expect it to give comparable reult. Thi i hown by proving that p p, while p q (p q). Here φ ψ i defined a : φ ψ. Becaue the pecific formulae ued in thi proof, we immediately prove that in inquiitive emantic double negation elimination doe not apply. Lemma 2. Uniform ubtitution doe not apply in inquiitive logic. Proof. p w : p w (2.12) p p (2.13) t : t p t (2.14) t : t p (2.15) t : t p (2.16) t : ( r t : r p r ) (2.17) t : ( r t : r p) (2.18) t : r t : r p (2.19) t : r t : w r : p w (2.20) w : p w (2.21) Step 2.20 ay that there in every ubet t of, there exit a ubet r o that all indice in r contain p. If we take t to be the mallet poible ubet, an arbitrary index, then r ha to be that ame index. So then we know that that index contain p. Becaue we choe thi index arbitrarily the ame i true for all indice in. Thu we know that every index in contain p, which i what i aid in and therefore that we can make the tep from 2.20 to Thi i alo the definition of p in 2.12, o therefore in inquiitive emantic p p.

13 2.2. FORMULAE IN INQUISITIVE SEMANTICS 9 p q p or q (2.22) w : p w or w : q w (2.23) (p q) (p q) (2.24) t : t (p q) t (2.25) t : t (p q) (2.26) t : t (p q) (2.27) t : ( r t : r p q r ) (2.28) t : ( r t : r p q) (2.29) t : r t : r p q (2.30) t : r t : r p or r q (2.31) t : r t : w r : p w or w r : q w (2.32) While at firt glance thi look identical to the proof of p = p, there an important difference. According to , (p q) if every ingle index in contain either p or q. Therefore every index can have a different valuation for p and q, a long a at leat one of them i claically true. According to , p q if every index in contain p, or every index contain q. Thi mean that for p q to be true, every index mut have the ame poitive valuation concerning p or q. So, p q (p q). p q i true when all indice contain p or all indice contain q. (p q) i true when all indice contain either p or q. It i clear that while p q (p q), the oppoite i not necearily the cae. Combined with the above proof of p = p, it i now clear that thi ytem i not cloed under uniform ubtitution. From the above proof it i alo clear that double negation elimination doe not apply in inquiitive emantic, for the ame reaon why p q (p q). Lemma 3. Double negation elimination doe not apply in inquiitive logic. Proof. If φ, then in every ingle index in, φ i true. The ame eem to be the

14 10 CHAPTER 2. INQUISITIVE SEMANTICS cae for φ: thi can be written a in 2.21 ( t : r t : r φ). Thi mean that for every ubet of, there exit a ubet in which every world contain φ. However alike the two eem, there i a crucial difference that make double negation elimination a problem. An example ituation i given in figure 2.3, for φ = p q. Here, in every ubet there i an index in which p q i true. However, it i not the cae that either all indice in contain p or all indice contain q. Therefore, p q in the example. By thi counterexample, we have hown that double negation elimination doe not necearily apply. p q p q Figure 2.3: (p q) but p q Note that double negation elimination doe apply for atom, ince thoe cannot be interpreted differently in different indice to be upported by. So p = p, but φ φ. Another very important difference between claical and inquiitive logic i the latter rejection of the law of the excluded middle. We will now look at the proof for thi, baed on Macarenha (2009). We how that φ φ doe not necearily hold, thereby proving that the ame i true for the law of excluded middle. Lemma 4. The law of no excluded middle doe not apply in inquiitive logic. Proof. For φ to be true if φ i an atom, every ingle world w in mut contain φ, or w : φ w. Figure 2.4a how thi ituation. If φ i not an atom, we know that we can reduce the formula uing definition 1 defining upport until only formulae of the form tate atom remain. For φ to be true, no world in can contain φ (or all world contain φ, which would have the exact ame reult. Becaue world behave claically, for them not containing φ i equal to containing φ). Thi i formalized a w : φ / w, hown in figure 2.4b.

15 2.3. KNOWLEDGE REPRESENTATION 11 Becaue of the way upport i defined, it i poible that φ φ. An example i given in figure 2.4c. When ome world in contain φ, while other world in do not contain φ, it i not poible to either ay φ or φ. Thi lead to φ φ. It hould be clear that becaue of thi, φ φ i not a tautology, and therefore the law of excluded middle doe not apply in inquiitive emantic. φ, ψ φ ψ, χ χ ψ, χ χ φ φ, χ ψ α φ φ, ψ (a) φ (b) φ (c) φ and φ Figure 2.4: Different ituation of upport in We have hown that ome of the theorem ued in claical logic cannot be applied when uing inquiitive logic. It hould be clear that thee diimilaritie can create major difference in what i conidered true in a ytem. Thi make it poible for inquiitive logic to deal with quetion in it related emantic. We will now proceed to give a little more information on how converation can be modeled in inquiitive emantic. 2.3 Knowledge repreentation Since we want to be able to ue inquiitive emantic to model real converation, we have to conider how a converation i repreented in the ytem that i ued. Thi hould keep track of what i aid, what each participant of the converation know, and what every participant know. Groenendijk and Roelofen (2009) and Ciardelli and Roelofen (2011) decribe a ytem that ha it bai in a common ground : a tate repreenting the common knowledge of the participant in a converation. In other word, the common ground repreent the knowledge that all participant have and of which they all know they all know it. At the beginning of a converation, the common ground will generally be

16 12 CHAPTER 2. INQUISITIVE SEMANTICS the maximal tate, containing every ingle poible index. During the converation, indice will be removed from the common ground, repreenting knowledge ince fewer poibilitie are left. Each participant peronal knowledge i repreented in the ame way: a tate repreenting what the participant know and know to know. The difference with the common ground i that thi tate generally already contain knowledge at the beginning of a converation: indice have already been removed from the tate. Therefore, a individual knowledge tate i a maximal tate that generally contain fewer indice than the tate repreenting the common ground. It i important to note that both the common ground and the individual knowledge tate are maximal tate: they contain all indice that are deemed poible by either the individual or the group. It i poible for thee knowledge tate to contain one or more maller tate. Thi could happen, for example, after an inquiitive entence i uttered. Thi caue the knowledge tate to divide into everal maller tate, the poibilitie that are poible anwer to the quetion. To avoid poible confuion, we will ue knowledge tate to indicate the maximal tate of one or more participant indice. State will be ued a before, to indicate a et of indice. Each entence from a participant i viewed a a propoal to change the common ground. If accepted, it might alo make a change to individual knowledge tate of the participant. If any of the participant reject the propoal it i ignored, and no change are made to any knowledge tate. During a converation, tate are removed from the common ground in accordance with the aertion and quetion uttered. Aertion uually remove indice, while quetion create a diviion between the indice in a knowledge tate, inviting other participant to make an aertion confirming one of the group. If the propoal i accepted, both the common ground and the peronal knowledge tate of the participant of the converation are updated. The latter i done to enure that the common ground really repreent the knowledge of all participant. Chronologically, the following i what happen when an propoal i voiced. Firt, every participant check the propoal with their own knowledge tate. If any participant find the information given i inconitent with their own knowledge, he in t able to update hi own knowledge tate. Doing thi would remove all indice from the knowledge tate, ince in the emantic omething i only true if it i true in every index in a tate: accepting omething cauing a contradiction a true would empty the knowledge tate. Becaue of thi, the participant ha to announce hi incapability to update hi tate. Thi cancel the update of all participant knowledge

17 2.3. KNOWLEDGE REPRESENTATION 13 tate and the common ground. Of coure, if the utterance i purely inquiitive, no participant hould be able to reject the propoal. An inquiitive entence doe not yet propoe to remove any indice (only it anwer would do that), and o hould alway be conitent. The only ituation in which an inquiitive propoal could be rejected i one concerning uch a topic that no index in the participant knowledge tate contain thi. Thi i hard to imagine and even harder to find an example of, therefore we will not conider it here. However, if every participant find the information in the utterance poible within their own knowledge, and no protet i made, all individual knowledge tate and the common ground are adapted to the new information. If the utterance i informative, any indice that do not comply with the poibilitie in the propoal are dicarded. If the utterance i inquiitive, every knowledge tate i divided into everal tate - the poibilitie. A poible next utterance may be informative and, after acceptation, remove one or more of thee poibilitie. After thi tep, no indice hould remain in the individual knowledge tate that have been removed from the common ground, ince that would create inconitencie. Example 2. To clarify thi abtract idea, we will look at an example of two people, let call them Andy and Bea, having a drink. In thi example we will take an incredibly imple world, in which the only thing that matter i whether thee people will drink coffee or tea. We alo aume it i not poible to drink more than one beverage, o both participant will have to chooe what drink they want. C n will mean participant n i having coffee, while T n will mean participant n i having tea. Our example converation conit of the following utterance: a) A: Would you like coffee or tea? (C 2 T 2 ) b) B: Tea, pleae. (T 2 ) c) B: How about you? (C 1 T 1 ) d) A: I d like a coffee. (C 1 ) In figure 2.5 we ee the knowledge tate and common ground. Note that neither participant ha every poible index repreented in their knowledge tate, becaue they already know what they themelve want to drink. After the firt entence, Andy quetion, the knowledge tate will change. Becaue the quetion i inquiitive and not informative, no indice will be removed, but a

18 14 CHAPTER 2. INQUISITIVE SEMANTICS C 1, T 2 C 1, T 2 T 1, T 2 C 1, T 2 T 1, T 2 C 1, C 2 C 1, C 2 T 1, C 2 (a) Knowledge tate of A (b) Knowledge tate of B (c) Common ground Figure 2.5: Situation at beginning of converation C 1, T 2 C 1, T 2 T 1, T 2 C 1, T 2 T 1, T 2 C 1, C 2 C 1, C 2 T 1, C 2 (a) Knowledge tate of A (b) Knowledge tate of B Figure 2.6: Situation after (a) (c) Common ground diviion i added a a ort of preparation for the anwer. The reult i hown in figure 2.6. After Bea aertion that he will have tea (which i according to her knowledge tate in which all indice contain her drinking tea), the indice in which thi i not the cae are removed from all three knowledge tate. Thi mean one of the previouly elected group i choen, and every index outide of thi group i removed from the knowledge tate. Thi reult in the knowledge tate depicted in figure 2.7. After the removal of indice from the tate, it i obviou that Andy now know the exact ituation. After all, there i only one index left in hi knowledge tate. But, ince Bea till doe not know what Andy want to drink, her knowledge tate till contain multiple indice. The common ground, depicting the common knowledge, therefore alo till contain multiple indice. For completene, we will how what happen in tep (c) and (d) of the converation in figure 2.8. Thee tep follow

19 2.3. KNOWLEDGE REPRESENTATION 15 C 1, T 2 C 1, T 2 T 1, T 2 C 1, T 2 T 1, T 2 (a) Knowledge tate of A (b) Knowledge tate of B Figure 2.7: Situation after (b) (c) Common ground along the ame line a (a) and (b). After thee tep, the common ground (and both individual knowledge tate) only have one index left. Thi i, for thi converation, the end, ince every participant know everything there i to know. It i alo poible that the converation end before there i only a ingle index remaining, depending on the goal of the participant. If Andy and Bea only goal wa to have it be known what Bea wanted to drink, the converation could have ended after (b). In thi chapter, we have looked at the definition of upport in inquiitive emantic. We defined poibilitie, propoition, and truth-et, along with other ueful definition, and we have een how they can be ued. Uing our definition of upport, we have alo looked at everal theorem from claical logic, and proved they do not all apply in inquiitive logic. Finally, we have een how inquiitive emantic ue all of thi to model converation. Thi will give u a bai for the following chapter, in which we will take a cloer look at inquiitive logic and at Grice maxim in inquiitive emantic.

20 16 CHAPTER 2. INQUISITIVE SEMANTICS C 1, T 2 C 1, T 2 T 1, T 2 C 1, T 2 T 1, T 2 (a) Knowledge tate of A C 1, T 2 (b) Knowledge tate of B C 1, T 2 (c) Common ground C 1, T 2 (d) Knowledge tate of A (e) Knowledge tate of B (f) Common ground Figure 2.8: Situation after (c) and (d)

21 Chapter 3 Inquiitive logic In thi chapter, we ll look at inquiitive logic, defined a the et of formula valid in inquiitive emantic by Ciardelli and Roelofen (2011). We will look at it dijunctive negative variant, and at the claification of the logic a an intermediate logic. 3.1 Entailment and the dijunction property Inquiitive logic, InqL, i the et of formula that are valid in inquiitive emantic. To properly define thi logic, we ll firt define entailment. We define entailment in term of upport. Definition 5. Entailment A et of formula Θ entail a formula φ if and only if any tate upporting all formula in Θ alo upport φ. (Θ InqL φ) A formula φ i valid iff it i upported by every tate. ( InqL φ) We can ay that two formula are equivalent if they entail each other, or equivalently, if their et of poibilitie, their propoition, are equal. Becaue a formula i valid iff it i upported by every tate, any formula that i true in every tate i in InqL. With thi knowledge, we can ee that InqL ha the dijunction property. If φ ψ i in InqL, then φ ψ i upported by all poible tate, and o by the maximal tate I, containing every poible index ((I) φ ψ). But then, uing our previou definition of upport, we know that (I) φ or (I) ψ. We can then conclude that φ or ψ mut alo be in InqL, and therefore, that InqL ha the dijunction property. 17

22 18 CHAPTER 3. INQUISITIVE LOGIC 3.2 Dijunctive negative variant Now we will look at the dijunctive negative variant of the logic. We will begin by giving the tranlation dnt(φ) of a formula φ. Then we will look at ome example to how that the dijunction of negation really i equivalent to the original formula. Definition 6. Dijunctive negative tranlation 1. dnt(p) = p 2. dnt( ) = 3. dnt(φ ψ) = dnt(φ) dnt(ψ) 4. dnt(φ ψ) = { (φ i ψ j ) 1 i n, 1 j m} where: dnt(φ) = φ 1 φ n dnt(ψ) = ψ 1 ψ m 5. dnt(φ ψ) = k 1,,k n { 1 i n (ψ k i φ i ) 1 k j m} where: dnt(φ) = φ 1 φ n dnt(ψ) = ψ 1 ψ m Now that we have thee definition, we will look at their working. One by one, we will give an example of how the dijunctive negative tranlation compare to the original formula.

23 3.2. DISJUNCTIVE NEGATIVE VARIANT 19 Example 3. dnt(p) p dnt(p) p w : p w dnt(p) p p t : t p t t : t p t : t p t : ( r t : r p r ) t : ( r t : r p) t : r t : r p t : r t : w r : p w It hould be clear that both p and dnt(p) are only true if and only if every ingle index in contain p. Therefore, they are equivalent. Example 4. dnt(φ ψ) p q dnt(p q) p q p or q w : p w or w : q w dnt(p q) dnt(p) dnt(q) dnt(p) or dnt(q) t : r t : w r : p w or t : r t : w r : p w Uing example 3 for the lat tep of the proof we can ee that p q i equivalent to dnt(p q). Here, too, it i clear that both p q and dnt(p q) are true if every ingle index in S contain p or if every ingle index contain q.do note that thi

24 20 CHAPTER 3. INQUISITIVE LOGIC example i relatively imple becaue p and q are atom. However, for thi equivalence it hould not make a difference if they were formula: it would merely make for a more complicated and confuing example. Example 5. dnt(φ ψ) (p q) a dnt((p q) a) (p q) a (p q) and a ( p or q) and a ( w : p w or w : q w) and w : a w dnt((p q) a) ( p a) ( q a) ( p a) or ( q a) ( p a) or ( q a) t : t ( p a) t or t : t ( q a) t t : (t p or t a) t or t : (t q or t a) t t : (t p and t a) or t : (t q and t a) t : (t p and t a ) or t : (t q and t a ) t : ( r t : r p and r t : r a) or t : ( r t : r q and r t : r a) Here, we ued (p q) for φ to how how the definition of φ i ued in the dijunctive negative tranlation. The tranition from the econd lat to the lat line i implified. Ued here in all place where it could be applied i: t φ ( r t : r φ r ) ( r t : r φ) r t : r φ The lat line, a interpreted uing natural language, ay that for every ubet of (with the mallet ubet being an index), at leat one index contain both p and a, or for every ubet at leat one index contain both q and a. Thi i the ame a

25 3.2. DISJUNCTIVE NEGATIVE VARIANT 21 every index in containing both p and a or both q and a. Seeing how thi i alo the truth condition for (p q) a, the two are equivalent. Example 6. dnt(φ ψ) (p q) a dnt((p q) a) (p q) a t : t (p q) t a t : (t p or t q) t a t : t p t a and t q t a t : w t : p w w t : a w and w t : p w w t : a w dnt((p q) a) (( a p) ( a q)) (( a p) ( a q)) t : t (( a p) ( a q)) t t : t (( a p) ( a q)) t : t (( a p) ( a q)) t : ( r t : r (( a p) ( a q)) r ) t : ( r t : r (( a p) ( a q))) t : r t : r (( a p) ( a q)) t : r t : r ( a p) and r ( a q) t : r t : c r : c a c p and c r : c a c q t : r t : c r : c (a ) c (p ) and c r : c (a ) c (q ) t : r t : c r : d c : d a d c : d p and c r : d c : d a d c : d q Thi example i long and omewhat hard to read. The lat line ay that for every ubet (again, the minimal ubet i an index) it ha to be o that if the index doe not contain a, then it alo cannot contain p, and that if an index doe not contain a, it alo cannot contain q. Thi applie to every index in. A we know, that i alo the inverted definition of implication. So a we can ee, (p q) a i equivalent to dnt((p] q) a).

26 22 CHAPTER 3. INQUISITIVE LOGIC The dnt i practical in inquiitive emantic becaue it matche the idea that a propoition i a et of poibilitie. When tranlated to the dijunctive negative form, thee eparate poibilitie are very eay to pot a the alternative in the dijunction. In the ame way, we know that all negative form mut be aertion, ince for any formula φ, φ cannot conit of more than one poibility. For an example of thi, ee figure 3.1. So the dnt can make it eaier to ee what poibilitie p, q q p, q q p p (a) Poibilitie for p q (b) Poibilitie for (p q) Figure 3.1: Difference between poitive and negative formula a formula embodie, which can be ueful for more complex formula. We ve looked at the dijunctive negative tranlation of the language, and een ome example of how it i ued. We will now go on to look at how inquiitive logic i related to other logic. 3.3 InqL and other logic We will now look at InqL poition relative to other logic, tarting with intuitionitic logic. A Ciardelli and Roelofen (2011) decribe, there i a imilarity between inquiitive and intuitionitic logic. Inquiitive logic evaluate the meaning of a formula through tate. Whether a tate upport a formula depend not only on what information i already available, but alo on information that may become available later in time. Take, for example, a quetion, where the poibilitie decribe poible future knowledge. In the ame way, intuitionitic logic evaluate the meaning through a point in a Kripke model. Whether the formula i true depend then not only on that point, but alo on the future acceible point, reembling future knowledge.

27 3.3. INQL AND OTHER LOGICS 23 Ciardelli and Roelofen (2011) prove that inquiitive upport and atifaction in a intuitionitic Kripke model M I coincide, o that for every formula φ and for every non-empty tate : φ M I, φ Becaue of thi, we know that InqL contain the intuitionitic propoitional logic, IPL. We alo know that InqL i itelf contained in claical propoitional logic, CPL. So we know that IP L InqL CP L. We will now look at where exactly InqL fall between thoe two logic. To do o, we introduce the notion intermediate logic. An intermediate logic i defined a a et of formula that contain IPL, doe not contain, and i cloed under both modu ponen and uniform ubtitution. A we ve een earlier, InqL i not cloed under uniform ubtitution. Therefore, it belong to the cla of weak intermediate logic, intermediate logic that aren t cloed under uniform ubtitution. So, if we have a weak intermediate logic L, we can only ay that φ and ψ are equivalent in L if φ ψ L Characterization of InqL We now have enough notion to provide a characteritic decription of inquiitive logic. Lemma 5. Weak intermediate logic L i InqL if: 1. for every formula φ, φ i in L equivalent to dnt(φ) and 2. L ha the dijunction property Proof. Let firt look at the firt of thoe two condition. Let φ be equivalent to dnt(φ) for all formula in weak intermediate logic L. Then if φ L, alo dnt(φ) L. A dnt-formula look like x 1 x k, and becaue InqL ha the dijunction property we know InqL mut contain at leat one of x 1,, x k. IPL and CPL agree on negation, and ince weak intermediate logic are between IPL and CPL, we know that two weak intermediate logic will handle negation in the ame manner. Becaue of thi, we know that becaue x i InqL, x i L. So we know that dnt(φ) L,

28 24 CHAPTER 3. INQUISITIVE LOGIC and therefore InqL L. For the econd part, let aume that L alo ha the dijunction property. We can then ue then ame reaoning a above to how that L InqL. If φ L, then dnt(φ) L, which mean x 1 x k L. If L ha the dijunction property, x i L for ome i. Since all weak intermediate logic treat negation the ame, x i InqL, a well a dnt(φ) and φ. So, L InqL, which mean if the condition are atified, L and InqL are equivalent. We have looked at InqL poition relative to other logic, and ingled out it characteritic feature. We will now go on to look at converational implication uing the converational maxim of Grice (1975) and adapted maxim of Groenendijk and Roelofen (2009)

29 Chapter 4 Converational Implicature In thi chapter we will look at Grice maxim for converational implicature, and how they tranlate to inquiitive emantic. Since inquiitive emantic ha a very pecific way of modelling converation, we will ee that ome adaptation to the maxim have to be made for them to be fully tranlated. Firt we will give a baic overview of Grice maxim of converation. continue with a dicuion of their function in inquiitive emantic. Then we 4.1 Grice maxim In 1975, Grice publihed hi article Logic and Converation (Grice, 1975). In thi work he defined a et of implicated guideline, the maxim, that people follow when participating in converation. To be able to compoe thee maxim, Grice aumed that the purpoe of converation i an effective exchange of information. Though thi i a very narrow pecification, it will erve for our purpoe, ince inquiitive emantic i alo ued to model information exchange. In hi article Grice talk about what he call converational implicature. In human converation, much can be aid without actually telling omeone. To take a an example the text from our introduction, if I were to ak Where doe Mr. Smith live? in normal converation, one would generally take it not jut a a prompt for the litener to give the anwer, but alo a a declaration that I do not know where Mr. Smith live. Grice therefore aid that in converation there alway i ome form of implicature. Hi maxim are guideline that, when followed, would remove thee implication, and in normal peech, we therefore do not alway follow all maxim. 25

30 26 CHAPTER 4. CONVERSATIONAL IMPLICATURE However, the maxim are ueful when trying to be a clear and unambiguou a poible. Grice main principle of converational implicature i called the Cooperative Principle. Thi principle, a defined by Grice (1975), i Make your converational contribution uch a i required, at the tage at which it occur, by the accepted purpoe or direction of the talk exchange in which you are engaged. Thi principle can be divided in everal categorie: Quantity, Quality, Relation and Manner. Thoe categorie are the bai for the actual guideline or maxim. All maxim are formulated auming that every participant goal i an effective exchange of information. The Maxim Definition 7. Maxim of Quantity 1. Make your contribution to the exchange a informative a i required for it purpoe. 2. Make your contribution to the exchange no more informative than i required. The maxim of Quantity make ure that every participant in a converation give a much information a he can, keeping in mind the goal of the converation. In addition, it prevent participant from giving too much information, which may be confuing or irrelevant for the current converational goal. The econd maxim i ometime left out, becaue it i poible to get the ame reult uing the maxim of Relation. Definition 8. Maxim of Quality 1. Do not ay what you believe to be fale. 2. Do not ay what for which you don t have adequate evidence. The maxim of Quality main purpoe i to prevent people from giving information that may not be true. It i obviou how thi would dirupt the converation goal, a untrue information can eventually lead to a contradiction in the knowledge of one or more of the participant. Here epecially the aumption that every participant i after effective information exchange i important. While lying i relatively common in converation, thi i only the cae when different converational goal are to be reached than aumed here.

31 4.1. GRICE S MAXIMS 27 Definition 9. Maxim of Relation 1. Be relevant. The maxim of Relation i the leat well-defined maxim, a the definition of relevance can provide multiple problem and quetion. For our purpoe it i enough to define relevance a the effect it may have on the converation: can it bring the converation cloer to it purpoe, then a contribution i relevant. Thi maxim and the econd maxim of quantity provide more or le the ame guideline: don t ay any more than you know i important, for it may only confue matter. Definition 10. Maxim of Manner 1. Be perpicuou. (a) Avoid obcurity of expreion. (b) Avoid ambiguity. (c) Be brief. (d) Be orderly. (e)... The maxim of Manner ha mot to do with the manner in which participant give their information, making it a clear and precie a poible. Thi maxim a given by Grice i incomplete, a other maxim might be needed to completely cover every way in which a tatement may be unclear, but the ubmaxim given are adequate for mot purpoe. All maxim are needed for maximally effective exchange of information, but not all are alway equally oberved, and not all are equally important. For example, it i generally le important to be a brief a poible than it i to be truthful. Grice (1975) himelf ha uggeted that the maxim of Quality might be the mot important maxim for effective converation. Other maxim could be added, epecially conidering the fact that not every human converation ha the purpoe of information exchange, but ome have the purpoe of being funny or influencing people to act in a pecific way. However, like tated above, the original Gricean maxim will do for our purpoe.

32 28 CHAPTER 4. CONVERSATIONAL IMPLICATURE 4.2 Inquiitive Implicature Now that we have given an overview of Grice maxim, we will continue with their application to inquiitive emantic. In thi analyi we will ue the work of Groenendijk and Roelofen (2009), who compoed converational rule for inquiitive emantic, baed on Grice maxim. Maxim of Quantity The maxim of Quantity ay every contribution hould be a informative a required for it purpoe. In converation, thi mean that the contribution hould add omething toward the converational goal. To achieve thi, a entence hould be either informative or inquiitive, both helping to (potentially) remove indice from the common ground and o increae knowledge. The other ubmaxim of Quantity ay a contribution hould not be more informative than needed. Thi i already covered by the maxim of relation, and therefore, we will not ue it here. Groenendijk and Roelofen (2009) call thi the Significance maxim. Maxim of Quality The maxim of quality ay not to contribute anything that you believe may be fale, or that you know i fale. Thi, in inquiitive emantic, can be een a never contributing a entence that i not informative for the contributor. A omething i only conidered true by a participant if it i true for every index in their own knowledge tate, he only know omething to not be fale if it i contained in all indice. So, informative entence hould only be uttered when every index in one knowledge tate contain that information. Since inquiitive entence never eliminate indice from the common ground, the above doen t apply for thoe entence. Intead, a participant hould not ue an inquiitive entence if the entence i not inquiitive in their own knowledge tate. Thi would mean that the participant could be more helpful by jut providing the informative tatement that i an anwer to their own quetion. Since that would be inefficient, quetion hould only be uttered if it alo add a diviion in the peaker own knowledge tate. Thee rule together form the Sincerity maxim of Groenendijk and Roelofen (2009).

33 4.2. INQUISITIVE IMPLICATURE 29 Maxim of Relation The maxim of relation only ay be relevant. Thi relevance can only be defined in relation to a former propoal or propoition. Following Groenendijk and Roelofen (2009), we will look at it tranlation to inquiitive emantic uing the notion of compliance. In converation each participant hould aim to be a compliant in their anwer a poible. Definition 11. Compliance A propoal φ i in compliance with ψ if: 1. Every poibility in [φ] i the union of a et of poibilitie in [ψ]. 2. Every poibility in [ψ] retricted to φ i contained in a poibility in [φ]. We will look at thee one by one. We will only conider the ituation where ψ, the firt propoal, i a quetion. Thi in t necearily the cae, but, a in natural converation, a relevant anwer to an aertion will uually be either agreement or rejection. After thi, there i no pecial need for compliance. Therefore, we will motly conider inquiitive propoal. The firt rule ay that every poibility in a propoal hould conit of one or more poibilitie in the propoal it i a reaction to. Becaue of thi rule, a propoal cannot add poibilitie. Becaue it may contain multiple of the poibilitie of it predeceor, it can be both an informative anwer, eliminating indice, or another quetion propoing an eaier to olve problem than the original quetion. Thi i an intuitive way to define compliance, ince in natural human converation thee rule are uually followed. To clarify, here i an example. Example 7. Let look at the earlier example of Andy and Bea having a drink. Imagine there being a third peron, Cate, who know abolutely nothing yet about Andy and Bea beverage preference. Cate tart the converation aking Do both of you want coffee?. Thi i a relatively complex quetion to anwer, eeing a the reponder ha to know both whether Andy want coffee and whether Bea doe. So now Andy ha multiple option. He can anwer that he would like a coffee, thereby eliminating all indice in which he d want tea from the knowledge tate. Or he can make the quetion le precie, to get an anwer from Bea, which could help them anwer the full quetion. In that cae, he could ay omething like Well, I don t know. Bea, do you want a coffee? Thi would merge the four poibilitie into jut two: thoe in which Bea drink coffee and thoe in which he drink tea. Anwering

34 30 CHAPTER 4. CONVERSATIONAL IMPLICATURE in either of thee way would be conidered compliant with Cate quetion, baed on the firt rule. In the econd rule, retriction of χ to φ i defined a the interection of χ and φ. If both propoal are quetion, thi can be implified a every poibility in [ψ] i contained in a poibility in [φ]. Thi rule exit to make ure no poibilitie can be removed without reaon. We can look at our example to ee how thi work. Example 8. Thi time, Cate ak Doe any of you want coffee?. Thi i a quetion of the form?c 1?C 2. Thi mean the propoal contain four ditinct poibilitie, like hown in figure 4.1a. Like in the previou example, we could have Andy anwer I don t know. Bea, do you want coffee? (?C 2 ). Thi would lead to the two poibilitie hown in figure 4.1b. However, according to the econd rule of compliance, thi i not poible. A i clear in the picture, there are two poibilitie in the propoal of figure 4.1a that are not contained within any poibility of the econd propoal. Becaue thee poibilitie are ignored, the econd propoal i not conidered in compliance with the firt. Making the quetion more difficult to anwer i not helpful or relevant in a converation. C 1, T 2 T 1, T 2 C 1, T 2 T 1, T 2 C 1, C 2 T 1, C 2 C 1, C 2 T 1, C 2 (a)?c 1?C 2 (b)?c 2 Figure 4.1: Example for the definition of compliance A we can ee the econd rule prevent the firt rule from being ued too retrictively. In our firt example, the firt rule would make it a poibility for Andy to create le poibilitie, thereby making it more difficult to anwer (maybe not in thi example, but in more difficult quetion thi could be a problem). The econd rule prevent thi by making it mandatory that every poibility in the firt quetion hould be in a poibility in the econd. A lat example (9) will how u a converation in which both rule are applied to ak a econd quetion.

35 4.2. INQUISITIVE IMPLICATURE 31 Example 9. Conider the following quetion. a) What do you want to drink? b) Do you want coffee, tea, or omething ele? Sentence (a) divide everal poibilitie: tea, coffee, water, lemonade, orange juice. Then the following quetion (b) merge ome of thee poibilitie, o that the only one remaining are coffee, tea, and other. Thi other poibility contain all that fall neither within coffee nor within tea. No poibilitie are ignored. The two rule of compliance alo help prevent over-informative anwer, becaue an anwer can never add indice to the knowledge tate. Over-informative anwer need to be avoided, ince the extra information increae the chance of the propoal conflicting with the individual knowledge tate of one of the participant, thereby being rejected. Maxim of Manner Grice maxim of Manner ay to be brief and orderly, and to avoid obcurity and ambiguity, and that more might be needed. Since not all of thee ubmaxim can be tranlated to inquiitive emantic (ambiguity, for example, i difficult to convey when modelling), we will look at ome of the rule introduced by Groenendijk and Roelofen (2009) that have to do with the manner of peech. The firt i the rule that participant hould ay more, and ak le, which could be claified a a rule about obcurity. The purpoe of thi rule i to have the participant reach their common goal fater, ince in general more indice will be eliminated by aertion than by quetion. Thoe elimination enable more pecific quetion to be aked, and o for the converation to reach it goal fater. So, we want participant to be a informative a poible. Participant hould try and maximie elimination of indice. Definition 12. Degree of informativene φ i at leat a informative a ψ iff in every tate where ψ i eliminative, φ i eliminative a well. Another rule hould be to properly update and maintain the knowledge tate. Every individual hould make ure to voice any contradiction of their own individual

36 32 CHAPTER 4. CONVERSATIONAL IMPLICATURE knowledge tate with a propoal, to make ure none can be ued to update the common ground. They hould alo update the common ground every time a propoal i accepted, a well a their individual knowledge tate. If any of the tate become inconitent, it may well be impoible to reach the converation goal. The third rule i that every participant hould trive to enhance the common ground to reach the converational goal. Thi ha much to do with the earlier dicued truthfulne and compliance, but it hould be noted that if participant do not try to enhance the common ground, the converation will likely end before the goal i reached. We have een both Grice maxim and the adaptation for inquiitive emantic. Mot maxim have a clear application in thi emantic, although ome, epecially thoe of manner, remain a bit vague.

37 Chapter 5 Dicuion In the lat chapter we looked at inquiitive emantic. More pecifically, we looked how it can be ued to model converation, and how Grice idea on converational implicature can be applied to it. We alo looked at the accompanying logic and it claification a a weak intermediate logic. In thi lat chapter we will dicu how thi ytem can contribute to further cientific reearch, both in general and in the field of artificial intelligence. 5.1 Inquiitive emantic application in reearch Since emantic can be een a a branch of linguitic, mot ue for inquitive emantic can be found in that field. Like mentioned in the introduction, ome linguit ay that quetion can not be modeled in a intuitive way uing claical emantic. Mot exiting technique for modeling quetion, and by extenion, for modeling human converation, can feel omewhat artificial. Inquiitive emantic can provide a new way to model language. Thi could help provide new inight on the way human language work. One could ay that the claical way of modeling converation i already ufficient. However, the intuitivene of inquiitive emantic would make it more acceible for both laypeople and reearcher from other field, and eaier to ue in other application. Becaue inquiitive emantic partly depend on the notion of compliance, it would be intereting to look at the complexity of determining the compliance of a repone ψ to a quetion φ. An algorithm to compute a et of compliant repone to a quetion ha been decribed by Ciardelli et al. (2009). The algorithm given by them conit mainly of tep to yntactically change the formula, with ome tep for checking 33

38 34 CHAPTER 5. DISCUSSION claical entailment or atifiability. Thi could mean that determining compliability i more complex than determining claical atifiability. A conequence could be that the way of modeling quetion a a conjunction of three tatement, for the time being, i more ueful for practical application. Another iue can be een when conidering open-ended quetion, which would be difficult to formulate in a open world. Only if the et of information i limited openended quetion could be formulated, and a et of poible repone completed. A olution would have to be found for ituation in which the information i not limited. Further reearch could alo focu on extending exiiting emantic to include thing like torytelling, intentional lie, and convincing a litener to do omething. So far emantic ha motly been about truthfulne of a tatement. Inquiitive emantic change thi by placing the focu motly on compliance, and in the ame way, new or extended ytem could be deigned to deal with other ue of language. 5.2 Inquiitive emantic in AI Inquiitive emantic could alo play a role in artificial intelligence reearch and application. One of the main iue in AI i one of language: the iue of how to get a computer to ue language like a human would. Reearch in emantic in general ha already made great progre in term of language recognition and generation. A famou example would be Siri, Apple virtual aitant application, that i able to recognize the uer quetion and anwer uing proper Englih. Semantic model are ued in many application to help the computer analyze natural language input for meaning, and o make human-computer interaction more intuitive. Since mot human-computer interaction conit of quetion and anwer, inquiitive emantic would be a logical choice to help expand the current poibilitie in natural language recognition. The way it can be ued to model quetion and tatement could provide the bai for more natural ue of language by computer. In uch a program, yntax analyi could firt recognize input a being a quetion. Then a model baed on inquiitive emantic could ue information from, for example, the internet, and combined with the right algorithm, pick the mot compliant repone to give to the uer. However, like we explained in the ection above, for now it i hard to imagine how any algorithm would work with an almot limitle ource of information like the internet; the algorithm would have to be limited to a cloedinformation world, like a helpdek for a commpany or a library databae earch.