Pragmatic Meaning and Non-monotonic Reasoning:

The Case of Exhaustive Interpretation

Robert van Rooij and Katrin Schulz†

Abstract. In this paper an approach to the exhaustive interpretation of answersis developed. It builds on a proposal brought forward by Groenendijk & Stokhof(1984). We will use the close connection between their approach and McCarthy’s(1980, 1986) predicate circumscription to describe exhaustive interpretation as aninstance of interpretation in minimal models, well-known from work on counterfac-tuals (see for instance Lewis (1973)). It is shown that by combining this approachwith independent developments in semantics/pragmatics one can overcome certainlimitations of Groenenedijk & Stokhof’s (1984) proposal. In the last part of thepaper we will provide a Gricean motivation for exhaustive interpretation buildingon work of van Rooij & Schulz (2004).

Keywords: Conversational Implicatures, Exhaustive Interpretation, Predicate Cir-cumscription, Relevance, Only knowing.

1. Introduction

The central aim of this paper is to find an adequate description ofthe particular way in which we often enrich the semantic meaning ofanswers. To illustrate the phenomenon, consider the following dialogue.

(1) Ann: Who passed the examination?Bob: John and Mary.

In many contexts Bob’s answer is interpreted as exhausting the predi-cate in question, hence, as stating that the set of John and Mary is theset of people who passed the examination. This reading is called theexhaustive interpretation of answers (see e.g. Groenendijk & Stokhof(1984), von Stechow & Zimmermann (1985)) and is what we will studyin this paper.1

But the term ‘exhaustive interpretation’ has not only been used inconnection with the interpretation of answers. Aspects of the mean-ing2 of sentences containing ‘only’ (compare ‘Only John and Marypassed the examination’), cleft constructions (‘It were John and Mary

† The names are listed in alphabetical order.1 As will become clearer in section 2, we will treat this reading as only a special

case of exhaustive interpretation.2 In this paper ‘meaning’ will be used as referring to the sum of semantic and

pragmatic meaning.

c© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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2 Robert van Rooij and Katrin Schulz

who passed the examination’) and free intonational focus (‘[John andMary]F passed the examination’), for instance, have been characterizedin this way as well. In this paper, however, we will limit ourselves to adescription of the exhaustive interpretation of answers. We will discusssemantic analyses of these other constructions only insofar they haveto deal with problems that arise with the exhaustive interpretation ofanswers as well.

In their dissertation from 1984, Groenendijk & Stokhof proposed avery promising approach to the exhaustive interpretation of answers.We will introduce this approach in section 3 and discuss its merits.However, Groenendijk & Stokhof’s (1984) description of exhaustiveinterpretation also has to face certain shortcomings. The main goalof the remaining sections is then to provide the necessary changes andadaptations to overcome these limitations.

In section 4 we will discuss the close relation between Groenendijk& Stokhof’s (1984) approach and McCarthy’s (1980, 1986) theory ofpredicate circumscription, one of the first formalisms of non-monotoniclogic.3 We will then switch to a description of exhaustive interpretationin terms of predicate circumscription and show that this already allowsus to address some of the problems Groenendijk & Stokhof (1984) hadto face.

In section 5 another modification is added: we will combine predicatecircumscription with dynamic semantics. Given the developments insemantics during the last 20 years, this is an alteration of the originalstatic approach of Groenendijk & Stokhof (1984) that would have beennecessary anyway. It will turn out that it solves some problems, alreadydiscussed by Groenendijk & Stokhof (1984) themselves, concerning, forinstance, the interaction of exhaustive interpretation and the semanticsof determiners.

In section 6 we will address the strong context-dependence of exhaus-tive interpretation. As we will see in section 2, exhaustive interpretationcan come in other forms than the reading we discussed for example(1). We will argue that this should be explained by taking a contextualparameter of relevance into account.

The question of how to describe the exhaustive interpretation of an-swers arises because classical semantics cannot handle the phenomenon.But if it is not due to semantics, where does exhaustive interpretationcome from? One answer to this question that seems to be particularlyattractive is to explain it using Grice’s (1989) theory of conversationalimplicatures. In section 7 we will sketch a formalization of parts of

3 Van Benthem (1989) was the first to explicitly point out this connection.

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Exhaustive Interpretation 3

Grice’s theory brought forward by van Rooij & Schulz (2004). Theyshow that when combined with a principle of competence maximiza-tion, this formalization can indeed explain exhaustive interpretation. Inthe present paper we will discuss in how far a parallel extension of theapproach we ended up with in section 5 can also be of help to furtherimprove on the descriptive adequacy of Groenendijk & Stokhof’s (1984)approach.

2. The Phenomenon

Before we can start thinking about how to formulate a general andprecise description of the exhaustive interpretation of answers, we firstneed to get a clearer picture of what we actually want to describe.Therefore, this second section is devoted to a closer investigation of theproperties of exhaustive interpretation.

2.1. Interaction with the semantic meaning of the answer

A first thing to notice is that the exhaustive interpretation does notalways completely resolve the question the answer addresses. Con-sider, for instance, example (2) (all sentences discussed in this sectionshould be understood as answers to the question ‘Who passed theexamination?’).

(2) Some female students.

This answer can be interpreted exhaustively as stating that just a fewstudents passed the examination and that they are all female. However,also on this reading the answer does not identify the students thatpassed the examination and therefore does not resolve the question.4

Hence, even though the exhaustive interpretation strengthens the tra-ditional semantic meaning of the answer – and therefore makes themarguably better answers – it does not turn all answers into resolvingones. This point is also nicely illustrated with the following example.

(3) John or Mary.4 This is true, in particular, for the notion of resolving questions introduced by

Groenendijk & Stokhof (1984). According to them, a question is resolved if the ex-tension of the question-predicate is fully specified. One may argue that resolvednessshould also depend on the information the questioner is interested in when askingher question (see Ginzburg (1995), van Rooij (2003), and section 6 of this paper forproposals along these lines). However, even if one modifies the notion of resolvingquestions accordingly, it will turn out that not in all contexts where we observe anexhaustive interpretation of answers like (2) it resolves the question.

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On its exhaustive interpretation this answer states that either onlyJohn or only Mary exhaust the set of people who passed the examina-tion. Again, in most contexts this information will not fully resolve thequestion asked.5 Another point that should be noticed in connectionwith this example is that its exhaustive interpretation is not completelydescribed by taking the exclusive interpretation of ‘or’.6 One wouldmiss the additional inference of the exhaustive reading that no-one elsebesides Mary and John passed the examination.

Careful attention should be payed also to the way exhaustive in-terpretation interacts with the semantics of determiners. Compare, forinstance, (4) and (5).

(4) Three students.

(5) At least three students.

The exhaustive interpretation of (4) allows us to conclude that not morethan three students passed the examination. (5), however, cannot beread in this way.7 So there is a difference between (4) and (5) that ex-haustive interpretation is sensitive to. However, it will not be adequateto propose that ‘at least’ simply cancels an exhaustive interpretation.(5) can give rise to the inference that nobody besides students passedthe examination, and, thus, can show effects of exhaustification.

Just as for (4), also for (6) we will not infer a limitation on thenumber of students that passed the examination, if it is interpretedexhaustively.

(6) Students.

Notice that also here it can be concluded that, besides students, no oneelse passed the examination. Thus, certain effects of exhaustive inter-pretation are present. In contrast to (6), the exhaustive interpretationof (7) implies additionally that not all students passed the examina-tion. So, again, something distinguishes (6) and (7) with respect toexhaustive interpretation.

5 Sometimes, however, a disjunction can be resolving. Consider, for instance, (i)reading Ann’s utterance as a polar question.

(i) Ann: Did Mary or John pass the examination?Bob: Yes, Mary or John passed the examination.

6 Under the exclusive interpretation of ‘or’, ‘A or B’ is true iff one of the disjunctsbut not both is true.

7 We only discuss here ‘at least’ as a modifier of the numeral. The occurrence of‘at least’ in (5) can also be read as particle, with a syntactical behavior similar to‘even’. This use is not discussed in the present paper. Readers who have problemsgetting the exhaustive interpretation for (5) should try ‘John and at least three ofhis friends passed the examination’.

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Exhaustive Interpretation 5

(7) Most students.

How can these observations be explained? We will propose in section5 that exhaustivity is sensible to certain properties of the determiner,namely whether it is weak or strong – i.e., whether it directly intro-duces a discourse referent or whether the discourse referent has to beconstructed afterwards – and this leads to the different interpretations.

2.2. The Context-dependence of Exhaustivity

The examples discussed above show how exhaustive inferences changedepending on the answer given. Interestingly enough, even the sameanswer (with the same question form) can in different contexts giverise to different exhaustive interpretations. First of all, it seems thatsome answers should not be interpreted exhaustively at all. A typicalexample is the dialogue given in (8).

(8) Ann: Who has a light?Bob: John.

Here, Bob’s answer is normally not understood as ‘John is the onlyone who has a light’. Instead, it seems that no information on top of itssemantic meaning is conveyed. We call this interpretation of answers themention-some reading, while we will refer to the one discussed in section2.1 as the mention-all reading. It appears that mention-some readingsoccur precisely in those contexts where the questioner is intuitively notinterested in the exact specification of predicate P and the semanticmeaning of the answer already provides her with all the informationshe needs.8

Except for the mention-all and mention-some readings, there also seemto be situations with intermediate exhaustive interpretations. In thesecases some of the typical inferences of mention-all readings are allowedbut not all of them.

Perhaps the best known limitation is one of domain restriction.There are contexts in which an answer to a question with question-

8 It is important to distinguish mention-some readings from the interpretationan answer gets, for instance, if the speaker uses a special (rising) intonation or adds‘as far as I know’. In contrast to mention-some readings, in the latter cases theinformation one receives is not exhausted by the semantic meaning of the answer.Instead it is additionally inferred that the information given in the answer exhauststhe knowledge of the speaker. Of course, this reading can also occur if the questioneris interested in a full specification of the question-predicate. See section 7 for morediscussion.

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predicate P specifies those and only those individuals that have prop-erty P - but only for a subset of all objects to which P may apply.Imagine Mr. Smith asking one of his employees:

(9) Mr. Smith: Who were at the meeting yesterday?Employee: John and Mary.

There is a reading of this answer implying that John and Mary are theonly employees of Mr. Smith who were at the meeting yesterday. Theremay have been others besides employees of Mr. Smith, but nothing isinferred about them. For the choice of interpretation it seems to berelevant, again, what is commonly known about the information Mr.Smith is interested in. Suppose, for instance, that it is mutually knownthat Mr. Smith would like to know whether one of his rivals from othercompanies was at this meeting. Then one would infer from (9) that Johnand Mary are the only rivals of Mr. Smith who were at the meetingyesterday.

Exhaustive interpretation is limited in other ways in so-called scalarreadings of answers (cf. Hirschberg, 1985). As in the example above,also here exhaustivity seems to apply only to parts of the question-predicate. Imagine Ann and Bob playing poker.

(10) Ann: What cards did you have?Bob: Two aces.

Here, Ann will interpret Bob’s answer as saying that he did not havethree aces or some additional two kings (a double pair wins over a singleone). Still, the answer intuitively leaves open the possibility that Bobadditionally had, for example, a seven, a nine, and the king of spades.Just as in the previous case, also here Ann’s interest for information isdifferent from the case in which an answer gets a mention-all reading.She is not interested in the exact cards that Bob had. She wants toknow, however, how good (with respect to an ordering relation inducedby the poker rules) Bob’s cards were. And the scalar reading makesBob’s answer complete in this respect.

To give a final example of a context where the force of exhaustiveinterpretation seems to change depending on relevance, consider (11).

(11) Ann: How far can you jump?Bob: Five meters.

It illustrates how – depending on the needs of the questioner – exhaus-tive interpretation can perceive the domain of the question-predicatewith different degrees of fine-grainedness. If it is commonly known that

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Exhaustive Interpretation 7

Ann wants to have precise information about Bob’s jumping capacities,the exhaustive interpretation of his answer will imply that he cannotjump a centimeter further than 5 meters. If, however, a rough indicationis sufficient, one may infer just that he cannot jump 6 meters far.

In this subsection we have discussed some examples where an answerdoes not obtain the strong interpretation that is traditionally associatedwith the name ‘exhaustive interpretation’. Sometimes only parts of themention-all reading were observed, sometimes nothing was added tothe semantic meaning of the answer at all. But in all cases it seems tobe relevance that is the contextual parameter on which the strength ofthe exhaustive interpretation depends. If the questioner is known notto be interested in certain information, then it will not be provided bythe exhaustive interpretation of the answer. For instance, in (8) it isclear that for the questioner it is sufficient to know of somebody whohas a light that she has a light. This interest is fully satisfied with thesemantic meaning of the answer given by Bob. We will take this obser-vation seriously and describe in section 6 exhaustive interpretation asdepending on the information the questioner is interested in. It will turnout that in this way we can account for domain restrictions, granularityeffects, scalar readings, as well as the mention-some interpretation.

2.3. Other types of questions

The examples discussed so far all were answers to some wh-questionwhere the question-predicate is of type 〈s, 〈e, t〉〉. The exhaustive inter-pretation is, however, not restricted to this class of answers. Thereare questions of other types whose answers also seem to show ex-haustiveness effects. For instance, there is a well-known tendency tointerpret conditional answers to polar questions, exemplified in (12), asbi-conditional.

(12) Ann: Will Mary win?Bob: Yes, if John doesn’t realize that she is bluffing.

Thus, one infers that Mary will win exactly in case John will not realizethat she is bluffing. Intuitively, in this reading the same is going on asin the cases of exhaustive interpretation discussed so far: the worldswhere Mary will win are taken to be exhausted by those where theantecedent of Bob’s answer is true. Therefore, it seems reasonable toexpect that a convincing approach to exhaustive interpretation shouldbe able to deal with this observation as well.

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There are various approaches to exhaustive interpretation around.However, the domain of application differs remarkably from theory totheory. As far as we know, none of the existing approaches can ac-count for all the observations discussed above. Moreover, none of thesetheories gives a satisfying explanation for why the scope of exhaustiveinterpretation should be restricted to those cases that they can actuallyhandle.

In this paper a unified approach to the exhaustive interpretation ofanswers is presented which is able to deal with the whole list of examplesdiscussed so far. This account essentially builds on a description ofexhaustive interpretation proposed by Groenendijk & Stokhof (1984)(abbreviated by G&S). We will therefore start by discussing their work.

3. Groenendijk and Stokhof’s proposal

Groenendijk & Stokhof (1984) propose to describe the exhaustive in-terpretation of answers to ‘Who’-questions by the following operationexhGS taking as arguments the generalized quantifier T denoted by theterm answer and the question-predicate P :9

DEFINITION 1. (The exhaustivity operator of Groenendijk & Stokhof)

exhGS =def λTλPλw.T (w)(P ) ∧ ¬∃P ′ : T (w)(P ′) ∧ P ′(w) ⊂ P (w)

Set-theoretically, the above formula applied to a generalized quan-tifier and a predicate allows the predicate only to select the minimalelements of the generalized quantifier. To illustrate, assume that Bob’sresponse to Ann’s question ‘Who knows the answer?’ is ‘John’. Ana-lyzed as a general quantifier ‘John’ denotes λwλP.P (w)(j), which istrue of some predicate P if in every world P denotes a set containing j.Applying exhGS to this function turns it into a generalized quantifierthat is true of P if in every world it denotes the minimal set containingj, which is the set {j}. Thus, it is correctly predicted that by exhaustiveinterpretation we can conclude from the answer ‘John’ that {j} is theset of individuals that passed the examination.

Reading term answers as generalized quantifiers in combination withthe exhaustivity operation defined above, allows us to account for theinterpretation effects observed in examples as (1), (2), (3), (4) and (7)discussed in sections 1 and 2. Actually, G&S can do even more. They

9 As mentioned in G&S (1984), this operator has much in common with Sz-abolcsi’s (1981) interpretation rule for ‘only’. Though similar in content, G&S’sversion provides the more transparent formulation.

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Exhaustive Interpretation 9

show that the above stated operator for terms can be generalized easilyto multiple terms and sentential answers.10

Although these results are very appealing, G&S’s exhaustivity operatorstill has been criticized. For instance, Bonomi & Casalegno (1993) haveargued that G&S’s analysis is rather limited because it can be appliedonly to noun phrases. To account for examples in which ‘only’11 asso-ciates with expressions of another category, they argue that we have tomake use of events. We acknowledge that the use of (something like)events might, in the end, be forced upon us. But perhaps not exactly forthe reason they suggest. Crucial for G&S’s analysis is that (ignoring theintensional parameter) their exhaustivity operator is applied to objectsof type 〈〈φ, t〉, t〉. It is normally assumed that noun phrases denotegeneralized quantifiers of type 〈〈e, t〉, t〉, which means that denotationsof noun phrases are in the range of the exhaustivity operator. However,it is also standardly assumed that an expression of any type φ can belifted to an expression of type 〈〈φ, t〉, t〉 without change of meaning. Butthis means that – after type-lifting – G&S’s exhaustivity operator canbe applied to the denotation of expressions of any type, and there isno special need for events. The only restriction is that one always hasto assume that the answer-form provides a semantic object of the type〈〈φ, t〉, t〉 to make exhaustive interpretation work.12

Although Bonomi & Casalegno’s (1993) criticism does not seem toapply, G&S’s analysis faces some other limitations. First, it is quiteobvious (and has been noticed by themselves) that they cannot accountfor mention-some readings, domain restriction, granularity effects, andscalar readings (see section 2.2). This is inevitable given the function-ality of exhGS . By taking only the predicate of the question and thesemantic meaning of the term in the answer as arguments, exhGS isnot rigid enough to account for differences that can occur involvingthe same question-predicate and the same answer. The limited func-tionality of the operation exhGS seems to be responsible also for other

10 Their general exhaustivity operator for n-ary terms looks as follows:exhn

GS = λTnλPnλw.Tn(w)(Pn) ∧ ¬∃P ′n : Tn(w)(P ′

n) ∧ P ′n(w) ⊂ Pn(w).

11 They discuss exhGS as a description of the semantic meaning of ‘only’, but,of course, this problem occurs with the same force in connection with exhaustifiedanswers.

12 The reason that we still might need (something like) events is that extendingexhGS to answers of questions as ‘What did you do last summer?’ is not at alltrivial. It seems that a possible-world approach provides not enough fine-structureto properly describe the ‘things’ one did last summer, and making use of events maybe one way to achieve this required fine-grainedness. But this is not a problem ofG&S’s approach to exhaustive interpretation but rather for the general conceptionof meaning in which this proposal is situated.

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problems of the approach (called ‘the functionality problem’ by Bonomi& Casalegno (1993)). Because G&S assign to (4) ‘Three students’ and(5) ‘At least three students’ the same meaning, exhGS predicts for thesepairs of answers the same exhaustive interpretation. However, as dis-cussed in section 2.1, intuitively the interpretations differ.13 Somethingsimilar happens with answers like (3) ‘John or Mary’ and (13)

(13) John or Mary or both.

Standard semantics takes both answers to be equivalent, but their ex-haustive interpretation differs. While (3) implies that not both, Johnand Mary, passed the examination, this is not true for (13). ExhGS

predicts the exclusive reading in both cases.The next problem (discussed in G&S, pp. 416-417) concerns the

way exhGS operates on its arguments. Assume that W is the class ofall models (possible worlds) for our language and let [φ] denote theintensional semantic meaning of expression φ. Hence, [φ] is a functionmapping elements w of W on the extension of φ in w (in case φ is asentence we use [φ]W to denote the set of models in W where φ is true).If we allow for group objects, interpret [‘passed the examination’] as adistributive predicate14 and [‘John and Mary’] in (14) as the quantifierλw.λP.P (w)(j⊕m), then the exhGS-reading of (14) claims [‘passed theexamination’] to denote the set {j ⊕m}.

(14) Ann: Who passed the examination?Bob: John and Mary.

Because [‘passed the examination’] is distributive, this cannot be ful-filled in any world: there can be no model w of the language where j⊕m ∈ [‘passed the examination’](w) but j 6∈ [‘passed the examination’](w).Hence, when applying exhGS to (14) Bob’s answer is interpreted asthe absurd proposition. This is inadequate given that (14) can beinterpreted straightforwardly in an exhaustive way.15

13 In both cases the application of exhGS implies that not more than three studentspassed the examination. This problem has also been noted by G&S themselves.

14 A predicate P with domain D is distributive in a set of models W if for allw ∈ W , (∀x, y ∈ D)([P ](w)(x) ∧ [P ](w)(y) ↔ [P ](w)(x⊕ y)).

15 There is a solution to this problem, already sketched by G&S, ibid. For inde-pendent reasons one is driven to allow the interpreter to choose freely between adistributive and non-distributive reading for predication to plural objects. If oneadditionally assumes that distributive predicates allow only for the second reading,(14) is interpreted as ∀x ≤ j ⊕m : P (x). Minimization of P relative to this answerdoes not give rise to complications - even without respecting meaning postulates.Later on (section 4.2) we will propose another solution. It has the advantage to carryover to another kind of problem that the answer sketched here cannot capture.

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Exhaustive Interpretation 11

Finally, negation is a problem for exhGS . Apply, for instance, thisoperation to Bob’s answer in (15).

(15) Ann: Who passed the examination?Bob: Not John.

Then Bob’s response is interpreted as implying that nobody passed theexamination: the smallest extension of predicate ‘passed the examina-tion’ such that the answer is true is the empty set. This is clearly nota possible reading for this answer.

The aim of this paper is to overcome the problems discussed above.We claim that this can be done without radically changing the basicidea behind G&S’s exhaustivity operator.

What do we understand this basic idea to be? According to G&S,to interpret an answer exhaustively means to minimize the question-predicate of the answer: from the fact that the answerer did not claimthat a certain object has property P it is inferred that the object doesnot have property P . Thus, the hearer makes the absence of informationmeaningful. She interprets it as negation. This we take to be an essentialinsight behind the approach of G&S.

However, G&S were not aware of the fact that this reasoning pat-tern was starting to get a lot of attention in artificial intelligence aswell, but now under the heading of negation as failure. It lead (amongother things) to the development of a whole new branch of logic: non-monotonic logic. When we now try to improve on the proposal of G&Swe can build on the work done in this area.16

16 A question one often hears in this context is ‘Do we really need non-monotoniclogic?’. Indeed, we do. Non-monotonicity is simply a property of exhaustiveinterpretation. Therefore, no matter how one describes exhaustive interpretation, itwill also be a property of the description. Recall that reasoning is non-monotone ifcertain inferences might be given up under the presence of more information. It iseasy to see that this holds for exhaustive interpretation. From the answer ‘John’ wecan conclude that Mary did not pass the examination. This inference is lost whenthe speaker also tells us that Mary passed.

(i) Ann: Who passed the examination?Bob: John and Mary.

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4. Exhaustivity as Predicate Circumscription

4.1. Predicate Circumscription

Only a few years before Groenendijk and Stokhof came up with theirdescription of exhaustive interpretation, McCarthy impressed the artifi-cial intelligence community by introducing Predicate Circumscription,one of the first formalisms of non-monotonic logic. McCarthy’s goalwas to formalize common sense reasoning. More specifically, PredicateCircumscription was intended to solve the qualification problem: if wewould use classical logic to derive every-day conclusions, we would needan “impracticable and implausible” (McCarthy, 1980, p. 145) number ofqualifications in the premises. For instance, if one wants to predict thatif we would throw our computers out of our windows, they would smashon Nieuwe Doelenstraat, one would have to specify that gravitationwill not stop working, the computers will not develop wings and flyaway etc. - in short: nothing extraordinary will happen. The solutionMcCarthy proposes is to strengthen the inferences one can draw froma theory by adding to the premises a normality assumption. It saysthat nothing abnormal is the case that is not explicitly mentioned inthe theory. Or, to restate it somewhat more abstractly, the extensionof certain predicates (the abnormality predicates) is restricted to thoseand only those objects that are explicitly stated by the premises tobe in the extension. To come back to the example above, if there isno explicit information about abnormalities in the gravitation of theearth the normality assumption adds the premise that the gravitationis working as normal. Thus, abnormality is negated as failure.

McCarthy (1986) formalizes this idea17 by defining a syntactic op-eration on a sentence (the premise) that maps it to a new second-ordersentence (the premise plus the normality assumption) in the followingway.

DEFINITION 2. (Predicate Circumscription)Let A be a second-order formula and P a predicate of some languageL of predicate logic. Then the circumscription of P relative to A is theformula Circ(A,P ) defined as:

Circ(A,P ) := A∧¬∃P ′(A[P ′/P ]∧∀x(P ′(x) → P (x))∧∃x(P (x)∧¬P ′(x))).

where A[P ′/P ] describes the substitution of all free occurrences of P inA by P ′.

17 This is a simplified version of his formalization.

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Exhaustive Interpretation 13

Looking at this formalization of Predicate Circumscription, our readerwill immediately recognize the following striking fact: G&S’s exhaus-tivity operation is – roughly speaking – just an instantiation of Mc-Carthy’s predicate circumscription! The circumscribed predicate is nowthe question-predicate, and the circumscription is relative to the propo-sition one gets by applying the generalized quantifier denoted by theterm answer to this predicate - or simply the proposition denoted bythe answer. This for our purposes so important parallelism has beennoticed first, as far as we know, by Johan van Benthem (1989).18

Predicate circumscription has a model-theoretic pendant: interpreta-tion in minimal models. Here, the semantics for classical logic is en-riched by defining an order on the set of models W : a model v is saidto be more minimal than a model w with respect to some predicateP , v <P w, in case they agree on everything except the interpretationthey assign to P and there it holds that [P ](v) ⊂ [P ](w). It can beshown that if W is the class of all models the P -minimal models of atheory A, hence the set {w ∈ [A]W |¬∃v ∈ [A]W : v <P w}, are exactlythe models where the circumscription formula Circ(A,P ) holds.19

This formulation of predicate circumscription – as interpretation inminimal models – is not a stranger to linguists. The Lewis/Stalnakerapproach to counterfactuals (see, for instance, Lewis (1973)) also makes

18 There are certain differences between exhGS and Circ that should be men-tioned. First, G&S took exhGS to be a description of an operation on semanticrepresentations while Circ(A, P ) is an expression in the object language. Second,Circ takes as arguments a predicate and a sentence, while exhGS applies to thepredicate and the sentence without the predicate. Circ, therefore, relies on lesssyntactic information. But, as Ede Zimmermann pointed out to us, it looks as if thereare cases where exhaustive interpretation relies on this information. Consider, forinstance, the answer ‘Men that wear a hat’ to a question ‘Who wears a hat?’, wherethe question-predicate P appears in the term answer part T . The circumscriptionof A = T (P ) w.r.t. P minimizes P in all occurrences of A and interprets the answeras implying that nobody wears a hat – which is obviously wrong. ExhGS onlyminimizes occurrences of P outside T and correctly predicts that exactly thosepeople wear a hat that are men that wear a hat. In this paper we will assume thatthe question-predicate does not occur in the term answer part.

19 This set of minimal models can be described relative to a set of alternativesof A, Alt(A), as well. If we say that v <Alt(A) w if and only if v is exactly like wexcept that {B ∈ Alt(A)|v ∈ [B]W } ⊂ {B ∈ Alt(A)|w ∈ [B]W }, we can define thefollowing set of minimal models: {w ∈ [A]W |¬∃v ∈ [A]W : v <Alt(A) w}. If P can bethought of as the background of A, this set is the same as the one described in themain text if we define Alt(A) as follows: {P (a)|d ∈ D & a names d}, and assumethat every individual has a (unique) name. This perspective clarifies the connection(and difference) between circumscription and alternative-semantics approaches tothe meaning of ‘only’, as the one of Rooth (1996).

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14 Robert van Rooij and Katrin Schulz

use of it. The application at hand differs mainly in the way the orderis defined.

4.2. The basic setting

It is this later, model-theoretic formulation that we will use to describeexhaustive interpretation. Here comes our basic definition.

DEFINITION 3. (Exhaustive interpretation - the basic case)Let A be an answer given to a question with question-predicate P incontext W . We define the exhaustive interpretation exhW

std(A,P ) of Awith respect to P and W as follows:

exhWstd(A,P ) ≡ {w ∈ [A]W |¬∃v ∈ [A]W : v <P w}

To illustrate the working of this interpretation function, let us goback to example (12) here repeated as (16).

(16) Ann: Will Mary win?Bob: Yes, if John doesn’t realize that she is bluffing.

In this case the question-predicate P = ‘Mary will win’ is of arity0.20 But this means that v <P w iff v is exactly like w, except thatwhereas w makes P true, v makes it false. Now it can be checked thatexhW

std(A → P, P ) is true only in those worlds where either both A andP are true, or both A and P are false. Worlds where A is false and Ptrue are ruled out because there are other worlds that verify A → P ,but do not make P true (worlds where both A and P are false) and,hence, are more minimal. The possibility that A is true and P is falseis excluded by the semantic meaning of the answer. Thus, by applyingexhstd the conditional answer gets a bi-conditional reading.21,22

The change from G&S’s approach to the one given in definition 3 israther subtle – mainly one of perspective. But, as we will see in the rest

20 We assume that the extension of an n-ary predicate P n is the set of n-ary tuplesthat verifies sentence P n(~x). If P 0 is true in w, it denotes {〈〉}, otherwise ∅.

21 This prediction is, of course, already made by G&S.22 As already suggested earlier, the meaning of exhaustive interpretation is closely

related with the meaning of ‘only’. However, it is standardly assumed that a sen-tence like ‘Only [John]F is sick’ does not semantically entail that John is sick. Tous this suggests that with respect to background property P , ‘Only A’ should beinterpreted as {w ∈ W |∃v ∈ exhW

std(A, P ) : w ≤P v}. It is interesting to observethat with respect to background 0-ary property P , the conditional ‘P, only if A’ isnow predicted to have the same meaning as the conditional ‘A, if P’. This approach,worked out in van Rooij & Schulz (to appear), is closely related to McCawley’s(1993) analysis of the meaning of ‘only’.

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Exhaustive Interpretation 15

of the paper, this model-theoretic description of exhaustive interpreta-tion proves to easily allow the amendments we have to make to dealwith the limitations of G&S’s approach. It also allows us to improveon exhGS directly. Remember our earlier discussion of applying therule of exhaustive interpretation to distributive predicates (enrichingthe domain with group objects). We discussed it using our very firstexample, repeated here as (17).

(17) Ann: Who passed the examination?Bob: John and Mary.

Let us calculate once more the exhaustive interpretation of Bob’sanswer, but now using exhstd. Again, Bob is understood as talkingabout a plural object j ⊕ m. To determine exhW

std(P (j ⊕ m), P ) wefirst eliminate all worlds where Bob’s answer is false. Then, we selectthose worlds where the question-predicate P is minimal. At first onemay think that these are the worlds where the extension of ‘passedthe examination’ consists only of j ⊕m. However, such worlds do notexist. The predicate is distributive and already G&S account for this byletting meaning postulates impose restrictions on the class of propermodels. Hence, there will only be worlds in W where the question-predicate is distributive, and the same is true for the minimal models.But then we obtain the right result! The minimal models are such thatbesides the plural object j ⊕ m also j and m are in the extension ofquestion-predicate ‘passed the examination’.

But why can we solve this problem just by taking exhstd instead ofexhGS? Did we not claim above that exhGS(T, P ) is roughly the sameas Circ(T (P ), P ) and that the latter is equivalent to exhW

std(T (P ), P )?Well, one has to be careful. Remember that the latter equivalence onlyholds if W is the class of all models. Meaning postulates impose restric-tions on W . Exhstd is sensible to this information because it quantifiesover this set. ExhGS , however, quantifies locally over alternative ex-tensions for the question-predicate. It does not check whether thesealternative are realized in some world. Only if the meaning postulatesare taken to be part of the answer, exhGS and Circ are forced torespect them and predict correctly.

To sum up, distributive predicates show that circumscribing just theanswer may not be enough. The exhaustive interpretation is sensibleto information available in the context set W , in particular to mean-ing postulates. Because exhstd quantifies over alternative worlds it canimmediately account for this dependence.23

23 The variable W makes our interpretation function very context dependent. If Wis understood as the respective common ground then all the information presented

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16 Robert van Rooij and Katrin Schulz

Actually, there are even more striking examples in favor of a notionof exhaustivity which respects meaning postulates and they do not relyon such particular premises as the group analysis of Bob’s answer in(17). The proposed formalization allows us also to account for somepuzzles connected with the meaning of ‘only’. For limitations of space,however, we cannot discuss this issue in detail here.24

5. Exhaustivity and dynamic semantics

Another problem of G&S’s approach that we discussed in section 2.1is that it makes incorrect predictions for answers like (5) and (6), hererepeated as (18b) and (18c).

(18) (a) Three Students.(b) At least three students.(c) Students.(d) Most students.

As we pointed out earlier, it is standardly assumed in generalizedquantifier theory (adopted by G&S) that ‘three students’ has the samesemantic meaning as ‘at least three students’. Because the operationexhGS (the same is true for exhstd) takes only the semantic meaning of

there will influence what counts as a minimal model in a particular context. It stillhas to be seen to what extend exhaustive interpretation is context sensitive in thissense. See also the discussion in section 7.

24 One particularly famous example that this approach can account for is thefollowing from Kratzer (1989).

(i) Bob: I only [painted a still-life]F .(ii) Lunatic: No. You also [painted apples]F .

The problem is, so to say, to account for the diagnosis: what is wrong with theresponse of the lunatic. We just want to note here that given a meaning postulatestating that if one paints a picture, then one also paints all of its parts, and assuminga semantic meaning of ‘only’ as proposed in footnote 22, we can straightforwardlyaccount for the inappropriatenes of the lunatic’s reaction. In making the set of eventsin which Bob participated yesterday as small as possible, the possibility that Bobalso cooked a dinner for Ann will be excluded - because there would be a smalleralternative history of what Bob did yesterday where no cooking took place and thatstill would make ‘Bob painted a still life’ true. But minimization will not be ableto exclude worlds where Bob painted a still-life but not all of its parts - because noalternative world will model such a case.

Notice, that even after exhaustive interpretation there may be different possibili-ties of what the parts of the still-life were – Bob did not say anything about what hepainted. Hence, it may be the case that Bob actually painted a still-life containingapples. Therefore, the lunatic cannot claim that Bob made a false statement becausehe did not mention the apples he painted.

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Exhaustive Interpretation 17

the answer and the question-predicate into account, it predicts for (18a)and (18b) the same readings. However, the exhaustive interpretation ofthe first answer gives rise, intuitively, to an at most inference, while theexhaustive interpretation of the latter does not. Something similar hasbeen observed comparing (18c) and (18d). Thus, there is a differencebetween these answers exhaustive interpretation is sensible to whichexhGS (and exhstd as well) fails to observe.

Different perspectives are possible on this dilemma. An interestingproposal is made by Zeevat (1994), who incorporates the ‘at most’inference (18d) comes with in the semantics of ‘most’. In this paper,however, we stick to the traditional analysis of this determiner.25 Othershave proposed that expressions containing ‘at least’ or bare nominalsshould not be interpreted exhaustively. However, as observed in sec-tion 2.1, also for these expressions we observe some exhaustivity effect.Hence, general cancellation is no option. We will propose instead thatexhaustive interpretation does take place but that it will not give riseto the at most inference.

There is a semantic distinction that can be made responsible for thedifferences in the exhaustive meanings: the distinction between weakand strong determiners. NPs with weak determiners such as ‘a man’,‘linguists’, ‘some1 girls’, ‘five girls’ or ‘at least five girls’ are not affectedby minimization, while NPs with strong determiners such as ‘all ducks’,‘most students’, and ‘some2 girls’ are. Adopting a standard assumptionof dynamic semantics (e.g. Kamp & Reyle, 1993), we treat only thelatter type of NPs as two-place generalized quantifiers. Weak quan-tifiers, instead, are not and directly introduce discourse referents. Foranaphoric reference to strong quantifiers, discourse referents have to beconstructed afterwards from the intersection of nucleus and restrictor.It turns out that if we take a dynamic perspective on semantics (hence,define semantic meaning as context change potential) and adopt thistreatment of weak and strong quantifiers we immediately can accountfor the differences in the exhaustive interpretations.

We will not introduce full-blooded dynamic semantics but restrictourselves to some of its essential features, leaving the exact implemen-tation to the reader’s favorite dynamic theory. We assume a dynamicinterpretation function that maps an information state σ and a sentenceφ to the new information state σ[φ]. An information state is a set ofpossibilities, i.e., a set of world-assigment pairs. Discourse referents areinterpreted as fixed variables of the assignments. The definition of theorder <P comparing the extension of the question-predicate is extendedto the case of possibilities by adding the condition that now also the

25 Still, our analysis will have some similarity with Zeevat’s proposal.

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18 Robert van Rooij and Katrin Schulz

assignments have to be identical to make possibilities comparable.26

Dynamic exhaustive interpretation is then defined as a context changefunction that selects minimal possibilities instead of worlds.

DEFINITION 4. (Dynamic Exhaustive Interpretation)Let A be an answer given to a question with question-predicate P incontext σ. We define the exhaustive interpretation exhσ

dyn(A,P ) of Awith respect to P and σ as follows:

exhσdyn(A,P ) ≡ {i ∈ σ[A]|¬∃i′ ∈ σ[A] : i′ <P i}.

How does this straightforward extension of exhstd to dynamic se-mantics solve the problems discussed above? The crucial point is thatthe interpretation of newly introduced discourse referents is alreadyfixed in the compared possibilities. Therefore, their denotation can-not be minimized with the extension of the question-predicate. Thevariable over which the generalized quantifier of a strong determinerquantifies is not fixed in the possibilities and therefore may be affectedby the minimization of the question-predicate. This can then lead tothe derivation of upper boundaries.

This will become much clearer after discussing some examples. Takefirst (18c), ‘Students’. Assume the following information state: σ ={i1, i2, ..., i8}, where ik = 〈wk, gk〉. In all possibilities we have the sameinterpretation for ‘students’, the set {a⊕b, a⊕c, b⊕c, a⊕b⊕c}. For theinterpretation of ‘passed the examination’ we have: P (i1) = {a, b, c, a⊕b, ...}, P (i2) = {a, b, a ⊕ b}, ..., P (i4) = {b, c, b ⊕ c}, ..., P (i8) = ∅ - wesimply take every possible distributive set given the three atoms {a, b, c}- hence, predicate P is assumed to be distributive. After updating withthe semantic meaning of the sentence ‘Students passed the examina-tion’, ∃X : S(X) ∧ P (X), we end up with an information state σ′

containing successors of the possibilities i1, i2, i3, and i4 whose variableassignments now are defined for X, the newly introduced discoursereferent.27 In σ′ there will be a possibility for every possible mappingof X to a group of students that passed the examination in one of theworlds w1, w2, w3, and w4. So σ′ contains, for instance, the possibility〈w1, X : a ⊕ b〉, because the object a ⊕ b is in the extension of P inw1. However, 〈w2, X : a ⊕ b ⊕ c〉 will not be an element of σ′. Giventhe assignment X : a ⊕ b ⊕ c, the answer would not be true in w2.The following tableau lists all possibilities in σ′ plus the way they areordered by <P , where 〈...〉1 → 〈...〉2 means that the second possibilityis P -smaller than the first.

26 Thus, we define 〈w, g〉 <P 〈v, g′〉 iffdef g = g′ and w <P v.27 The others are excluded by the truth conditions of the answer.

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Exhaustive Interpretation 19

〈w1, X : a⊕ b⊕ c〉 〈w1, X : a⊕ b〉 〈w1, X : a⊕ c〉 〈w1, X : b⊕ c〉

? ? ?

〈w2, X : a⊕ b〉 〈w3, X : a⊕ c〉 〈w4, X : b⊕ c〉

To determine the exhaustive interpretation of answer (18c) we col-lect the minimal elements of this ordering (marked by a box in thepicture) and obtain the set: {〈w1, X : a⊕ b⊕c〉, 〈w2, X : a⊕ b〉, 〈w3, X :a⊕c〉, 〈w4, X : b⊕c〉}. This interpretation still allows for the possibilitythat all students passed the examination - even though it would beexcluded that anybody else than students passed the examination. Thereason is that after updating σ with exhdyn(∃X : S(X)∧P (X), P ) thereis still a possibility that takes the world to be w1. And this is so becausethere will be no possibility in σ[∃X : S(X)∧P (X)] where the extensionof P is smaller than in w1 and which still maps X to a⊕ b⊕ c. Such apossibility would not make the answer true. Hence, we correctly predictno minimization for newly introduced discourse referents.28

However, doing the same calculation with ‘Most students’, the ex-ample (18d), will lead to a different result. Because strong determinersdo not immediately introduce discourse referents, we obtain as thesemantic meaning of the answer in context σ the set {i1, i2, i3, i4}. Buti2, i3, i4 are all <P -smaller than i1. Thus, after exhaustive interpretationwe end up with a new information state containing only i2, i3 and i4.The possibility that all students passed the examination is excluded.

Dynamic semantics also helps to account for the difference in exhaustiveinterpretation of (18a) ‘Three students passed’ and (18b) ‘At least threestudents passed’ (or something like ‘Three or more students passed’).Within dynamic frameworks (e.g. Kamp & Reyle, 1993) it is standardto represent (18a) as ∃X : S(X) ∧ cardinality(X) = 3 ∧ P (X). Thisformula has the same ‘at least three’ truth conditions that we obtainwith the classical generalized quantifier interpretation of numerals.In particular, this sentence is true if four students passed, becausethen there is still a set of three students that passed. Thus, from atruth-conditional perspective we could have represented the semanticmeaning of (18a) as well by ∃X : S(X) ∧ cardinality(X) ≥ 3 ∧ P (X).Dynamically, however, the two formulas are not equivalent: the former

28 Notice, by the way, that after exhaustifive interpretation, if we refer back to thenewly introduced discourse referent, we are talking about all students that passedthe examination. This is on a par with intuition.

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20 Robert van Rooij and Katrin Schulz

introduces discourse referents that denote groups of exactly three in-dividuals, while the groups introduced by the latter formula might belarger. As a consequence, if we apply exhdyn, the former formula getsthe ‘exactly three’ reading, while the latter does not. This suggests thatthe former one correctly represents (18a), while the latter formula is thenatural representation of answer (18b). And indeed, that was proposedby Kadmon (1985) (for related, but still somewhat different reasons).Hence, adopting Kadmon’s analysis of the two determiners ‘three’ and‘at least three’ allows us to account for their different behavior underexhaustive interpretation.29

To sum up the discussion in this section so far: the behavior of deter-miners is not a problem that forces us to give up the circumscriptionaccount for exhaustive interpretation or to propose that certain deter-miners have to come with special cancellation properties with respect tothis modus of interpretation. It suffices to make the operation sensibleto dynamic information.

In the last part of this section we will discuss how dynamic informationmay also help to solve another part of the functionality problem ofexhGS . Remember example (13) ‘John or Mary or both’. The applica-tion of exhGS to this answer excludes the last disjunct, hence, predictsthat Either John or Mary is the only one who passed the examination.But even though this answer can be interpreted exhaustively, implyingthat nobody besides John or Mary passed the examination, the possi-bility that both of them passed should not be excluded. Similarly, thesentence ‘John owns 3 or 5 cars’ is on our exhaustivity analysis falselypredicted to mean that John owns exactly 3 cars (the question is ‘Howmany cars does John own?’ and we assume an at least interpretationof numerals). What we like to end up with, however, is the predictionthat John owns either exactly 3 cars, or exactly 5.

Intuitively, the problem seems to be that by exhaustively interpret-ing an answer we are not allowed to ignore any possibility explicitlymentioned by the speaker. Gazdar (1979) accounted for this intuitionby proposing that a disjunctive sentence triggers the clausal impli-cature that every disjunct has to be possible, which can cancel thescalar implicature that gives rise to the standard exclusive reading.Our solution of the closely related problem connected with determiners

29 Kamp & Reyle (1993), in fact, would not represent a sentence like (18b) by∃X : S(X)∧ cardinality(X) ≥ 3∧ P (X), but rather by ∃X : X = λy[S(y)∧ P (y)]∧cardinality(X) ≥ 3. For our purposes, however, this does not matter. They stillpredict that (18b) directly introduces a discourse referent and that is all we needfor our analysis to go through.

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Exhaustive Interpretation 21

suggests a different approach to the problem: while ‘John or Marypassed the examination’ and ‘John or Mary or both passed the ex-amination’ have the same truth conditions, their dynamic semanticmeanings are, again, different. Maria Aloni (2003), for instance, arguesfor independent reasons that the first sentence should be representby something like ∃q : ∨q ∧ (q = ∧P (j) ∨ q = ∧P (m)), where ‘q’is a propositional discourse marker and ‘∨’ and ‘∧’ have their usualMontagovian meanings. Notice that this formula has the same truthconditions as the standard representation of the sentence: P (j)∨P (m).Following Aloni’s lead, we should then, of course, represent ‘John orMary or both passed the examination’ by ∃q : ∨q ∧ (q = ∧P (j) ∨ q =∧P (m) ∨ q = ∧(P (j) ∧ P (m))), which also gives rise to the sametruth conditions. Still, the semantic meanings of the two formulas differ,because the latter allows for a verifying world-assignment pair wherethe assignment maps q to the proposition that both John and Marypassed the examination, while the former formula does not. In almostexactly the same way as in example (18c) explicitly discussed above,this difference in dynamic semantic meaning has the effect that the twoformulas give rise to different exhaustive interpretations: the former,exhσ

dyn(∃q : ∨q ∧ (q = ∧P (j) ∨ q = ∧P (m)), P ), allows only forpossibilities (and thus worlds) in which either only John or only Marypassed the examination; the latter, exhσ

dyn(∃q : ∨q ∧ (q = ∧P (j)∨ q =∧P (m) ∨ q = ∧(P (j) ∧ P (m))), P ), also allows for possibilities whereboth passed the examination. In a similar way we can account forthe exactly-reading of ‘John owns 3 or 5 cars’, if we represent it by∃X : Car(X) ∧Own(j,X) ∧ (card(X) = 3 ∨ card(X) = 5).

6. Exhaustivity and Relevance

One problem of G&S’s approach that our operation exhdyn still inheritsis that it cannot account for the contextual dependence of exhaus-tive interpretation we have observed in domain restricted exhaustiveinterpretations, the scalar readings, the mention-some readings, anddifferences in the fine-grainedness of the interpretation (see section 2.2).The crucial observation made when discussing these readings were that(i) in all these cases exhaustive interpretation was not substituted bysome other interpretation but simply weakened, and (ii) this weakeningcan be characterized as follows: those inferences of the strong fine-grained mention-all reading of exhaustive interpretation were no longerpresent that were commonly known in the context of utterance to benot relevant for the questioner. This leads us to adopt the followingstrategy towards this problem: we extend our definition of exhaustive

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22 Robert van Rooij and Katrin Schulz

interpretation by making it dependent on what counts as relevant forthe questioner. As it turns out, we then can correctly describe theintended variance in readings.

Let us first try to understand what it means to be relevant infor-mation for the questioner and in how far it may play a role for theexhaustive interpretation of answers. If somebody poses a question,she is (normally) in need of certain information. A simple standardway to describe this information is by a set DP of propositions.30

For the questioner it is relevant to know which of these propositionsactually holds. The semantic meaning of a question is also standardlydescribed as a set of propositions, the appropriate, complete, or re-solving answers to the question (see Hamblin, 1973; Karttunen, 1977;G&S, 1984). It seems rational to assume that for reasons of efficiencythere might be a difference between the information asked for explic-itly by the questioner and the information needed, described by DP .For instance, assume that Ann is interested in who of John, Mary,and Peter passed the examination. Thus, DP in this case is the set{{v ∈ W |[P ](w)∩{j,m, p} = [P ](v)∩{j, m, p}}|w ∈ W}. The questiondirectly corresponding to this DP is ‘Who of John, Mary and Peterpassed the examination?’. But it is arguably better for Ann to ask‘Who passed the examination?’ A complete answer to this questionwould provide her with more information than she needs, but thatdoes not bother her. However, the second question is shorter and thusspares her effort.

If it is commonly known what counts as relevant for the questioner, itwould be reasonable for the answerer Bob to take this information intoaccount as well and exhaustively specify only this part of the syntacticquestion-predicate that is relevant. Instead of listing all individuals thatpassed the examination, he only mentions whom of John, Mary, andPeter did. This spares him effort. Then, of course, a rational hearer willrespect this factor as well when interpreting Bob’s utterance and willnot conclude from the answer ‘John.’ that John was the only individualthat passed the examination, but rather that he was the only one ofJohn, Mary, and Peter who did so. And this is exactly what seems tobe going on in the case of domain restricted exhaustive interpretation.

Before we can come to a general formalization of this relevance-dependence of exhaustive interpretation, one further question has tobe addressed: Does relevance already affect the interpretation of thequestion (thus, does Ann’s question ‘Who passed the examination?’semantically mean ‘Who of John, Mary and Peter passed the exam-ination?’) or is relevance independent contextual information? In the

30 This is a simplification of the notion of decision problem.

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Exhaustive Interpretation 23

first case the description of exhaustive interpretation we have given caneasily be made sensible to relevance by proposing that the operationdoes not work on the syntactic question-predicate but rather on thepredicate that the question is really about. The only thing that wehave to do is to clarify how this predicate can be calculated given thesemantic meaning of the question. If, however, the semantic meaningof the question is not affected by what is known about DP , then wecannot use this shortcut and have to incorporate relevaow fnce as afourth argument in our definition of exhaustive interpretation.

This paper is not about the semantics of questions. Therefore, wewill not make a decision on this subject and shortly sketch how onecan proceed in both cases distinguished above.

6.1. The indirect approach

There are certain arguments that speak in favor of taking the semanticmeaning of questions to depend on relevance. For instance, it seems thatthis factor also influences the interpretation of embedded questions. Anextensive discussion of the pros and cons on this issue can be found invan Rooij (2003), followed by an approach to the meaning of questionsthat takes relevance into account. Also according to this approach themeaning of a question is a set of propositions – but now their semanticsis underspecified with respect to contextual relevance. We will adoptthis proposal here.

Given this position towards the semantics of questions we can makeexhstd

31 dependent on relevance simply by manipulating its arguments.We propose that it does not apply to the syntactic predicate of thequestion Q but to that predicate whose extension the questioner isreally interested in. We define it to be a minimal property X such thatknowing the extension of X would resolve the question Q, whereby Xis at least as minimal as Y , X ⊆ Y , iff ∀v : X(v) ⊆ Y (v). FollowingG&S, we take ‘knowing X’ to mean knowing which of the followingpropositions is true: QX = {{v ∈ W |X(v) = X(w)}|w ∈ W}. Thus,someone knows X if she can specify for every object whether it hasproperty X or not. ‘Knowing the extension of X resolves Q’ is nowunderstood as the following relation between QX and Q: ∀q ∈ Q∃q′ ∈QX : q′ ⊆ q, or informally, to know the extension of X has to imply toknow which elements of Q are true.

With this definition of the property the question is about we cancorrectly account for the different readings of exhaustive interpretation

31 To avoid unnecessary complications, we continue with exhstd in this section.However, the changes that will be proposed for this operation can be easily appliedto exhdyn as well.

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24 Robert van Rooij and Katrin Schulz

distinguished in section 2.2. For instance, if the whole extension ofthe syntactic predicate P of the question is relevant, van Rooij (2003)predicts that the meaning of the question is exactly such a partitionQ[P ] as defined above. In this case, no smaller property than P itselfwill exist such that QX solves Q[P ]. Hence X = [P ] and the speaker willby exhaustively specifying X specify P . Thus, we predict a mention-allreading.

How to account for exhaustive interpretation when domain restric-tion or the level or required granularity is at issue is straightforward.For a degree-question like (11) ‘How far can you jump?’, for instance,the question-predicate can in some contexts range over meters, ratherthan centimeters. We therefore address scalar readings next. Rememberthe poker game example, (10) ‘What cards did you have?’. Again thereexists a uniquely determined minimal X, i.e., the X that contains inevery world those and only those cards that contribute to the value ofthe syntactic question-predicate P in terms of the card game. Exhaus-tifying this set tells us that the speaker had no additional cards thatwould increase the value of the cards she mentioned explicitly. Becausein every world w X(w) is a subset of [P ](w), she may have had other,irrelevant, cards. About them, nothing can be inferred.

Finally, the mention-some case. Here, intuitively, the questioner doesnot care about what she learns about the extension of the syntacticpredicate P , as long as she learns for one thing in its extension thatit has property [P ]. Q is predicted by van Rooij (2003) to be the set{{w ∈ W |d ∈ [P ](w)}|d ∈ D}. Any predicate that in each world wapplies to exactly one object in [P ](w) and nothing else will qualifyas a predicate X that Q is about. In this case, X is not uniquelydefined. But for the interpreter the choice does not matter. In any caseone learns from the answer that some subset of the extension of Pconsists exactly of the things mentioned in the answer - nothing moreand nothing less. But that’s exactly what one wants for an answer asin (8).

6.2. The direct approach

Assume that the semantic meaning of the question does not dependon relevance. How, then, can it influence the exhaustive interpretationof answers? A solution to this problem that still keeps the principalsetting of our approach the same is to propose that the order ≤P onwhich the selection of minimal models is based depends on relevance.In some sense this is what we have done in the indirect approach aswell. We proposed that the predicate, or property, used in exhstd shouldbe one that is partly defined in terms of relevance. Because the order

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Exhaustive Interpretation 25

<P compares worlds with respect to the extension they assign to thispredicate, the order becomes sensible to relevance as well. Now, wehave to change the definition of the order to make it, besides on P ,also directly dependent on some measure of relevance.

We say that a world w1 is more minimal than a world w2, w1 <relP w2,

if the proposition that all the objects in [P ](w1) indeed have propertyP is less relevant than the proposition saying the same for world w2.

DEFINITION 5. (The relevance order)

w1 <relP w2 iffdef λw.[P ](w1) ⊆ [P ](w) <R λw.[P ](w2) ⊆ [P ](w)

By substituting this new order in the definition of exhstd we obtain arelevance dependent description of exhaustive interpretation.

DEFINITION 6. (Relevance Exhaustive Interpretation)Let A be an answer given to a question with question-predicate P incontext W . We define the exhaustive interpretation exhW

rel(A,P ) of Awith respect to W and <rel

P as follows:

exhWrel(A,P ) ≡ {w ∈ [A]W |¬∃v ∈ [A]W : v <rel

P w}

This leaves us with the problem to define the order ≤R comparingthe relevance of propositions. Fortunately, a lot of work has been doneon this topic in decision theory that we can make use of. The accountwe propose here simplifies this work quite a bit, for we do not need full-blooded decision theory for our concerns. Remember that we describedthe information a questioner is interested in by a set of propositionsDP . The questioner wants to know which one of these propositionsis true. We define the utility value of a proposition p by how much ithelps to select one of these propositions in DP as true. Let P(q|p) bethe quotient [q]W

[p]W. Hence, P is a simplified measure of the probability

of q given that p is true.32

DEFINITION 7. (The utility value)

UV (p) = maxq∈DPP(q|p)−maxq∈DPP(q|W )

32 It is perhaps useful to point out that if DP is a singleton set consisting onlyof a ‘goal’ proposition h, the utility value we assign to a proposition comes down– according to definition 7 – to the standard statistical notion of relevance, and isalso very similar to what Merin (1999) defines as the relevance of this proposition.The notion given in definition 7 has also been used by Parikh (1992) and van Rooij(2003) for linguistic purposes.

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26 Robert van Rooij and Katrin Schulz

The order <R, comparing the relevance of propositions, can then bedefined in terms of this utility value.

Now everything is in place and we can start to test the proposal.Assume that the speaker wants to know what exactly the extension ofthe syntactic question-predicate P is. DP is then the set of propositionsthat exactly specifies the extension of P . But this means that v <rel

P wiff, everything else being equal, [P ](v) ⊂ [P ](w) and, thus, iff v <P w.Hence, exhW

rel reduces to exhWstd and we predict a mention-all reading.

Mention-some answers can be treated as well. As already discussedin section 2.2, answers get mention-some or non-exhaustive interpreta-tions in cases where it is clear that the addressee only has to know forsomeone in the extension of P that she has property P . For example (8)‘Who has a light?’, for instance, it is normally enough for Ann to knowsomeone who has a light. She just wants to know who to ask for light-ening her cigarette. Let us discuss a concrete example. Let D = {j, m},W = {w1, w2, w3}, and [P ](w1) = {j}, [P ](w2) = {m}, [P ](w3) ={j, m}. Given what we have said about the questioner, DP would inthis situation be the set: {{w1, w3}, {w2, w3}}. To determine the order<rel

P we first have to calculate the utility values of the propositionsλw.[P ](wi) ⊆ [P ](w) for i = 1, 2, 3. It turns out that UV (λw.[P ](w1) ⊆[P ](w)) = UV (λw.[P ](w2) ⊆ [P ](w)) = UV (λw.[P ](w3) ⊆ [P ](w)) -the order collapses totally, i.e., it does not make any difference whichproposition is given as answer. Hence, exhW

rel(P (j), P ) = [P (j)]W : ourexhaustification operator adds nothing to the semantic meaning of theanswer. Thus – and in contrast with other analyses – it is not thatexhaustivity is absent in mention-some cases, our account says that itdoes not do anything!

The other effects of exhaustive interpretation observed in section 2.2can be treated in terms of exhW

rel(A,P ) successfully as well. Consider thegranularity effect, for instance. If Ann is known to be only interestedin the amount of (full) meters that Bob can jump, a world u wherehe can jump 5.00 meters, a world v where he can jump 5.50 meters,and a world w where he can jump 5.80 meters are all equally relevant,u ≈rel

P v ≈relP w. It follows that by taking Bob’s assertion ‘I can jump

five meters’ exhaustively, we predict that in this case Ann can concludeonly that he cannot jump six meters, but not that he cannot jump fivemeters and 10 centimeters.33

33 Of course, granularity effects can play a similar role in the exhaustive interpre-tation of answers to degree-questions. Consider Mr. Smith asking ‘How many of theemployees were at the meeting yesterday?’ If Mr. Smith is known to be interestedin the exact number of employees that were at the meeting, we can infer, or so wepredict, from the answer ‘Most employees’ that only a small majority of employeeswere at the meeting. If his interests are not so specific, or if the answerer is taken

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Exhaustive Interpretation 27

7. Exhaustive interpretation as conversational implicature

We will now leave the excursion to relevance at this point and continuewith the dynamic version exhdyn of our proposal. As the reader mayhave noticed, some inferences we have analyzed under the heading ofexhaustive interpretation have also often been explained as conversa-tional implicatures, in particular as scalar implicatures. To give twoexamples, the exclusive interpretation of ‘or’ in (3): ‘ John or Mary’,and the inference that not all students passed the examination fromthe exhaustive interpretation of (18d) ‘Most students’ (the questionwhich (3) and (18d) address is again ‘Who passed the examination?’)are standard scalar implicatures.

It turns out that the description of exhaustive interpretation pro-posed in the previous sections correctly predicts also many other scalarimplicatures. To give an example, it also generates for (19b) the ‘scale’reversal inference (19c).

(19) (a) Ann: In how many seconds can you run the 100 meters?(b) Bob: I can run the 100 meters in 12 seconds.(c) Bob can’t run the 100 meters in 11 seconds.

The reason for this ‘scale’ reversal is that in contrast to predicateslike ‘Bob owns n many children’ and ‘Bob can jump n meters far’,the question-predicate of (19a) behaves monotone increasingly in num-bers:34 if n is in the extension of the predicate and m > n, then m isin its extension as well.

Particularly pleasing is the observation that the approach to exhaus-tive interpretation defended here can also account for the well-knownproblematic cases of implicatures of complex sentences. For instance,using exhGS or exhdyn one can derive for multiple disjunctions as inanswer (20) the inference that only one of the disjuncts is true (hence,only one of John, Mary, and Peter passed the examination).

(20) John, Mary, or Peter.

For example (21) we correctly predict that the interpreter can inferthat John ate either only the apples or some but not all of the pears.

(21) Ann: What did John eat?Bob: John ate the apples or some of the pears.

to address a less specific question, however, we can plausibly infer only that not allemployees were at the meeting.

34 Something that has to be guaranteed by meaning postulates.

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28 Robert van Rooij and Katrin Schulz

Given this observation it is not very surprising that at differentplaces in the literature it has been suggested to explain exhaustive in-terpretation as a pragmatic phenomenon using Grice’s theory of conver-sational implicatures (see, for instance, Harnish (1976), G&S (1984)).The central problem of such an approach is that there is no thoroughlysatisfying formalization of Grice’s theory and, hence, no precise de-scription of the conversational implicatures an utterance comes with.But without such a rigorous description we cannot say whether Grice’stheory indeed does account for some enrichment of semantic meaning.In particular, we cannot make such a claim for the exhaustive interpre-tation of answers. Thus, before we can see whether Grice’s theory canbe used to explain exhaustive interpretation, we first need to formallydescribe at least parts of the conversational implicatures an utterancecomes with.

In van Rooij & Schulz (2004) a new formalization of the Griceanreasoning leading to scalar implicatures is proposed.35 We will adaptthat approach to the formal situation at hand and add some smallimprovements.

The following Gricean principle has – in different forms – often beentaken to be responsible for scalar implicatures. It combines Grice’s firstsubclause of the maxim of quantity with the maxims of quality andrelevance.

The Gricean Principle

In uttering A a rational and cooperative speaker makes amaximal relevant claim given her knowledge.

In the special case we are interested in here, where the utterance givenis an answer to some previously asked question, the principle comesdown to saying that the speaker will not withhold information fromthe audience that would help to resolve the question she is answering –she provides the best (i.e. most informative) answer she can, given herknowledge.

Our goal is to formalize the inferences an interpreter can derive ifshe takes the speaker of some sentence A to obey this principle. Thesolution proposed in van Rooij & Schulz (2004) is closely related toMcCarthy’s predicate circumscription and makes essential use of ideasdeveloped by Halpern & Moses (1984) and generalized by van der Hoek

35 Which, in turn, was motivated by Schulz’ (2003) epistemo-logical analy-sis of clausal implicatures to account for the well-known problem of free-choicepermissions.

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Exhaustive Interpretation 29

et al. (1999, 2000) on the concept of only knowing. We describe thepossibilities where the speaker obeys the principle as those where sheknows the sentence A she uttered to be true but knows as least aspossible about the predicate in question besides what is conveyed by heranswer. Hence, as in the case of predicate circumscription, the enrichedinterpretation of answer A is described by selecting minimal models.Now, however, the selection takes place among those possibilities wherethe speaker knows A and the order that determines minimality does notcompare the extension of the question-predicate, but rather how muchthe speaker knows about this extension.

To formalize the idea, we have to refer to the knowledge state ofthe speaker. We will adopt a standard modal logical way of modelingknowledge. Let W be a set of models/possible worlds of our language.36

We add to W an accessibility relation R that connects every element wof W with a subset R(w) of W . This subset represents what the speakerknows in w. Then we can say that sentence KA, ‘the speaker knowsA’, is true in w (with respect to W and R) if A is true in every world inR(w). For the dynamic case we define that KA is true in a possibilityi = 〈w, g〉 (with respect to W and R) if for every v in R(w), A is truein j = 〈v, g〉. Because we want to model knowledge we demand thatw is an element of R(w). In this way we warrant that if the speakerknows A, the sentence is actually true.

Now, we can define a pragmatic interpretation function that givesus besides the semantic meaning also the conversational implicaturesdue to the Gricean Principle. Assume that �P,A is the order that com-pares how much the speaker who uttered A knows about the question-predicate P .

DEFINITION 8. (Interpreting according to the Gricean Principle)Let A be an answer given to a question with question-predicate P incontext σ. We define the pragmatic interpretation griceσ(A,P ) of Awith respect to P and σ as follows:

griceσ(A,P ) =def {i ∈ σ[KA]|∀j ∈ σ[KA] : i �P,A j}

Of course, this definition will only be of use if we can also give anexplicit definition of the order �P,A, hence, describe what it meansthat in one possibility the speaker knows more about the extensionof the question-predicate than in another. But when is this the case?Informally, what we want to express is that in one world the speakerhas more knowledge about P if she knows of more individuals that

36 We allow multiple occurrences of the same interpretation function of thelanguage in W .

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30 Robert van Rooij and Katrin Schulz

they have property P . Thus, i1 �P,A i2 if for every world v2 consideredpossible by the speaker in w2, she distinguishes some possibility v1 inR(w1) where the extension of P is smaller than or equal to the extensionof P in v2 (and g1 = g2).37 But wait! It may be the case that thespeaker makes in her utterance a claim about the extension of P whichdepends on some other facts. For instance, if she answers ‘If they askedthe same questions as last year then Peter passed the examination’ tothe question ‘Who passed the examination?’. Of course, in this casewe expect a speaker that obeys the Gricean principle also to tell uswhether – as far as she knows – they asked the same questions as lastyear. Therefore, we define the order as follows:38

DEFINITION 9.For v1, v2 ∈ W we define

v1 ≤∗P,A v2 iffdef 1. [P ](v1) ⊆ [P ](v2) and

2. for all non-logical vocabulary θ occurring in Abesides P : [θ](v1) = [θ](v2);

v1 ≡∗P,A v2 iffdef v1 ≤∗

P,A v2 and v2 ≤∗P,A v1.

DEFINITION 10. (Comparing relevant knowledge)For w1, w2 ∈ W we define

w1 �P,A w2 iffdef ∀v2 ∈ R(w2) : ∃v1 ∈ R(w1) : v1 ≤∗P,A v2

For i1 = 〈w1, g1〉, i2 = 〈w2, g2〉 ∈ σ we define

i1 �P,A i2 iffdef w1 �P,A w2 & g1 = g2

i1 ∼=P,A i2 iffdef i1 �P,A i2 & i2 �P,A i1

Now that we have with grice at least a part-wise description ofthe conversational implicatures an utterance comes with, we can see

37 Some readers may complain that in this way we do not respect knowledge thespeaker might have about some individuals not having property P . This is true.The question is, however, in how far this information is relevant for the exhaustiveinterpretation of answers. According to this approach it is not. Empirical studieshave to show whether this claim is correct.

38 ≤∗P,A is stronger than the order ≤P that we have used so far. If one would sub-

stitute the latter in the definition of �P,A, then grice would minimize the knowledgeof the speaker about the extension of all non-logical vocabulary, which is inadequatefor our purposes. In van Rooij & Schulz (2004) an even stronger order was used.There, condition 2 of definition 9 was dropped and non-logical vocabulary besidesP did not play any role for the order. Then, however, one misses for the answer‘If they asked the same questions as last year then Peter passed the examination’the intuitive inference that the speaker does not know whether they asked the samequestions as last year.

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Exhaustive Interpretation 31

whether the part of Grice’s theory we have formalized can explain theexhaustive interpretation of answers. Unfortunately, it turns out thatthis is not the case. To illustrate the problem, let us calculate whatgrice predicts for example (22).

(22) Ann: Who passed the examination.Bob: Mary.

Hence, let us determine griceσ(P (m), P ). To make things even simpler,we assume that in σ there are only four different possibilities: i1 =〈w1, g〉, i2 = 〈w2, g〉, i3 = 〈w3, g〉, and i4 = 〈w4, g〉 with R and P definedas given in figure 1.39

•w1 : P = {m, p}

6R

�� � w1 : P = {m, p}

•w2 : P = {m}

6R

�� � w2 : P = {m}

•w3 : P = {m}

6R

��

��

w3 : P = {m}v : P = {m, p}

•w4 : P = {m, p}

6R

��

��

w4 : P = {m, p}u : P = {m}

?

@@

@@R

-

Figure 1.

What we would like to predict in such a situation is that all otherindividuals (in our example there is only one other individual: Peter (p))did not pass the examination. To calculate griceσ(P (m), P ) accordingto definition 8, the first thing we have to do is to select those possibilitiesi in σ where the speaker knows that P (m) is true. In turns out that thisis the case for all elements of σ. In a second step we select among thosethe possibilities where Bob knows least about the question-predicate P .The order tells us that the speaker knows more in w1 than in w2, w3,and w4, and that in the latter three worlds he knows equally much.Hence: griceσ(P (m), P ) = {i2, i3, i4}. A closer look reveals that this

39 From possible worlds w, represented by points, arrows annotated by R lead tothe knowledge state R(w) of the speaker in this world. The arrows in the middleof the figure symbolize the ordering relation �P,A. Notice that in this example theworlds w2 and w3 differ, because in w2 the speaker has a more definite opinion aboutthe extension of P than in w3.

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32 Robert van Rooij and Katrin Schulz

interpretation allows the interpreter to derive from Bob’s answer theconversational implicature that he does not know that Peter passedthe examination (i.e. griceσ(P (a), P ) |= ¬KP (p)). But we are not ableto derive the desired inference that Peter did, in fact, not pass theexamination. Hence, we have to conclude that the Gricean Principle,at least in the formalization given above, cannot explain exhaustiveinterpretation.

Actually, many students of conversational implicatures will find thisa rather pleasing result. It has often been argued in the literature thatthe conversational implicatures due to the Gricean Principle40, shouldbe generated primarily with the weak epistemic force we predict (see,among others, Soames (1982), Leech (1983), Horn (1989), Matsumoto(1995), and Green (1995)). Hence, the conversational implicature ofBob’s answer is indeed claimed to be that he does not know for otherpeople than Mary that they passed the examination. Only in contextswhere the speaker is assumed/believed to be competent/an authorityon the subject matter under discussion, these authors propose, one canderive the stronger inference that what the speaker does not know tohold indeed does not hold (hence, in the example the desired inferencethat Peter did not pass the examination).

For our approach this would mean that we should be able to obtainthe exhaustive interpretation by calculating grice with respect to theset σ of possibilities where the speaker Bob is competent/an authorityon the question she is answering. However, in van Rooij & Schulz (2004)it is shown that this will not lead to an adequate description of theexhaustive interpretation of answers (or their scalar implicatures). Theproblem is that some sentences that can be interpreted exhaustively(or give rise to scalar implicatures) cannot stem from a speaker thatis at the same time competent/an authority and obeys the GriceanPrinciple. Take, for instance, the answer in (3), here repeated as (23)

(23) Ann: Who passed the examination?Bob: John or Mary.

In this paper we have proposed to analyze the often observed exclusiveinterpretation of ‘or’ as due to exhaustive interpretation, and at manyplaces in the literature the inference that not both disjuncts are trueat the same time has been claimed to be a scalar implicature. However,the approach sketched above will not predict any conversational impli-catures for such disjunctive answers. The reason is that a competentspeaker should know which of the disjuncts is true and, if obeying the

40 In particular, conversational implicatures due to the first subclause of themaxim of quantity.

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Exhaustive Interpretation 33

Gricean Principle, should have given this information. The fact thatBob nevertheless did not do so shows that he either is not competent orhas disobeyed the principle. In either case no conversational implicaturecan be derived.

To overcome this problem but nevertheless stay faithful to the intu-ition that competence/authority plays a decisive role for the derivationof exhaustivity effects/scalar implicatures, van Rooij & Schulz (2004)propose to maximize the competence of the speaker when interpretinganswers. However, it is only maximized in so far as this is consistentwith taking the speaker to obey the Gricean Principle.41

There is an obvious way to extend the function grice such that itfollows this idea. We introduce a second order on the set of possibilitiesthat compares the competence of the speaker in different possibilities(a possibility is higher in the order if the speaker is more competent).Then we select maximal elements with respect to this order – but nowonly among those possibilities where the speaker obeys the GriceanPrinciple, i.e., among the elements in griceσ(A,P ).

Let vP,A be the order that compares competence. The interpretationfunction defined below tells an interpreter what she can infer if she takesthe speaker, first, to obey the Gricean Principle, and, second, to be ascompetent on the question she answers as is consistent with the firstassumption.

DEFINITION 11. (Adding Competence to the Gricean Principle)Let A be an answer given to a question with question-predicate P incontext σ. We define the pragmatic interpretation epsσ(A,P ) of A withrespect to P and σ as follows:

epsσ(A,P ) =def {i ∈ griceσ(A,P )|∀j ∈ griceσ(A,P ) : j 6=P,A i}

= {i ∈ σ[KA]|∀j ∈ σ[KA] : i �P,A j∧(i ∼=P,A j → j 6=P,A i)]]}

Again, to make this definition useful we have to define the order<P,A properly. We propose that in a world w2 the speaker is as least

41 In this respect, our analysis bears resemblance with Gazdar’s (1979) proposalthat clausal implicatures with weak epistemic force can cancel scalar ones that havestrong epistemic force; Sauerland’s (2004) method of strengthening implicatureswith weak epistemic force, and especially Spector’s (2003) defense of exhaustiveinterpretation. In contrast to their analyses, however, ours is more general, fullymodel-theoretic, and based on standard methods of non-monotonic reasoning.

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34 Robert van Rooij and Katrin Schulz

as competent as in world w1 if in w1 the speaker considers as least asmany extensions possible for question-predicate P as in w2.42

DEFINITION 12. (Comparing competence)For w1, w2 ∈ W we define

w1 vP,A w2 iffdef ∀v2 ∈ R(w2) : ∃v1 ∈ R(w1) : v2 ≡∗P,A v1

For i1 = 〈w1, g1〉, i2 = 〈w2, g2〉 ∈ σ we define

i1 vP,A i2 iffdef w1 vP,A w2 & g1 = g2

To illustrate the working of the new strengthened interpretationfunction eps, let us reconsider our example (22). We calculate epsσ(P (m), P ),where σ is defined as in figure 1. Remember that we want to obtainthat Bob’s answer implies that Peter did not pass the examination.Because we already know griceσ(A,P ), the only thing that still has tobe done is to select among the possibilities in this set those where ac-cording to vP,A the speaker is maximally competent. Unsurprisingly, i2is the unique vP,A-maximum in griceσ(P (m), P ) = {i2, i3, i4}. Hence,epsσ(P (m), P ) = {i2}. But, as figure 1 shows, in i2 the desired conclu-sion ¬P (p) holds! So, at least for this example the new interpretationfunction combining Gricean reasoning with a principle of maximizingcompetence predicts correctly.

Of course, we would like to establish that eps can account for theexhaustive interpretation of answers in some generality. At least itwould be pleasing if eps does not perform worse in describing exhaustiveinterpretation than does exhdyn. It turns out that both notions areindeed closely related.43

FACT 1. If A and φ do not contain modal operators and σ is the setof all possibilities for the language used then

epsσ(A,P ) |= φ iff exhσdyn(A,P ) |= φ.

Hence, if the answer given does not contain modal operators and thereis no information in the context, then both interpretation functions pre-dict the same modal-free inferences. Until now, all examples discussed

42 Here, again, we deviate a bit from the approach in van Rooij & Schulz (2004).There it is proposed to compare only what the speaker knows about objects thatdo not have property P . Because of the way how grice is defined, it does not makeany difference for eps which of the two definitions is chosen.

43 The proof of this fact goes very much along the same lines as the proof of asimilar claim given in van Rooij & Schulz (2004).

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Exhaustive Interpretation 35

in this paper were of this kind. Thus, all the pleasing predictions madeby exhdyn are inherited by eps.

However, our richer modal analysis allows us, additionally, to de-scribe exhaustivity effects that could not be accounted for in terms ofexhdyn. One kind of example are sentences that contain modal expres-sions, like belief attributions or possibility statements as in the answer‘John and perhaps also Mary’. This advantage of the Gricean derivationof exhaustive interpretation given above is explicitly discussed in vanRooij & Schulz (2004) and we will not repeat that discussion here. Inthis section we only want to indicate (as also discussed in the papermentioned) how the modal approach can help us to solve a last problemof G&S’s (1984) approach that has not been addressed so far: theexhaustive interpretation of negative answers.

Remember our discussion at the end of section 3: exhGS predicts wronglythat the answer ‘Not John’ to question ‘Who passed the examination?’means that nobody passed the examination. The same prediction ismade for all (other) cases in which the predicate occurs under negationin the answer. To solve this problem, von Stechow & Zimmermann(1984) propose to modify G&S’s exhaustivity operator by selecting inthese cases not the minimal extensions of the predicate, but ratherthe maximal ones. In terms of our framework this means that now thespeaker gives not the exhaustive extension of question-predicate P , butof its complement, P̄ , instead. Thus, in case P occurs negatively in A,we should not look for exhσ

dyn(A,P ) but rather for exhσdyn(A, P̄ ). Then,

the answer ‘Not John’, for instance, would be interpreted as implyingthat, except for John, everybody passed the examination. According tomost of our informants, however, this kind of exhaustive interpretationis the exception, rather than the rule. They report instead that negativeanswers give rise to the conclusion that the semantic meaning of theanswer is the only information the speaker has about the question-predicate. Interestingly enough, the same intuition is reported also forother answers, for instance, if the speaker uses special intonation orresponds ‘Well/As far as I know, Peter’.44

A very welcome side-effect of the Gricean explanation given to ex-haustive interpretation in this section is that we can correctly describethis interpretation when we only apply the function grice, hence, takethe speaker to obey the Gricean Principle, but not maximize her compe-tence. This suggests the following explanation for the non-exhaustiveinterpretation of the answers discussed above. The speaker is alwaystaken to fulfill the Gricean principle and, hence, grice is applied to the

44 See also footnote 7.

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36 Robert van Rooij and Katrin Schulz

answer. We normally also take the answerer to be competent45 and,hence, apply eps. However, the answerer can cancel this additionalassumption by either mentioning that she is not competent or simplydeviating from the standard form of answering a question (by usingnegation, special intonation, etc.). In this way we can correctly pre-dict the weakening of exhaustive interpretation to ‘limited-competence’inferences for such answers.

8. Conclusion and outlook

In this paper we propose a description of the exhaustive interpreta-tion of answers in terms of relevance-dependent dynamic predicatecircumscription. The main concept we use is that of interpretationin minimal models, which we took from AI-research.46 It constitutesthe fundament of our formalization of exhaustive interpretation andholds the whole paper together. The second backbone of our proposalis dynamic semantics. It provides us with the semantic framework inwhich we embedded minimal interpretation. And finally, we use stan-dard conceptions of relevance to bring communicational interests ofthe agents into play. Brought together, these three independent lines ofresearch allow us to account for many observations on the phenomenonof exhaustive interpretation. And we can do so in terms of one generalmode of interpretation.

In the last section of the paper we used a proposal made in vanRooij & Schulz (2004) to provide a pragmatic explanation of exhaustiveinterpretation. It is shown that one can describe this reading of answersby assuming, first, that the speaker obeys the Gricean Principle and,second, that she is competent on the question she answers (as faras this is consistent with the first assumption). This analysis clarifiesthe connection between exhaustive interpretation, scalar implicatures,and Grice’s theory of conversational implicatures. Scalar implicatures(of answers) are taken to be inferences due to exhaustive interpreta-tion. Exhaustive interpretation, however, is explained as the result of aGricean-like reasoning about rational behavior of cooperative speakers.

45 This seems to be a natural default assumption, given that only in such situationsit makes perfect sense to ask a question to a certain addressee.

46 We only know of one (other) attempt to use circumscription for (some of) thedata we discuss in this paper: by Wainer in his dissertation (1991). When applyingcircumscription to utterances directly, he came across some of the same problemsthat we discussed for G&S’s proposal. For this reason he opts, in the end, for a seconddescription in which stipulated abnormality predicates are circumscribed. One of themain goals of this paper was to show that the direct approach without additionalabnormality predicates can be pushed much further than Wainer assumed.

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Exhaustive Interpretation 37

Furthermore, this Gricean derivation also allows us to account for acertain contextual weakening of exhaustive interpretation that cannotbe explained using relevance as due to cancellation of the competenceassumption.

Because we restrict ourselves in this paper to the exhaustive inter-pretation of answers to overt questions, we can account for scalarimplicatures in these contexts only. This may be taken to be a seri-ous limitation of the proposed approach. But, first, the fact that wefocused on answers to overt questions does not necessarily mean thatexhaustive interpretation occurs only under these circumstances. Thisrestriction was forced upon us mainly because we did not have sufficientempirical data to support a general statement about the contexts inwhich exhaustive interpretation occurs. Furthermore, with respect toscalar implicatures one of the central issues in the recent literature onthis subject is the context-dependence of these inferences. In particular,it has been claimed that questions can play an important role for thepresence of scalar implicatures (see, for instance, Hirschberg (1985) andvan Kuppevelt (1996)). Further research on this subject has to clarifyin which contexts we do observe scalar implicatures, and whether theycoincide with the contexts of exhaustive interpretation.

Another interesting question for further research is whether the givenformalization of the Gricean Principle can be extended to a generalimplementation of Grice’s maxims of conversation. Consider, for in-stance, the second subclause of the maxim of quantity. This subclauseis taken to be the driving force behind another class of pragmaticinferences: those to the most stereotypical interpretation. For instance,that we normally interpret ‘John killed the sheriff’ as meaning thatJohn murdered the sheriff in a stereotypical way, i.e. by knife or pistol,is often explained with reference to this maxim. Inferences to the stereo-type/normal case (called I-implicatures by Atlas & Levinson (1981),and R-implicatures by Horn (1984)) are often analyzed as being insome sense opposite to scalar implicatures.47 Against this backgroundit is interesting to observe that the minimal model approach can beused naturally to account for the later inferences as well.48 The onlything that we have to change is how we instantiate the ordering. In

47 The intuition being that while in case of scalar implicatures some stronger claimis excluded, in case of inference to the stereotype some stronger claim is assumed tohold.

48 In fact, these are the inferences non-monotonic reasoning was originallymade for. See, for instance, McCarthy’s motivation for introducing PredicateCircumscription as briefly discussed in section 4.1.

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38 Robert van Rooij and Katrin Schulz

this case it is not relevance – of which predicate minimization andentailment are special cases – but rather plausibility (or expectedness)that is maximized. Thus, now we have to assume that v ≺ w iff v is amore plausible world than w is, and the interpretation of A w.r.t. ≺,{w ∈ [A]W |¬∃v ∈ [A]W : v ≺ w}, results then just in the set of mostplausible worlds that verify A. In the future we would like to see inhow far this formalization can account for the wide range of I or Rimplicatures described by Atlas & Levinson and Horn as due to thismaxim,49 and how they interact with scalar implicatures.

Of course, the observation that both types of implicatures may becaptured by very similar interpretation rules does not make them nec-essarily the same phenomenon. In AI there has been an intense debateon the interpretations that non-monotonic reasoning formalisms canreceive. One of the distinctions made there seems to show up here again.We have described exhaustive interpretation as a rule of negation asfailure in the message: from the fact that the speaker did not say p for acertain class of propositions p, the interpreter infers that (the speakercannot exclude) ¬p. Already McCarthy (1986) mentioned such rulesof language use as examples of circumscription in action. Such a rulemay also govern the I or R implicatures: if the speaker did not mentionthat the situation is in a certain way abnormal, then the interpreter canconclude that it is normal. But here we do not have to take the detourvia language use. It may also simply be the case that the interpreterconcludes to the stereotype interpretation because it is for her the nor-mal state of affairs given the information she has (including the messageof the speaker). Note that this is not an admissible interpretation ofexhaustive readings: if we learn that Mary has property P , only in veryexceptional cases will general knowledge about how the world normallyis allow us to infer that John does not have property P .

Hence, in summary, while the inference of negation as failure inher-ent in exhaustive interpretation is most plausibly due to rule-governedconversational behavior (which may be conventional or not), the infer-ence to the stereotypical interpretation does not need to be anchoredin language use.

Acknowledgements

The work on exhaustive interpretation goes back several years for bothauthors, but got a new boost when we realized the similarity between

49 In several papers, e.g. Asher & Lascarides (1998), a sophisticated method ofnon-monotonic reasoning is used to account for some of these inferences.

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G&S’s rule of exhaustive interpretation and McCarthy’s predicate cir-cumscription. Early versions of this paper were first presented in Am-sterdam and later at the Fifth International Workshop on Computa-tional Semantics in Tilburg (January 2003) and at several other work-shops, summerschools (ESSLLI, NASSLLI), and lectures. We wouldlike to thank the audiences at those meetings for their questions andcomments. We also would like to thank Jacques Wainer for sending ushis dissertation and Richmond Thomason for pointing us to it. Mostof all, however, we would like to thank Johan van Benthem, JeroenGroenendijk and Martin Stokhof for critical readings of earlier versionsof this paper. Finally, we would like to thank two anonymous refereesfor their useful suggestions.

References

Aloni, M. (2003), ‘On choice-offering imperatives’, In: P. Dekker & R. van Rooy(eds.), Proceedings of the Fourteenth Amsterdam Colloquium, Amsterdam, pp.57-62.

Asher, N. and A. Lascarides, (1998), ‘Bridging’, Journal of Semantics, 15: 83-113.Atlas, J. and S. Levinson (1981), ‘It-Clefts, Informativeness and Logical Form’, In

Radical Pragmatics, P. Cole (ed.), Academic Press, New York, pp. 1-61.Benthem, J. van (1989), ‘Semantic parallels in natural language and computa-

tion’, In. H. D. Ebbinghaus et al. (eds.), Logic Colloquium ’87, Elsevier SciencePublishers, Amsterdam, pp. 331-375.

Bonomi, A. and P. Casalegno (1993), ‘Only: association with focus in eventsemantics’. Natural Language Semantics, 2: 1-45.

Gazdar, G. (1979), Pragmatics, Academic Press, London.Ginzburg, J. (1995), ‘Resolving questions, I’, Linguistics and Philosophy, 18: 459-

527.Green, M. (1995), ‘Quantity, volubility, and some varieties of discourse’, Linguistics

and Philosophy, 18: 83-112.Grice, H. P. (1967), ‘Logic and Conversation’, typescript from the William James

Lectures, Harvard University. Published in P. Grice (1989), Studies in the Wayof Worlds, Harvard University Press, Cambridge Massachusetts, pp. 22-40.

Groenendijk, J. and M. Stokhof (1984), Studies in the Semantics of Questions andthe Pragmatics of Answers, Ph.D. thesis, University of Amsterdam.

Halpern, J.Y. and Y. Moses (1984), ‘Towards a theory of knowledge and ignorance’,Proceedings 1984 Non-monotonic reasoning workshop, American Association forArtificial Intelligence, New Paltz, NY, pp. 165-193.

Hamblin, C.L. (1973), ‘Questions in Montague English’, Foundations of Language,10: 41-53.

Harnish, R. M. (1976), ‘Logical form and implicature’, In: T. G. Bever et al. (eds.)An integrated theory of linguistic ability, Crowell, New York, pp. 313-391.

Hirschberg, J. (1985), A theory of scalar implicature, Ph.D. thesis, PennsylvaniaState University.

Hoek, W. van der, J. Jaspers and E. Thijsse, (1999), ‘Persistence and minimality inepistemic logic’, Annals of Mathematics and Artificial Intelligence, 27: 25-47.

exhLPNewer.tex; 1/09/2004; 9:09; p.39

40 Robert van Rooij and Katrin Schulz

Hoek, W. van der, J. Jaspers and E. Thijsse, (2000), ‘A general approach to multi-agent minimal knowledge’, In: M. Ojeda-Aciego et. al. (eds.) Proceedings JELIA2000, LNAI 1919, Springer-Verlag, Heidelberg, pp. 254-268.

Horn, L. (1984), ‘Towards a new taxonomy of pragmatic inference: Q-based and R-based implicature’. In: Schiffrin, D. (ed.), Meaning, Form, and Use in Context:Linguistic Applications, GURT84, Georgetown University Press, Washington,pp. 11-42.

Horn, L. (1989), A Natural History of Negation, University of Chicago Press,Chicago.

Kadmon, N. (1985), ‘The discourse representation of NPs with numeral determiners’,In: S. Berman et al. (eds.), Proceedings of NELS 15, GLSA, Linguistics Dept.University of Massachusetts, Amherst, pp. 207-19.

Kamp, H. and U. Reyle (1993), From Discourse to Logic, Kluwer, Dordrecht.Kratzer, A. (1989), ‘An investigation of the lumps of thought’, Linguistics and

Philosophy, 12: 607-653.Karttunen, L. (1977), ‘Syntax and semantics of questions’, Linguistics and Philoso-

phy, 1: 3-44.Kuppevelt, J. van (1996), ‘Inferring from topics. Scalar implicatures as topic-

dependent inferences’, Linguistics and Philosophy, 19: 393-443.Leech, G. (1983), Principles of Pragmatics, Longman, London.Lewis, D. (1973), Counterfactuals, Blackwell, Oxford.Matsumoto, Y. (1995), ‘The conversational condition on Horn scales’, Linguistics

and Philosophy, 18: 21-60.McCarthy, J. (1980), ‘Circumscription - a form of non-monotonic reasoning’,

Artificial Intelligence, 13: 27 - 39.McCarthy, J. (1986), ‘Applications of circumscription to formalizing common sense

knowledge’, Artificial Intelligence, 28: 89-116.McCawley, J. (1993), Everything that linguists always wanted to know about Logic∗,

The University of Chicago Press, Chicago.Merin, A. (1999), ‘Information, relevance, and social decisionmaking’, Logic, Lan-

guage, and Computation, Vol. 2, In L. Moss et al. (eds.), CSLI publications,Stanford, pp. 179-221.

Parikh, P. (1992), ‘A game-theoretical account of implicature’, In: Y. Vardi(ed.), Theoretical aspects of Rationality and Knowledge: TARK IV, Montery,California.

Rooth, M. (1996), ‘Focus’, In: S. Lappin (ed.), The Handbook of ContemporarySemantic Theory, Blackwell Publishers, Malden Massachusetts, pp. 271-297.

Rooij, R. van (2003), ‘Questioning to resolve decision problems’, Linguistics andPhilosophy, 26: 727-763.

Rooij, R. van and K. Schulz (2004), ‘Exhaustive interpretation of complex sentences’,Journal of Logic, Language, and Information, in press.

Rooij, R. van and K. Schulz (to appear), ‘Only: meaning and implicatures’.Sauerland, U. (2004), ‘Scalar implicatures of complex sentences’, Linguistics and

Philosophy, 27, 367-391.Schulz, K., (2003), ‘You may read it now or later’, Master thesis, University of

Amsterdam.Szabolcsi, A. (1981), ‘The semantics of topic-focus articulation’, In J. Groenendijk

et. al. (eds.), Formal methods in the study of language, Mathematical Centre,Amsterdam, pp. 513-540.

Soames, S. (1982), ‘How presuppositions are inherited: A solution to the projectionproblem’, Linguistic Inquiry, 13: 483-545.

exhLPNewer.tex; 1/09/2004; 9:09; p.40

Exhaustive Interpretation 41

Spector, B. (2003), ‘Scalar implicatures: Exhaustivity and Gricean reasoning?’, In:B. ten Cate (ed.), Proceedings of the ESSLLI 2003 Student session, Vienna.

Stechov, A. von and T.E. Zimmermann (1984), ‘Term answers and contextualchange’, Linguistics, 22: 3-40.

Wainer, J. (1991), Uses of nonmonotonic logic in natural language understanding:generalized implicatures, Ph.D. thesis, Pennsylvania State University.

Zeevat, H. (1994), ‘Questions and Exhaustivity in Update Semantics’, In: Bunt et al.(eds.), Proceedings of the International Workshop on Computational Semantics,Institute for Language Technology and Artificial Intelligence, Tilburg.

Institute for Language, Logic and Computation (ILLC)University of AmsterdamNieuwe Doelenstraat 151012 CP AmsterdamThe Netherlands{R.A.M.vanRooij,K.Schulz}@uva.nl

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