Pragmatic Meaning and Non-monotonic Reasoning: The Case of Exhaustive Interpretation

Transcription

1 Pragmatic Meaning and Non-monotonic Reasoning: The Case of Exhaustive Interpretation Robert van Rooy and Katrin Schulz Abstract. In this paper, we study the phenomenon of exhaustive interpretation. We discuss an analysis of exhaustification as interpretation in preference structures using work in non-monotonic reasoning. We first will discuss the proposal of Groenendijk & Stokhof (1984) and some problems it has to face. Then, the similarity of this account to predicate circumscription as introduced by McCarthy is discussed, an observation already made by van Benthem (1989). Finally the idea of interpretation in preferred models and independent developments in semantics/pragmatics are brought together to overcome the previously discussed limitations. Exhaustification is defined as a dynamic update function, circumscribing the predicate the question is about relative to the given answer. This predicate is defined in terms of relevance. Keywords: Circumscription, Conversational Implicatures, Exhaustive Interpretation, Only, Relevance 1. Introduction. The phenomena The topic of this paper is the phenomenon of exhaustive interpretation, a systematic and specific way in which we sometimes enrich the semantic meaning of assertions. The following dialogue fragment gives a standard example. (1) Paul: Who knows the answer? Paula: John and Mary. In most contexts Paula s answer is interpreted as {John, Mary} = {x : x knows the answer in w 0 }, i.e., that John and Mary are the only people who know the answer. 1 Although the sentence John and Mary know the answer doesn t bear this information if occurring in isolation (hence, one could argue, it is not conveyed by its semantic meaning in the strict sense of the word), used as answer it is understood as giving all those and only those who know the answer. This is called a mention all interpretation of an answer. As will become clear soon, we will analyze it as a special form of exhaustive interpretation. There exists broad consensus in the literature that exhaustification works on the information structure of the discourse and that this is The names are listed in alphabetical order. 1 At least, if the answerer does not signal, for instance by her use of intonation, that the answer is only a partial one. c 2003 Kluwer Academic Publishers. Printed in the Netherlands. exhlp-r.tex; 22/09/2003; 9:00; p.1

2 2 Robert van Rooy and Katrin Schulz the reason why this kind of interpretation is present especially in the context of questions. But it may also be applied to sentences containing free focus that are not direct answers to open questions. In this paper we will concentrate on exhaustive interpretations of direct answers. As is well-known, the effects of exhaustification and the meaning effects of the focus particle only are very similar. Although we don t think that the meaning of only can be reduced completely to that of exhaustification, 2 we will discuss semantic and pragmatic analyses of only when they concern at least in our eyes features that they have in common with exhaustive answers. The structure of this paper is as follows. We start with some considerations regarding the character and behavior of exhaustive interpretation. Afterwards, we will introduce Groenendijk & Stokhof s exhaustivity operator and discuss its empirical adequacy. Then we try, step by step, to overcome some difficulties of this analysis without loosing any of its specific pleasant properties. In doing so, we will make use of semantic theories we cannot fully introduce here. In particular, we assume at least a rough acquaintance with dynamic semantics. We will proceed step by step because we think it will serve the understanding of the formalization of exhaustive interpretation we propose. Doing so allows the reader to easily identify which property of the defined operation serves which purpose, and us to motivate each of its features. However, this way of representing has its drawback: our central notion, exhaustive interpretation, will be redefined over and over again. Please, keep in mind that each version of exh (coming with it s own suffix) highlights a particular aspect of this modus of interpretation. They do not stand for different notions of exhaustivity. Finally, some short remarks on notation and formal apparatus. If not said explicitly we interpret expressions of a language with respect to a set W of models of the language, often also called worlds. The meaning of an expression ψ is denoted by [ψ] W, which is a function from models to the interpretation ψ obtains in the respective model. We will also use the set-theoretic representation of truth functions, for instance, if ψ is a sentence [ψ] W is often understood as the set of models in W that make ψ true. Furthermore, if not explicitly mentioned, A is used as metavariable for the logical form of an answer and P stands for the logical form of the question predicate. Sometimes these metavariables are also used for 2 Above all, exhaustivity misses the less than expected/hoped for inference only comes with. But they have different effects to answers of mention-some questions as well (cf. van Rooy, 2002). exhlp-r.tex; 22/09/2003; 9:00; p.2

3 Exhaustive Interpretation 3 the respective interpretations, but what is meant will be clear from the context Mention-all readings As already stated above, in the mention-all reading of exhaustivity, the most studied, but, as we will see, not its only variant, an answer is interpreted as giving all and only those individuals in the extension of question predicate P. Let us look at some more examples to get better acquainted with this type of interpretation (all sentences discussed in this section should be understood as Paula s answers to Paul s question Who knows the answer?). (2) Some men. In most contexts, this answer is interpreted as stating that just a few persons know the answer and that they are all adult males - even though Paula doesn t give the information which men it are. This type of answer is of some relevance, because sometimes it is, with and without exhaustive interpretation, not the best answer one could give here. If the questioner wants to receive the information that will enable him to identify all the individuals that satisfy the question predicate, (2) is simply insufficient. Not in all contexts where it is observed, exhaustivity makes answers complete, i.e., make them fully resolve the question. This observation is a major challenge if it comes to explanations of why we interpret exhaustively. (3) John or Mary. Looking at this response to the question Who knows the answer? we meet the well-known phenomenon of exclusive interpretation of or: one inference from the interpretation of the answer is that either John knows the answer or Mary does, but not both. This observation is often explained in terms of Grice s (1967) theory of conversational implicatures. Following his proposal, an interpreter can draw additional information (conversational implicatures) from the assumption that the speaker follows certain rules (maxims of rational and cooperative behavior) in conversation. The exclusive interpretation of or is proposed to be due to the first submaxim of quantity (Q 1 ) which demands the speaker to give all relevant information he has. But this is not the only additional inference we observe in connection with examples as (3). Intuitively, one also infers that no one else knows the answer besides John or Mary. This latter inference we certainly want to attribute to exhaustification. The question, then, is, whether exhlp-r.tex; 22/09/2003; 9:00; p.3

4 4 Robert van Rooy and Katrin Schulz one has to distinguish between two different interpretation mechanisms to be at work, or whether exhaustification is also responsible for the former, scalar, inference. In the absence of evidence to the contrary we think it is better to look for a unifying account of these inferences. As we will see in this section, there seems to be in general a close connection between exhaustivity and conversational implicatures, especially inferences analyzed as due to the first submaxim of quantity. (4) Three students. This seems a quite uninteresting example. Normal use allows to conclude that exactly three students know the answer. So what s new? The interesting question here is how the process of exhaustification interacts with the meaning of determiners. Compare: (5) At least three students. In contrast to (4), in (5) 3 there is no inference that only three students know the answer. For the latter example, there is no limitation on the set of students that know the answer at all. However, it is excluded that anyone else but students know the answer. Some kind of exhaustive inference is still observed. So there is a difference between (4) and (5) exhaustification is sensitive to. Also in (6) we don t observe a limitation on the number of students that know the answer. (6) Students. For the sentence to be true, it has to be the case that at least some students know the answer, but nothing more can be inferred about the exact number. Notice that also here it can be inferred that, besides students, no one else knows the answer. In contrast, (6) is normally understood as conveying additionally that not all students know the answer. (7) Most students. Here, again, we meet a classical scalar implicature. How can these differences be explained? We will propose in section 3.3 that exhaustivity is sensible to certain properties of the determiner, namely whether it is 3 In case at least modifies the numeral. At least can also occur as particle, with a syntactical behavior similar to even. We cannot discuss this occurrence in the present paper. Readers who have problems getting the exhaustive interpretation for (5) should try The professor and at least three of her students know the answer. exhlp-r.tex; 22/09/2003; 9:00; p.4

5 Exhaustive Interpretation 5 weak or strong i.e., whether it directly introduces a discourse referent or that it has to be constructed afterwards The Context-dependence of Exhaustivity The examples discussed above show how exhaustive inferences change depending on the answer given. Interestingly, exhaustivity can also be quite ambiguous with respect to the same answer (and the same question form). First of all, it seems that exhaustification should sometimes not be applied to an answer at all. A typical example is dialogue (8). (8) Paul: Who has a light? Paula: John. Here, Paula s answer is normally not understood as John is the only one who has a light. We call this interpretation of answers, in contrast to the one discussed until now, their mention some reading. Actually, it seems that no additional information besides the semantic meaning is conveyed. This absence seems to occur in those contexts where the hearer (i.e. questioner) is intuitively not interested in the exact specification of predicate P and the semantic meaning of the answer already provides him with all the information he needs. 4 Except for the mention all and mention some readings, there also seem to be situations with intermediate exhaustive interpretations. In such situations this operation is present, but its use seems to be limited in one way or another. Perhaps the most observed limitation is one of domain restriction. There are contexts in which an answer to a question with this questionpredicate specifies those and only those individuals that have property 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 where it is interpreted as John and Mary are the only employees of Mr. Smith who were at the meeting yesterday. There may have been others at the meeting besides employees of Mr. Smith, but nothing is inferred about them. For the choice of interpretation it seems to be relevant, again, what is commonly known 4 Still, one could argue that also here an additional inference takes place: one of the best ways to get a light is from Paula. It is not unreasonable to assume that this inference is due to exhaustive interpretation. exhlp-r.tex; 22/09/2003; 9:00; p.5

6 6 Robert van Rooy and Katrin Schulz about the information Mr. Smith wants to get. It may also be the case that it is mutually known that Mr. Smith would like to know whether one of his rivals from other companies was at this meeting. In that case, the above discussed form of limited exhaustification doesn t take place. Exhaustification is limited in other ways in so-called scalar readings of answers (cf. Hirschberg, 1985). As in the example above, also here exhaustivity seems to apply only to parts of the question predicate. Imagine Paul and Paula playing poker. (10) Paul: What cards did you have? Paula: Two aces. Here, Paul will interpret her answer as saying that she didn t have three aces or some additional two kings (a double pair wins over a single one). Still, the answer intuitively leaves open the possibility that Paula additionally had, for example, a seven, a nine, and the king of spades. Just as in the previous case, also here Paul s interest for information is different from the case in which an answer gets a mention-all reading. He is not interested in the exact cards that Paula had. He wants to know, however, how good (with respect to an ordering relation induced by the poker rules) Paula s cards were. And he takes Paula s answer to be complete in this respect. One could try to model the chameleon-like character of exhaustivity in terms of contextual cancellation. The problem is, that (i) it is absolutely unclear what serves as cancellation trigger here. For instance, we could take exhaustification as a conversational implicature, as suggested by Groenenedijk & Stokhof (1984). As mentioned above, Grice claimed these inference to be derivable based on certain assumptions about the speaker, such as his rationality and cooperativity etc. If, Grice goes on, additional information in the context negates these assumptions and hence, they cannot be consistently made, then also the inferences based on them will not be derived and the conversational implicatures are cancelled. Intuitively, nothing seems to characterize the contexts discussed in this section that seems to qualify as contradicting Gricean assumptions. And (ii) at least in examples (9) and (10) some kind of exhaustive inference is going on - hence general cancellation of exhaustive interpretation cannot account for those cases. On the other hand, intuitively, the chosen kind of exhaustive interpretation in the cases discussed above seems to correlate with what kind of information is really of relevance in the specific situations. For instance, in (8) it is clear that for the questioner it is fully sufficient to know of somebody who has a light that she has a light. We will exhlp-r.tex; 22/09/2003; 9:00; p.6

7 Exhaustive Interpretation 7 take this observation seriously and describe in section 4 exhaustivity as depending on the information the questioner is interested in Other types of questions The examples discussed so far all take question predicate P to be of type s, e, t. There are other types of questions, and answers to them that also show exhaustification effects. We want to mention here at least one kind of inference that we think exhaustivity is responsible for. There is a well-known tendency to interpret conditionalized answers to Yes/No-questions, exemplified in (11), as bi-conditional. (11) Paul: Will Mary win? Paula: Yes, if John doesn t realize that she is bluffing. Thus, Mary will win exactly in case John will not realize that she is bluffing. Intuitively, the same is going on as in the other cases of exhaustive interpretation: the cases were Mary will win are taken to be completely described by the antecedent in the answer. Hence, if we would be able to analyze exhaustivity correctly, we should also be able to account for this inference. 5 There are various theories of exhaustification, or Q 1 -implicatures, around. However, the domain of application differs remarkably from theory to theory. As far as we know, none of the existing theories can account for all the effects discussed above. Moreover, none of these theories gives a satisfying explanation for why the scope of exhaustification should be restricted to the cases that they can actually handle. In this paper a unified approach to exhaustification is presented which is able to account for the whole list of examples discussed so far. As already mentioned, exhaustivity is modeled as a dynamic version of circumscription, in which minimizing predicates plays a crucial role. The idea to describe exhaustivity as minimizing the extension of 5 Interestingly, some linguists have tried to analyze also this inference known as conditional perfection as an implicature due to first submaxim of quantity - though with limited success. Van der Auwera (1997) proposed to account for it by means of scales like if A, then C and if B, then C, if A, then C. For this to work, either the propositions have to be very special or conditionals have to be treated in a special way. For instance, take A, B and C to be logically independent of each other and assume the material implication analysis of conditionals. In that case the inferred If A, then C and not (if A, then C and if B then C) is only verified in situations in which A and C are false and B is true. This would obviously be inadequate. exhlp-r.tex; 22/09/2003; 9:00; p.7

8 8 Robert van Rooy and Katrin Schulz the predicate of the question is due to Groenendijk & Stokhof (1984) (abbreviated by G&S). We will therefore first discuss their proposal. 2. The proposal of Groenendijk and Stokhof Groenendijk & Stokhof (1984) propose to describe the special way in which we interpret answers to (implicit or explicit) Who-questions by the following operation using 2 nd order quantification (assuming that B, the background, is an individual predicate and F, the focus, a generalized quantifier): 6 DEFINITION 1. (The exhaustivity operator of Groenendijk & Stokhof) exh GS = λf λbλw.f (w)(b) B (F (w)(b ) B (w) B(w)) Set-theoretically, the above formula applied to a generalized quantifier and a predicate allows the predicate only to select the minimal elements of the generalized quantifier. To illustrate, assume that Paula s response to Paul s question Who knows the answer? is John and assume that our domain contains 3 individuals: John, Bill and Mary. Analyzing answer John as a generalized quantifier gives λp λw[p (w)(j)], in setdenotation: {{j}, {j, b}, {j, m}, {j, b, m}}. The formula resulting from applying exh GS to λp λw[p (w)(j)] is λp λw.[p (w) = {j}], in set denotation {{j}}. Thus, if John is given as the true exhaustive answer to the question, we can conclude that {j} is the set of individuals that know the answer. Reading term-answers, or focus-marked terms, as generalized quantifiers in combination with the exhaustivity operation defined above, allows us to account for the minimization effects observed in examples as (1), (2), (3), (4) and (7) discussed above. Actually, G&S can do even more. They show that the above stated operator for terms can be generalized easily to multiple terms and sentential answers. 7 Although these results are very appealing, G&S s exhaustivity operator has still been criticized. Bonomi & Casalegno (1993) have argued that G&S s analysis is rather limited because it can be applied only to noun phrases. To 6 As mentioned in G&S (1984), this operator has much in common with Szabolcsi s (1981) interpretation rule for only. Though similar in content, the form in which this latter rule is stated is much more complicated than that of G&S s exhaustivity operator. 7 Their general exhaustivity operator for n-ary terms looks as follows: exh n GS = λf nλb nλw.f n(w)(b n) B n(f n(w)(b n) B n(w) B n(w)). exhlp-r.tex; 22/09/2003; 9:00; p.8

9 Exhaustive Interpretation 9 account for examples in which only 8 associates with expressions of another category, they argue that we should make use of events. We acknowledge that the use of (something like) events might, in the end, be forced upon us. But perhaps not exactly for the reason they suggest. Crucial for G&S s analysis is that (in the extensional case) their exhaustivity operator is applied to objects of type φ, t, t. It is normally assumed that noun phrases denote generalized quantifiers of type e, t, t, which means that denotations of noun phrases are in the range of the exhaustivity operator. However, it is also standardly assumed that an expression of any type φ can be lifted to an expression of type φ, t, t without change of meaning. But this means that after typelifting G&S s exhaustivity operator can be applied to the denotation of expressions of any type, and there is no special need for events. The only restriction is that one always has to assume the answer-form as providing a semantic object of the type φ, t, t to make exhaustivity work. The reason that we still might need (something like) events is that extending exh GS to answers of questions as What did you do last summer?, is not trivial if we assume that properties are modeled as functions from worlds to sets of individuals. It seems we need more fine-structure, and making use of events may be one way to achieve this. Thus, if exh GS cannot straightforwardly be applied to expressions that denote (before type-lifing) n-ary properties, this is not so much a problem for the operator by itself, as it is for the way in which these properties are modeled in standard possible world semantics. Although Bonomi & Casalegno s (1993) criticism doesn t seem to apply, G&S s analysis faces some other limitations. First, it is quite obvious (as also noticed by themselves) that they cannot account for mention-some readings, domain restriction, and scalar readings. This is inevitable, because, by taking only the semantic meaning of the explicit predicate of the question and the term in the answer as arguments, their input to exhaustification is not sensible enough to account for differences that can occur involving the same question-predicate and the same answer. Second, even for the mention-all case their analysis makes too strong predictions. We have, for instance, the following problem (also noticed in G&S, 1984, pp ). If we allow for group objects, interpret [know the answer] W as a distributive predicate 9 and John and Mary in (12) as the group j m, then the exh GS -reading of (12) claims [know the answer] W to be the set {j m}. 8 They discuss only, but, of course, this problem occurs with the same force in connection with exhaustified answers to questions of other types. 9 A predicate P with domain D is distributive in a set of models W if it holds in W that ( x, y D)(P (x) P (y) P (x y)). exhlp-r.tex; 22/09/2003; 9:00; p.9

10 10 Robert van Rooy and Katrin Schulz (12) Paul: Who knows the answer? Paula: John and Mary. Because [know the answer] W is distributive, this can t be fulfilled in any world: there can be no model w of the language where j m [know the answer] W (w) but j [know the answer] W (w). Hence, we end up with the empty information state. 10 Also in another sense exh GS is too strong. It excludes not only everything besides the things explicitly given by the answer from the extension of P, it also minimizes the denotations of the given terms in the answer. For instance, as already mentioned, from exh GS applied to (7) Most students we will additionally infer that not all students know the answer. That s a very welcome result for this example, because we get the scalar implicature associated with most as intended. But, unfortunately, we get this kind of minimization also for (5) At least three students and (6) Students, and here we don t want it. 11,12 Moreover, applied to John or Mary or both know the answer, exh GS wrongly excludes the possibility that both of John and Mary fulfill the predicate. Finally, negation is a problem for exh GS. The simplest problematic case is this: the answer John didn t know the answer to question Who knows the answer? is wrongly predicted to mean that nobody knows the answer: the smallest extension of predicate Know the answer such that the answer can be true is the empty set. The aim of this paper is to overcome the difficulties mentioned above. 10 There is a solution to that problem, already sketched by G&S, ibid. For independent reasons one is driven to allow the interpreter to choose freely between a distributive and non-distributive reading for predication to plural objects. If one additionally assumes distributive predicates to be true only of atomic objects and interpret John and Mary as a group, (12) can only have the semantic meaning x j m : P (x). Minimization of P relative to this answer doesn t give rise to complications - even without respecting meaning postulates. Later on (section 3.2) we will propose another solution. It has the advantage to carry over to another kind of problem that the answer sketched here cannot capture. 11 This problem has also been noted by G&S themselves. 12 The reader may object that this strict minimization is too strong also for the cases where we observe an upper bound implicature. One might argue that for (7), Most students, for instance, we only exclude the all case, and we don t observe a further minimization. Hence, one should make the mechanism sensible to which stronger alternatives are lexicalized. We tend to think that this is not the case (why should the meaning of most be just more than half?), although we have no clear intuitions here. A second possible objection is that whatever one chooses as lower boundary of upward monotone quantifiers, one will end up with exactly this value after exhaustification. For instance if some would be interpreted as more than one, exhaustification would give for Some men the result exactly two men. But this problem becomes much less acute if one assumes the lower boundaries to be vague. exhlp-r.tex; 22/09/2003; 9:00; p.10

11 Exhaustive Interpretation 11 We claim that this can be done without radically changing the basic idea behind G&S s exhaustivity operator. What do we understand this basic idea to be? G&S take the observation seriously that exhaustive updates are non-monotone with respect to the given answer: certain inferences of an exhaustive interpretation are lost when additional information is given by the answer. Look at the following possible responses to Who knows the answer?: (13) (a) John and Mary. (b) John or some other people, too. (c) John, if not Mary too. (d) John and perhaps also Mary. In every example, some inferences that can be drawn after updating one s information state with the exhaustive reading of only the nonitalized part are no longer present if the answer is extended by the italic part. From the exhaustive interpretation of the answer John one can conclude that Mary doesn t know the answer. This inference, however, is not conveyed by any of the examples in (13). We face here a classical scheme of non-monotonic reasoning: the absence of information is made meaningful. From the fact that the speaker didn t claim that p holds for some proposition p the interpreter infers that p is not the case - negation as failure. The specific linguistic twist given to this well-known scheme is that exhaustivity is negation as failure in the message. Closure operators implementing negation as failure have been thoroughly investigated in logic and artificial intelligence (AI) under the heading of non-monotonic reasoning. And we can make use of this work. In this paper we will address the problems for G&S s exhaustivity operator as discussed above subsequently in a number of steps. We will extend G&S s notion such that it can (1.) respect meaning postulates; (2.) account for the observations made with respect to the quantifiers; (3.) deal with modal and disjunctive answers; and (4.) predict correctly for negative answers. Afterwards we come to the final point: making exhaustification context dependent. Before we change G&S s analysis of exhaustification, we will first reformulate it by making use of a standard method of non-monotonic reasoning: circumscription or interpretation in minimal models. exhlp-r.tex; 22/09/2003; 9:00; p.11

12 12 Robert van Rooy and Katrin Schulz 3. Exhaustivity as dynamic Circumscription 3.1. Circumscription Only a few years before Groenendijk and Stokhof finished their dissertation, McCarthy impressed the artificial intelligence community by introducing Circumscription, one of the first formalisms that can account for certain non-monotonic inferences. McCarthy s goal was to formalize common sense reasoning. More specifically, Circumscription was intended to solve the qualification problem: if we would use classical logic to derive every-day conclusions we would need an impracticable and implausible (McCarthy, 1980, p. 145) number of qualifications in the premisses. For instance, if one wants to predict that if we would throw our computers out of our windows, they would smash on Nieuwe Doelenstraat, one would have to specify that gravitation will not stop working, the computers will not spread their wings and fly away etc. - in short: nothing extraordinary will happen. The solution McCarthy proposes is to strengthen the inferences one can draw from a theory by adding to the theory the assumption that nothing abnormal is the case that is not explicitly mentioned in the theory. Or, to restate it somewhat more abstractly, the extension of certain predicates (the abnormality predicates) is restricted to those and only those objects that have the property explicitly given by the premises. To come back to the example above, without additional information about abnormalities in the gravitation of the earth we will just assume that it will work as it normally works. McCarthy (1986) formalizes this idea 13 by defining a syntactic operation on a sentence that maps it to a new second order sentence in the following way. DEFINITION 2. (Predicate Circumscription) Let A be a second order formula and P a predicate of some language L. Then the circumscription of P relative to A is the formula CIRC(A, P ) defined as: CIRC(A, P ) := A P (A[P /P ] x[p (x) P (x)]). where A[P /P ] describes the substitution of all free occurrences of P in A by P. Looking at this formalization of Circumscription, our reader will immediately recognize the following striking fact: G&S s exhaustivity 13 To be very precise, it is a simplified version of one of his formalizations. exhlp-r.tex; 22/09/2003; 9:00; p.12

13 Exhaustive Interpretation 13 operation is nothing more than an instantiation of McCarthy s predicate circumscription! One just takes the circumscribed predicate to be the predicate of the question and the circumscription to be relative to the proposition one gets by applying the generalized quantifier to the predicate of the question - or simply the proposition of the answer. This parallelism has been noticed first, as far as we know, by Johan van Benthem (1989). 14 There is a minor technical difference between predicate circumscription and G&S s exhaustivity operator which should not be overseen. The latter takes besides a predicate a generalized quantifier as argument, the answer without the predicate in question. Circumscription takes the predicate and a sentence - hence relies on less information. In this respect it is a more spartanic notion. 15 This is especially welcome because it spares us the trouble to explain how to get the generalized quantifier which exh GS needs in case a sentential answer is given. Even though, as the discussion above shows, the formalization G&S use for the description of exhaustive interpretation has been studied before, there is one point in which their work is without question original. This is the way in which they bring the formalism to use to describe a concrete linguistic phenomenon. A prominent problem in applying nonmonotonic reasoning techniques and especially circumscription is the question how to instantiate the variables, i.e. to determine to which predicates circumscription should be applied. 16 G&S link the to be circumscribed variable to the information structure of the discourse: it s the question predicate (or the background) that is circumscribed 14 We only know of one (other) attempt to use circumscription for (some of) the data we discuss in this paper: by Wainer in his dissertation (1991). Wainer contrasts two ways of doing so: (i) a direct method, according to which a sentence is represented as usual, and the additional inferences are due to predicate minimization, which corresponds to the use made of circumscription in the present article; and (ii) an indirect way, according to which the potential implicatures are coded explicitly (by means of McCarthy s abnormality predicates). Although G&S s exhaustivity operator is not mentioned in his dissertation and he does not relate circumscription to the information structure of the discourse, Wainer noticed some of the same problems for his first method as we discussed in section 2. For this reason he opts, in the end, for the second (Gazdarian-like) indirect method of deriving implicatures, although he admits that the first method is intuitively more appealing. One of the main goals of this paper is to show that the direct method can be pushed much further than Wainer assumed. 15 See Rooth s (1996) related discussion where he compares his own alternative semantics to interpret focus with the structured meanings approach: the former relies on less information. 16 Notice that Circumscription can also be applied to a sentence and two predicates. We think this is important for the analysis of certain implicatures in complex sentences, but we won t delve upon this issue here. exhlp-r.tex; 22/09/2003; 9:00; p.13

14 14 Robert van Rooy and Katrin Schulz and the circumscription is relative to the newly provided information. The circumscription formula has a semantic pendant: interpretation in minimal models. Here, the semantics for classical logic is enriched by defining an order on the set of models: a model v is said to be more minimal than a model w with respect to some predicate P, v < P w, in case they agree on everything except the interpretation they assign to P and here it holds that P (v) P (w). It can be shown that the P -minimal models of a theory A, hence the set {w [A] W v [A] W (v < P w)}, are exactly the models where the circumscription formula CIRC(A, P ) holds. Once one starts to think of circumscription from a semantic point of view, the idea to abstract away from the specific kind of ordering this mode of interpretation is based on readily suggests itself (cf. Shoham, 1988). Generalizing interpretation in minimal models has stimulated the development of one of the best studied branches of non-monotonic reasoning: reasoning in preference structures. The new non-monotonic inferences that can be drawn from a theory are then simply the set of sentences which are true in the most preferred worlds that verify this theory. Actually, semanticists are quite familiar with this kind of reasoning: the Lewis/Stalnaker approach to conditional sentences also makes crucial use of such preference structures The basic setting It s the semantic variant, i.e. interpretation in minimal models, that we will prominently use in this paper. Definition 3 gives our basic notion of exhaustive interpretation. DEFINITION 3. (Exhaustive interpretation - the basic case) exh W std(a, P ) {w [A] W v [A] W (v < P w)} To illustrate the working of this interpretation function, let s go back to example (11) here repeated as (14). (14) Paul: Will Mary win? Paula: Yes, if John doesn t realize that she is bluffing. In this case the question-predicate P is of arity But this means that v < P w iff v is exactly like w, except that whereas w makes P true, 17 We assume that the extension of an n-ary predicate P n is the set of n-ary tuples that verifies sentence P n ( x). If P 0 is true in w, it denotes { }, otherwise. exhlp-r.tex; 22/09/2003; 9:00; p.14

15 Exhaustive Interpretation 15 v makes it false. Now it can be checked that exh W std (A P, P ) is true only in those worlds where either both A and P are true, or both A and P are false. Worlds where A is false and P true are ruled out because there are other worlds that verify A P that don t make P true. Thus, by applying exh W sd the conditional answer gets a bi-conditional reading. 18,19 There are several reasons for our decision to think of exhaustive interpretation primarily from a model-theoretic point of view instead of as a syntactic operation on classical second-order theories. First, it allows us to describe exhaustification as a modus of interpretation 20 which enables us to separate the semantic from the pragmatic information conveyed by a sentence. Second, the model-theoretic perspective allows us to make generalizations later on in the paper which would not have occurred to us otherwise. But the most important reason to adopt this model-theoretic perspective on exhaustive interpretation is that it allows us to improve on G&S s account in a way that can be explained best by going back to one of the problems we noticed in connection with their proposal. Remember our earlier discussion of applying exhaustification to distributive predicates (enriching the domain with group objects). We discussed it using our very first example, repeated here as (15). (15) Paul: Who knows the answer? Paula: John and Mary. Let us once more consider our previous argumentation, but now using the model-theoretic pendant of circumscription. Again, Paula is understood as talking about a plural object j m. We first select only models were Paula s answer is true, hence, both John and Mary are among the people who know the answer. Then, we keep only the minimal worlds verifying the answer with respect to the question predicate. At first one may think that these are the worlds where the extension of knows the answer consists only of j m. However, such models are not 18 This prediction is, of course, already made by G&S. 19 As already suggested earlier, the meaning of exhaustification is closely related with the meaning of only. However, it is standardly assumed that a sentence like Only John F is sick does not semantically entail that John is sick; this inference is due to a presupposition or implicature. This suggests that with respect to background property P, Only A should be interpreted as {w W v [A] W (v < P w)}. It is interesting to observe that with respect to background 0-ary property P, the conditional P, only if A is now predicted to have the same meaning as the conditional A, if P. We don t know of any other analysis of only that has this pleasing result. 20 G&S complained already that this is missing in their approach. exhlp-r.tex; 22/09/2003; 9:00; p.15

16 16 Robert van Rooy and Katrin Schulz acceptable models of the language. The predicate is distributive and already G&S account for this by letting meaning postulates impose restrictions on the class of proper models. Hence, there will be only models in the set we minimize over where the question predicate is distributive and so will be the minimal models. But then we obtain the right result! The minimal models are such that besides the plural object j m also j and m are in the extension of question predicate Know the answer. But why can we solve this problem just by taking exh W std instead of exh GS? Didn t we claim above that as far as exh G&S corresponds with CIRC both notions are interchangeable? Indeed, there is nothing wrong with this result. The point is that the completeness claim is made for circumscription with respect to a whole theory, i.e., every information available. Exh GS, however, only circumscribes the answer, ignoring everything else known about the predicate, in particular its meaning postulates. Therefore, exh GS considers alternative extensions for P that are not proper for this predicate. Only if we move the meaning postulate under the scope of circumscription, i.e. in the answer, we obtain the same result we get by interpreting in minimal models. To sum up, distributive predicates show that circumscribing just the answer may not be enough. The scope of the operation has to be extended - in particular if meaning postulates have to be respected. We accounted for this in terms of model restriction 21 and added a third argument to exh std : a set of models, W, with respect to which minimality is defined. Hence, we will talk about the exhaustive interpretation of the answer A minimizing the question predicate P with respect to a set of models W. The variable W makes our interpretation function very context dependent. If W is understood as the respective common ground then all the information presented there will influence what counts as a minimal model in a particular context. It still has to be studied to what extend exhaustive interpretation is context sensitive in this sense. Actually, there are even more striking examples in favor of our approach to a notion of exhaustivity which respects meaning postulates and they do not rely on such questionable premisses as the group analysis of Paula s answer. The proposed formalization allows us also to account for some puzzles connected with the meaning of only. For limitations of space, however, we cannot discuss such examples in detail here The alternative would be to explicitly encode additional relevant information in the circumscribed theory. 22 One of the famous examples discussed in this connection is the following from Kratzer (1989). exhlp-r.tex; 22/09/2003; 9:00; p.16

17 3.3. Dynamic Circumscription Exhaustive Interpretation 17 Next, we consider the problem with the minimization of quantifiers. Let s start with repeating the observation we made in section 1.2. concerning the examples (4), (5), (6), and (7), repeated here as (16) (16) (a) Three Students. (b) At least three students. (c) Students. (d) Most students. It is standardly assumed in generalized quantifier theory (adopted by G&S) that three students has the same semantic meaning as at least three students. Because standard exhaustification takes only the semantic meaning into account, it predicts that both give rise to the same readings. However, sentences in which the former occur seem to give rise to an at most implicature, while the latter do not. Something similar has been observed comparing (16c) and (16d). (16d) should be minimized, but (16c) should not. So, again there is a difference exhaustive interpretation is sensible to which our account fails to observe. Different perspectives are possible on this dilemma. An interesting proposal is made by Zeevat (1994), who incorporates the implicature most comes with in its semantics. In this paper, however, we stick to the traditional analysis of this determiner. Others have proposed that (i) Paula: 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 the response of the lunatic. This problem is discussed extensively in the literature on only. We just want to note here that given a meaning postulate stating that if one paints a picture, then one also paints all of its parts, and assuming that the semantic meaning of only is the exhaustive interpretation of the sentence with respect to the background, our formalization of exhaustivity can straightforwardly account for the inappropriatenes of the lunatic s reaction. In making the set of events in which Paula did participate yesterday as small as possible, the possibility that Paula cooked a dinner for Paul will be excluded - because there would be a smaller alternative history of what Paula did yesterday where no cooking took place and that still would make Paula painted a still life true. But minimalization will not be able to exclude that Paula painted also the parts of her picture - because no alternative world will model such a case. Notice, that there may be different possibilities of what the parts of the still-life were - Paula didn t say anything about what she painted. Hence it may be the case - even after exhaustification - that Paula actually painted a still life containing apples. Therefore, the lunatic cannot claim that Paula made a false statement by pointing out that she was actually painting apples. We hope to discuss the effects of taking meaning postulates for the meaning of only in more detail at another place. exhlp-r.tex; 22/09/2003; 9:00; p.17

18 18 Robert van Rooy and Katrin Schulz at least cancels the at most implicature or that exhaustification should not apply to expressions containing this modifier or to bare nominals. However, we argued in section 1.1 that also for these expressions some form of exhaustification still takes place. Hence, general cancellation is no option. We will propose instead that exhaustification does take place but that it will not have any effect on the claimed number of students who know the answer. There is a semantic distinction that can be made responsible for the differences in the exhaustive meanings: the distinction between strong and weak determiners. We propose that NPs with weak determiners such as a man, linguists, some 1 girls, five girls or at least five girls are not affected by minimization, NPs with strong determiners such as all ducks, most students, and some 2 girls are. Adopting a standard assumption of dynamic semantics (e.g. Kamp & Reyle, 1993), we treat only the latter type of NPs as two-place generalized quantifiers. Weak quantifiers, instead, are not and directly introduce discourse referents. For anaphoric reference to strong quantifiers, discourse referents have to be constructed afterwards from the intersection of nucleus and restrictor. The difference can then be modeled easily by taking the directly introduced discourse referents out of the scope of the circumscription. Hence, what we have to do now is combining circumscription with dynamic semantics. We will not introduce full-blooded dynamic semantics but restrict ourselves to some of its essential features, leaving the exact implementation to the reader s favorite dynamic theory. To implement the concept of discourse referents in our setting, we no longer interpret on worlds but on world-assignment pairs. We will call such pairs possibilities and we will use i, i, i 1,... to refer to them. Discourse referents are then interpreted as fixed variables of the assignments. Instead of ordering worlds, we now have to order possibilities, but this is simply done by extending the definition of the order between worlds by the condition that now also the assignments have to be identical. Finally, we need an update function σ[ψ] from information states σ (classes of possibilities) and formulas ψ to new information states, to keep track of the assignments to introduced discourse referents, defined in the standard way. Now we come to the definition of dynamic exhaustification. DEFINITION 4. (Dynamic Exhaustification) exh σ dyn(a, P ) {i σ[a] i σ[a](i < P i)} This formulation adds nearly nothing new to exh W std. The information state σ takes the place of our old variable W, which means that exhlp-r.tex; 22/09/2003; 9:00; p.18

19 Exhaustive Interpretation 19 all previously collected information will be respected in the circumscription. The small but crucial detail is the extension of the ordering to assignments. This will make it more dfficult for possibilities to be comparable in case the answer introduces discourse referents. In turn, the notion of minimality becomes weaker compared with a parallel sentence with a generalized quantifier. This will become much clearer after discussing some examples. Take first (16c). Assume the following information state: σ = {i 1, i 2,..., i 8 }, where i k = w k, g k. In all possibilities we have the same interpretation for students, the set {a b, a c, b c, a b c}. For the interpretation of know the answer we have: K(i 1 ) = {a, b, c, a b,...}, K(i 2 ) = {a, b, a b},..., K(i 4 ) = {b, c, b c},..., K(i 8 ) = - we simply take every possible distributive set given the three atoms {a, b, c} - hence, predicate K is assumed to be distributive. After updating with the semantic meaning of the sentence Students know the answer, ( X)(S(X) K(X)), we end up with an information state σ containing successors of the possibilities i 1, i 2, i 3, and i 4 whose variable assignments now are defined for X, the new introduced discourse variable. 23 In σ there will be a possibility for every possible mapping of X to a group of students that know the answer in one of the worlds w 1, w 2, w 3, and w 4. So σ contains, for instance, the possibility w 1, X : a b, because the object a b is in the extension of K in w 1. However, w 2, X : a b c will be no element of σ. Given the assignment X : a b c, the answer wouldn t be true in w 2. The following tableau lists all possibilities in σ plus the way they are ordered by < P, where means that the first possibility is P -smaller than the second. w 1, X : a b c w 1, X : a b w 1 X : a c w 1, X : b c w 2, X : a b w 3, X : a c w 4, X : b c Now, we collect the minimal elements of this ordering (marked by a box in the picture) and get the set: { w 1, X : a b c, w 2, X : a b, w 3, X : a c, w 4, X : b c }. Given this interpretation it is still possible that all students know the answer - even though it would be excluded that anybody else than students know the answer. The reason is that after update there is still a possibility in the information state that takes the world to be w 1. And this is so because there will be 23 The others are excluded by the truth conditions of the answer. exhlp-r.tex; 22/09/2003; 9:00; p.19

20 20 Robert van Rooy and Katrin Schulz no possibility where the extension of P is smaller than in w 1 and which still maps X to a b c. Such a possibility wouldn t make the answer true. Hence, we correctly predict no minimization for newly introduced discourse referents. 24 However, doing the same calculation with Most students, the example (16d), the situation looks different. Because strong determiners don t immediately introduce discourse referents, there will be no discourse information that makes a successor of i 1 incomparable with every successor of a posibility where K has a smaller extension than in w 1 and hence in i 1. Thus, after exhaustive interpretation we end up with a new information state containing only sucessors of i 2, i 3 and i 4. The possibility that all students know the answer is excluded. So, we see that discourse information, coded in the assignments, can effect the outcome of exhaustivication. The more information the assignments contain the more difficult it is for possibilities to be, after update, comparable, which in turn weakens the notion of minimality. Finally we will calculate the results of exhaustive interpretation of (16a) and (16b). Within dynamic frameworks (e.g. Kamp & Reyle, 1993) it is standard to represent numerals as follows: [[two men know]] = ( X)(M(X) cardinality(x) = 2 K(X)). Although this formula introduces a discourse referent which denotes a set of exactly 2 individuals, truth-conditionally the sentence is still predicted to have an at least reading: if 3 men know, there is still a set of individuals of two men that know. Thus, from a truth-conditional perspective we could have represented the sentence as well by ( X)(M(X) cardinality(x) 2 K(X)). Dynamically, however, the two formula are not equivalent: the former introduces discourse referents that denote groups of exactly 2 individuals, while the groups introduced by the latter formula might be larger. As a consequence, if we apply exh σ dyn, the former formula gets the exactly 2 reading, while the latter does not. This suggests that the former one correctly represents the numeral sentence, while the latter formula is the natural representation of a sentence like At least 2 men know. And indeed, that was proposed by Kadmon (1985) for related, but still somewhat different reasons. Hence, adopting Kadmon s analysis of the two determiners allows us to account for their different behavior under exhaustification. To sum up the discussion in this section: the behavior of determiners is not a problem that forces us to give up the circumscription account for 24 Notice, by the way, that after exhaustification, if we refer back to the newly introduced discourse referent, we are talking about all students that know the answer. This is on a par with intuition. exhlp-r.tex; 22/09/2003; 9:00; p.20