JOURNAL OF SEMANTICS Volume 16 Number 3
CONTENTS JEAN-PIERRE KoENIG AND GAIL MAUNER A-Definites and the Discourse Status of Implicit Arguments
207
REGINE ECKARDT Normal Objects, Normal Worlds, and the Meaning of Generic Sentences
237
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A-definites and the Discourse Status of Implicit Arguments JEAN -PIERRE KOENIG and GAIL MAUNER
Abstract This
Understanding discourse requires determining which eventualities are described, keeping track of the entities which participate in those eventualities, and determining the roles they play. In many cases, our understanding can rely on the morphosyntactic mention of participants. But in others, some of the participants involved are not explicitly mentioned. This paper is concerned with the effect this difference in morphosyntactic expression has on the semantic representation of parti"' cipants. More specifically, the central issue on which we focus is whether the semantic interpretation of arguments that are not morphosyntactically expressed differs from that of explicit indefinite arguments with which they can be paraphrased salva veritate. Our claim is that once the discourse potential of expressions is taken into account, the semantics of implicit arguments and their indefinite explicit paraphrases do differ. We show that this difference supports the fundamental hypothesis of Discourse Repre sentation Theory (hereafter DRT, see Kamp 1981; Heim 1982; and K.amp & Reyle 1993, among otheci) that the semantic information: that sentences
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paper focuses on the semantics of implicit arguments and compares it with that of explicit indefinites with which they can be truth-conditionally paraphrased. It is shown that once the discourse-potential of expressions is taken into account, the semantics of implicit arguments differs from their indefinite explicit counterparts. They are shown to be semantically identical to a particular kind of non-quantificational NP (a-definites} which are characterized by their inability to serve as antecedents for future reference. A model of this behavior of implicit arguments, it is argued, follows naturally from the underlying assumption of Discourse Representation Theory that semantic representations must include two kinds of information. a set of available discourse markers and a set of predicative conditions. Because implicit arguments satisfy a predicate's argument positions -without introducing discourse markers into the Discourse Representation Structure of a sentence, they cannot serve as the antecedent of definite pronouns. When they do enter into anaphoric relations it is not through discourse markers equality clauses, but instead is the result of either lexical identification of variables (via semantic detransitivization or meaning postulates} or of an accommodation process which involves bridging and/or factoring interferences.
208 A-definites and the Discourse Status
of
Implicit Arguments
encode is two-pronged. It consists of both predicative conditions discourse markers.1
and sets of
I IMPLICIT AR GU MENTS
A ship was sunk b. A ship sank c. A ship was sunk by someone d. ...to collect settlement money from the insurance company.
(r) a.
Psycholinguistic work supports this common linguistic hypothesis.The data presented in Mauner, Tanenhaus & Carlson {I99S) suggest that the verb sunk in the short passive sentence in example (ra), but not the verb sank in the intransitive sentence in example (Ib), includes an agent in the representation of its argument structure.Mauner et al. show that subjects find less sensical-and take longer to process-rationale clauses like (rd) when they follow intransitive sentences like (Ib) than when they follow short or agentive passive sentences like (ra) and (xc) respectively. They reason that this behavioral difference between surface similar intransitive and short passive sentences stems from the fact that only the short passive verb sunk in (1a) includes an agent in its argument structure that can serve ·
1 We are not the 6rst to notice that short passives and agentive passives whose agent adjunct is by
so�� are
not semantically equivalent. McCawley {1988) argues that the twO structlll'es have different semantic interpretations and proposes that the logical subject of short passives is an unspecific pronoun which never surfaces {UNSPEC in his terminology). By contraSt to McCawley's model-theoretic distinction, we focus in this paper on the discourse-potential differences between implicit arguments and their explicit indefinite truth-conditional paraphrases.
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Linguists typically assume that the verb sunk in the short passive sentence in {Ia) includes an implicit actor argument {see Roeper I987; Williams 1987, among others).The representational nature of this implicit argument varies somewhat from framework to framework, but this variation is orthogonal to the issues addressed in this paper.We assume as our starting point that implicit arguments are included iri. the argument structures of verbs that introduce them, however one represents them, as theta-grids {Chomsky 1981), argument-structure lists of various sorts (the ARG-ST list in HPsG) (Davis & Koenig 2000; Manning & Sag 1998), or as a set of positions within a logical structure (Van Valin & Lapolla 1997). For our purposes, what is crucial is that implicit arguments are part of the representation of lexical items that would license their explicit morphosyntactic expression, even though no explicit material in either the surface string or its structural analysis might clue readers in to their presence.
Jean-Pierre Koenig and Gail Mauner
209
as the antecedent of the unexpressed PRO subject of the rationale clause. Since readers must anchor the anaphoric PRO subject of collect, the absence of a possible anchor when the rationale clause follows a non-agentive
intransitive verb like sank leads to niore anomaly judgments and increased processing time. Mauner & Koenig (1999) and Mauner & Koenig (to appear) provide evidence for the lexical source of implicit agents in the argument structures of (short) passive, but not intransitive and middle verbs. They show that (short) passive and middle verbs differ ln. the availability of an implicit
The \ra5e was sold immediately b. The vase sold immediately c. The vase was sold by someone d. ...to raise some money for charity. (3) a. #The vase was sold immediately, but nobody sold it. b. #The vase sold immediately, but nobody sold it. (2)
a.
Despite their conceptually equivalent agent entailments,· the short passive and middle forms of sell differ in the availability of an agent for the interpretation of the unexpressed PRO subject of rationale clauses, just like the short passive and intransitive sunk and sank did. Only the former can felicitously combine with the rationale clause in (2d). This intuitive contrast was confirmed experimentally. Main verbs of rationale clauses took longer to process when they followed middle sentences than whey they followed short pas�ive sentences. Furthermore, when rationale clauses were preposed, the sanie processing differences showed up as soon as the main clause verb was recognized (see Mauner & Koenig, to appear).A plausible interpretation of these results is that the middle and short passive forms of sold differ in their argument structure, even though they both describe events· which include an agentive seller participant. Linguistic and psycholinguistic evidence thus converge to suggest that whether an agent is encoded in the argument structure of a short passive (intransitive/ middle) verb is lexically encoded in the verb's entry and immediately affects sentence processing. The common presence of an agent in the lexical representation of sunk in the short and agentive passive sentences in (1a) and (1c) raises the question of the semantic relation between these two sentences.Both entries for sunk
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agent, when verbs are equated for whether or not they describe events whose occurrences entail an agentive participant. Consider examples (2a) and (2b).Both forms of sold describe events in which an agentive participant sold the vase. Vases do not literally sell themselves. This intuition is confirmed by the semantic anomaly of both (3a) and (3b) where an agent entailment is explicidy contradicted.
210 A-defin.ites and the Discourse Status of Implicit Arguments
(4) (3x) (3y) (shp '(y)/\sink '(x, y) ) Two classically truth-conditionally equivalent sentences can differen tially contribute to the on-going representation of the discourse. The truth conditional equivalence of (ra) and (1c) is therefore no guarantee that their 'meaning'-i.e. their conventional contribution to the interpretation of their utterances-are equivalent. As we show in this paper, this distinction between meaning-as-truth-conditions and meaning simpliciter is emphati-· cally confirmed by the behavioral differences of short and agentive passives in discourse. More specifically, our claim is that once the discourse potential of expressions is included in the de6.nition of the meaning of expressions, the semantics of (1a) and (rc) do differ; the (discourse-related) semantic contribution of implicit agents is distinct from that of their explicit paraphrases. We thus argue for the following two hypotheses. 2 As a reviewer pointed out, this is only true if the sortal gender restriction included in the pronoun someone is disregarded. Of course, this Cllveat does not apply to the formula in (4). See fn. s for additional qualifications.
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include an agent argument and the two sentences are truth-conditionally equivalent; they are true in the same set of models.2 Should we conclude from these two facts that the semantic contribution of implicit agents (and implicit arguments more generally) is identical to that of by-phrases whose complement NP is an indefinite pronoun? Indeed, (1a) and (1c) are often assumed to be semantically equivalent. Their common semantic inter pretation can be roughly paraphrased by the first-order formula in (4) (see Fodor & Fodor 1980 and Dowty 1981, among others). This putative semantic identity between implicit arguments and their explicit indefinite counterparts seems necessary within traditional model-theoretic approaches in which meaning is directly tied to truth-conditions. If the meaning of a sentence equals the conditions under which it is true, then the fact that (1a) and (1c) are true in the same sets of models entails that they are semantically equ�valent. I£ furthermore, the meaning of a (subsentential) expression equals its contribution to the truth-conditions of the sentence in which it is included, the identity of the non-agent related subparts of (1a) and (1c) suggests that the semantic contribution of implicit· and explicit indefinite agents must be identical, too. This traditional position seems confirmed by the rationale clause data we just discussed: implicit and explicit indefinite agents can equally easily control the unexpressed subject PRO of a rationale clause. But, as work in Discourse Representation Theory and File Change semantics has shown (see Kamp 1981 and Heim 1982), traditional truth conditions do not exhaust the semantic contributions of sentences and subsentential expressions.
Jean-Pierre Koenig and Gail Mauner
2II
Implicit arguments are not semantically equivalent to existentially quantified variables. Hypothesis
I
:z. Implicit arguments satisfJ the argument positions of predicates without introducing a discourse marker into the Discourse Representation Structure for sentences in which they occur.
Hypothesis
2
Our main Hypothesis 2 assimilates the semantic contribution of implicit arguments to a subclass of non-quantificatiohal NPS and pronominals discussed in Koenig (1999). We summarize ftrst this work's analysis of non-quantificational NPS. As Koenig suggests, non-quanti£cational NPS or pronominals hl.ve (at least) three main distinct functions within DRT: 1.
They can introduce new discourse markers to which subsequent NPS can refer back;
They satisfy one of a main predicate's arguments; 3· They introduce a restriction on the referent of the discourse marker that they introduce.
2.
These functions coalesce for typical (indefinite) NPS. Consider the simple discourse in example (s).
(s)
A man smiled. He was happy.
a man fulftls all three functions. It satisfies the sole actor argument role of the predicate denoted by smiled. It also introduces a discourse marker to which the pronoun he in the following sentence can
The indeftnite
NP
refer. Finally, it restricts the referent of this discourse marker to be a man. These three functions are perspicuously represented in the linear format for representing Discourse Representation Theory ofZeevat (1989), as shown in the semantic translation for sentence (6a) given in (6b).3 3
To simplify.'&' represents a dynamic version of traditional auth-conditional conjunction. See Eijck. & .Kamp (1997) for details on the various possible semantic interpretations of such 'merge' operaton. The distinction between the introduction of discourse marken and satisfying the argument positions of predicates parallels the distinction between Discourse Entities and Discourse Roles in the psychological model of reading proposed in Garrod & Sanford ( 1990). We leave an explicit discussion of the similarities between Discourse Representation Theory and Garrod & Sanford's psychological model to a more appropriate forum. (See Bosch 1988 for a critical discussion of Garrod & Sanford's model that echoes the issues we discuss in this paper.) van
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THE THREE FUN CTIONS OF NP S AND PR ONOMINALS
212.
A-definites and the Discourse Status of Implicit Arguments
(6) a. A man smiled. b. x& man' ( x) &smiled' (x) c. 'a' translates as '>J'.>tQ (x &P ( x) & Q (x ))' 'man' translates as '.>tv(man' ( v))' 'smiled' translates as '.>tu(smiled' (u))' As
(7) a. He was happy. b. z&z = x& happy' (z) The use of discourse markers as a representational device separate from predicative conditions and quantificational structure thus serves to encode explicitly the varying contributions to the ongoing representation of the discourse of distinct kinds of NPs, in particular the distinction between quantificational, indefinite, and definite NPs. What has not been traditionally recognized is that not all NPS fulfil all three functions or alternatively that not all non-quantificationallexical NPs or pronouns divide up into either indefinite or definite NPS. In particular,. the French subject clitic on (Koenig 1999) and Hungarian Predicate Modifier NPS (Farkas I 997) do not fulfil the first function. •
• German man seems to paralle l French on, as example (i) illustrates (W. Wolck, p.c.� We leave a detailed comparison of these to another occasion. (i) #Man; hat die Priisidentin erschossen INDEF have.PR the president shoot.PPT Er; kam aus Bayern He come.PST &om Bavaria 'Someone; shot the (woman) president. He; comes &om Bavaria.'
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one can easily see, (6b) derives from the semantic translations of a , man, and smiled represented in (6c) through ,8-reduction. Furthern:lore, the semantic translation of a consists of three clauses, each encoding one of the three functions of non-quantificational NPS. The first clause explicitly marks the introduction of a new discourse marker. This discourse marker will serve as an anchor for subsequent discourse markers which are coreferential with it. The second clause restricts the model-theoretic interpretation of this newly introduced discourse marker to entities which are included in the interpretation of the predicate man' (i.e. to men). The final clause restricts it further to those men who smiled, thereby making the referent of the discourse marker x satisfy the sole argument position of the predicate smiled'. The relevance of the new discourse marker x manifests itself in sentence (7a). The pronoun he in it again bears three functions. It introduces a new discourse market z, it constraints its referent to be identical to that of a previously introduced discourse marker x, and forces it to be in the interpretation of the predicate happy'.
Jean-Pierre Koenig and Gail Manner
2.13
Consider the minimal pair in (8)-{9). (8) #On; a assassine la presidente. INDEF have.PR kill.PPT the president ll; etait du Berry, parait-il he be.PST from.the Berry, seem.PR-it 'Someone; murdered the (woman) president. He; comes from the Berry, it seems' (9) Quelqu'un; a assassine la presidente. INDEF have.PR killPPT the president ll; etait du Berry, parait-il he be.PST from.the Berry, seem.PR-it 'Someone; murdered the (woman) president. He; comes from the Berry, it seems' ·
(10)
a. y&president '(y) &y p & murdered'(x, y) &z t&from '( z, b) x&Jrom '( z, b ) b. x &y&president '(y) &y p&murdered '(x, y) &z =
=
=
=
Of paramount importance for the anaphoric behavior of on is the absence of the discourse marker x in (1oa), which quelqu'un introduces in (1ob). This lexical semantic diff erence reflects the fact that on satisfe i s the agent 5
French subject clitic on has several uses which can be roughly translated as English �n, ptopk, or This p:Lper is only concerned with the last use. This :L-definite use is typially described as the indefinite use of on in rnditional gramnurs. See Koenig (1999) for :l review of the evidence th:lt these uses inst=ti:lte different grammatial constructions. Note, fin:illy, wt Koenig ( I 999) c:ills on m 'ulm-indefinite'. We now think the term ':L-definite' more :Lppropri:ltely reflects the fact th:lt the discourse behavior of on lies outside of the opposition between indefinite md definite pronouns. someone.
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QueLJu'un and on are both felicitously translated by the English indefinite pronoun someone. But whereas quelqu'un can be the antecedent of a definite pronoun in a subseqllte'nt sentence, on cannot. For all interests and purposes on is inert in discourse: it cannot serve as the anchor of an anaphoric element (see section 4 for qualifications). We call words or phrases such as on 'a-definites', indicating by that term pronominals or lexical NPS that are neither definite nor indefinite: the status of the discourse marker they introduce is irrelevant, since they do not introduce one.5 We reserve the term 'indefinite' for NPS such as quelqu'un, i.e. to lexical NPS or pronouns which introduce a discourse marker that is not anchored to a previously introduced entity. Representing the difference between a-defmites and indefinites is easy in the linear representation of Discourse Representation Structures proposed in Zeevat (1989) since it explicitly encodes the difference between the various functions of non-quantificational NPS. (1oa) and (1ob) are the two translations, into this semantic metalanguage, of sentences (8) and (9) respectively.
214
A-definites and the Discourse Status of Implicit Arguments
In Chierchia's approach, the intersentential coreference possibilities of indefinites follows from the fact that each sentence includes in its semantic translation a variable which functions as a place-holder for subsequent sentences (p in (ua)). The semantic translation of (7a), namely (rrb), then replaces the 'subsequent discourse' variable p (via ordinary ,8-reduction, cancellation, and a dynamic definition of conjunction which is not relevant -to our purposes), yielding (IIc). The variable x in the second sentence now falls under the scope of the initial existential quantifier. Thus, the 'dynamism' of indefinite reference is handled through assigning widest scope to existential quantifiers at their point of introduction into the discourse. No special representational status is assigned to the semantic translation of non-quantificational NPs. By contrast to the analysis of a-definites we outlined above, no distinction is made between the existential ¥A
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argument position of tuer without introducing a new element into the discourse representation. As a consequence, whereas the discourse marker introduced by il 'he' in the second sentence can be interpreted as coreferential with the· one which quelqu'un introduces through the discourse markers equality clause z = x in (rob), the referent of z must be identified with that of another discourse marker (arbitrarily named t) in (roa). Thus, the lexical properties of on make the intended interpretation of (8) impossible. It is worth noting that this variation in the lexically specified discourse properties of pronouns is expected given DRT's hypothesis that sentence meaning in natural language includes information about 'active/inactive' discourse markers as well as information about the 'world'. If non quantificational NPS and pronominals have the three aforementioned functions, we expect some NPS or pronominals to fulfil some, but not all, of these functions. The contrast between on and quelqu'un provides evidence for the possible dissociation of the role and reference functions of nominals and thus confirms the claim that non-quantificational NPs and pronominals are functionally diverse. Moreover, the discourse inertness of a-definites like on is not as easily modeled in all approaches to the dynamic aspects of meaning. In fact, the behavior of on favors a traditional DRT-style approach over some other approaches that make use of dynamic predicate logic a la Chierchia (r995a). In this latter class of approaches, the protracted availability of discourse markers for coreference reduces to a dynamic interpretation of existential quantifiers. {6a) and (7a), for example, would be translated as (rra) and (ub) respectively, in Chierchia's logic, and their discursive sequence as (uc) (simplifying for expository purposes): (rr) a. >.p3x ( man' (x ) t\Smiled' ( x) A) ) b. >.q ( happy' ( x ) A ¥q ) c. >.q3x( man' ( x ) Asmiled' (x) A happy' (x ) A¥q)
Jean-Pierre Koenig and Gail Mauner
2.IS
of free variables and the fact that these variables correspond to active discourse markers. The distinctive behavior of indefinites and a-definites like French on therefore cannot be modeled: if both receive an interpretation logically equivalent to traditional existential quantification, they must both be interpreted as variables bound by an existential quantifier.6 By contrast, the two-pronged semantics of DRT affords us the representational means of distinguishing between a-definites and indefinites. Responsibility for the existential interpretation of free variables falls upon the embedding function which maps Discourse Representation Structures onto their model-theoretic interpretations. Availability of a free variable for subsequent anaphoric reference, on the other hand, depends on the presence of an accessible discourse marker corresponding to the argument position the variable satisfies.7 Because the two semantic con sequences of the presence of non-quantifi.cational NPs or pronominals are dissociated, the fact that only the first consequence is relevant for a-definites (but not (in)definites) is easily accounted for. It reduces to whether or not the NP or pronominal introduces a discourse marker, as the contrast between the semantic translations of a-definite on and indefinite quelqu'un in ( 1 2) illustrates. Only quelqu'un lexically specifies that the variable x also constitutes a new discourse marker.
interpretation
3 THE SEMANTICS OF IMPLICIT ARGUMENTS
Now that we have introduced the crucial semantic distinction between indefinites and a-defmites, let us return to the semantic interpretation of implicit arguments, and, in particular, the implicit agents of short passives. Our claim is simple: implicit arguments are a-defi.nites. They satisfy one of • Interestingly, not all forms of Dynamic Predicate Logic lack the representational means to distinguish between indefinites and a-definites. The original formulation of Dynamic Predicate Logic in Groenendijk & Stokhof {1991) is not subject to this limitation because of its attribution of a dynamic interpretation to both open and {existentially) closed formulas. Conversely, not all formulations of Discourse Representation Theory preserve the representational distinction between existentially interpreted variables and the introduction of new discourse markers for these variables. As discussed below, the recasting of DRT within Constructive Type Theory proposed in Piwek & Krahmer {to appear) is one such example. More important than the particular rradition on which a proposal draws is its presemtion of the two-pronged nature of semantic interpretation which rraditional DRT stresses. As long as it does, it can model the distinction between indefinites and a -definites. 7 See section 4 for qualifications and discussi ons of examples which involve accommodation in the sense of lewis {1979�
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'on' translates as '..XQ(human ' (v) & Q(v))' b. 'quelqu'un' translates as '..XQ (x&human ' (x) &Q ( x))'
( 12) a.
2.16
A-definites and the Discourse Status of Implicit Arguments
a predicate's argument positions, but do not introduce a discourse marker for that position. As a result, the referent of an implicit argument cannot serve as the antecedent of a pronoun in a subsequent clause. Sentences (I3a) and {I4) illustrate.8 {IJ)
#The first female president; was murdered ej. Hej was from the Berry, it seems. b. y&president'(y) &y = p & murdered' (x,y) &z&z = t & fiom ' (z, b) (I4) #The Russian space agency; issued pressurized suits ej. Hej was still unhappy. . a.
(I 5) a. The president; was murdered by someonej. Hej was from the Berry, it seems. b. x&president ' (y) & y
=
p & murdered' (x,y) & z & z
=
x & fiom 1 (z, b)
The discourse marker for the next sentence subject pronoun he, namely z, can now be equated with this previously occurring and accessible discourse marker. Hence the felicity of (Isa). In brief, the two hypotheses we set forth at the beginning of this paper regarding the semantic contribution of implicit arguments appropriately model the contrast between implicit and explicit agents in short passive sentences. If a-definites, be they explicit or implicit, do not introd"!lce discourse markers, subsequent NPS cannot refer back to them through discourse marker equality clauses. We similarly correctly predict that {I6c) can follow {I6b), but not (16a), since {I6h) alone introduces a discourse marker, say x, with which the referent of he can be identified (see {I6d)). {I6)
a.
b.
A A
ship was sunk. ship was sunk by someone.
8 Giv6n's (1995) claim that agents in short passive clauses have low topic persistence is very similar to what we call discourse inertness of a-definites except that we claim that the topic persistence of a-definites is not low, but zero!
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Semantically, sentence {IJa) is identical to the French sentence in (8), as its semantic translation in {I3h) {to be compared to {Ioa)) demonstrates. No discourse marker corresponding to the verb's implicit agent is introduced in the Discourse Representation Structure. The same holds true for the implicit recipient argument of issued in (I4). Since no discourse marker is introduced for this argument, he in the following clai.tse cannot be anchored to it. By contrast, agentive passives do introduce a discourse marker for the argument position of their governing verbs' agents. Thus, the agentive passive counterpart to sentence {IJa) in {Isa) receives a translation which crucially includes a discourse marker introducing clause, namely x, for the variable which satisfies the agent argument position, as indicated in {Ish).
Jean-Pierre Koenig and Gail Mauner
217
c. He was French. d. z & French'(z} & z = x Despite its apparent success, our account of the discourse inertness of implicit agents seems to run afoul of the linguistic data presented in Manzini (1983), Roeper (1987}, and Williams (1987} and the psychological evidence reported in Mauner, Tanenhaus, & Carlson (1995) and Mauner & Koenig (to appear), all of which suggest, as noted above, that the implicit agents of short passives can antecede the unexpressed PRO subject of rationale clauses. Sentence (17), repeated from (1), illustrates. A
ship was sunk to collect a settlement from the insurance company. b. *A ship sank to collect a settlement from the insurance company.
a.
these scholars argue, the unexpressed PRO subject of collect is controlled by the implicit agent of sunk, which contrasts with what happens in the case of intransitive verbs, as illustrated in (17b). The implicit argument of sunk does not seem, contrary to our claim, to be insert in discourse since it can antecede what is generally regarded as an anaphor of some sort, namely the unexpressed PRO subject of to VP infinites. A similar issue arises in the case of our explicit a-definite argument, French subject clitic on. On does not exclude all instances of anaphoric reference. The reference of on can, in particular, be the target of intra sentential reflexive binding. As (18} illustrates, the agent argument position that on satisfies can be bound to the patient argument position of lave 'to wash' via the reflexive clitic marker se. As
(18) On
s'
est encore lave dans ma salle de bain. INDEF REFL be.PPI' again wash.PPr in my room of bath 'Somebody washed himself/herself in my bathroom again.'
We claim, in accord with the assumptions of most syntacticians, that cross-sentential pronominal coreference differs from both subject PRO anaphoric identification and intrasentential reflexive binding. In particular, the latter two do not involve discourse marker equality clauses and are thus compatible with our hypothesis that implicit and explicit a-definites do not introduce discourse markers. Informally speaking, the anaphoric interpreta tion of the PRO subject of rationale clauses, as well as intrasentential reflexive binding, follows from lexical processes or meaning postulates as we will show in detail. To stress the lexical source of the coreferentiality of · the referents of the a-definite and anaphor, we will talk of lexical identification of argument positions for both cases.
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(17)
.uS
A-definites and the Discourse Status of Implicit Arguments
(19) 'se laver' translates as
'>.v(wash'(v, v))'
According to this proposal, then, reflexivization marks the semantic detransitivization of the predicate wash'. In so doing, it removes the need for the identification of the variables corresponding. _to the predicate's agent and patient arguments through discourse markerS equality 'clauses: the identification is part of the lexical meaning· of the derived predicate. The absence of discourse markers for the argument that on satisfies is not an impediment to the lexical identification of the two variables. Unfortunately, a generalization of this approach to implicit a-definites and the PRO subject of rationale clauses is not forthcoming. The difficulty can best be seen by considering the analysis of rationale clauses proposed in Carpenter (1997). Carpenter assigns the following meaning to the infinitive marker to which introduces rationale clauses. (2o) (21)
to =>AP.AlL\x.purpose(R (x) ) (P(x ) ) a. Francis practiced to defeat Brooke. b. purpose{practice(f), defoat( £b))
{Carpenter's example)
The R variable in (2o) corresponds to the me:ming of the main clause that the rationale clause modifies, while the P variable corresponds to the meaning of the VP complement of to. The resulting semantic translation of a 9 Chierchia's analysis of the Italian data involves the use of sorted variables. See Koenig (1999) for arguments that Chierchia's analysis does not generalize to French on or to a-definites in general.
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To account for the data involving on, we can adapt the analysis of similar data for the Italian arbitrary si discussed in Chierchia {1995b). Chierchia notices that the external argument of Italian arbitrary si cannot be the antecedent of a cross-sentential pronoun, but can be the antecedent of an intrasentential reflexive or reciprocal anaphor.9 He suggests an analysis of reflexive binding in terms of semantic argument reduction, after Bach & Partee (198o), an analysis recently revived by Pollard & Sag {1992) and Reinhart & Reuland {1993). A VP like hit himselfin this approach is analyzed semantically as >.x(hit'(x,x)). The fact that the predicate's two argument positions are 6lled by the same variable insures that the arb sortal restriction on the external argument which the Italian arbitrary si construction imposes is met by the anaphor as well as its antecedent. This analysis can easily be adapted to the analysis we put forth to account for similar data involving French on (see (r8)above). We only need assume that the translation of the predicate se laver is as shown in (19,); the meaning of (r8) is then derived compositionally as before via ordinary .8-reduction.
Jean-Pierre Koenig and Gail Mauner
2.19
sentence contammg a rationale clause is illustrated in (21a) and (21b). (Italicized material corresponds to R and underlined material to P.) The presence of the lambda-abstracted variable x in (2o) insures-given the syntactic category of to not represented in (2o)-that the unexpressed PRO subject of the rationale clause is bound to the surface subject of the VP it modifies. Of course, this is the wrong result in the case of short or agentive passives, as the sentences in example (22) illustrate. The house is not the agent of pay; rather, the unexpressed seller (22a) or Jane (22h) is. (22)
The housei was sold [PROi to pay the inheritance tax]. b. The housei was sold by Jane [PROi to pay the inheritance tax].
a.
(23) For every eventuality which involves a property ..XX P (x) as one of its semantic arguments, there is a corresponding eventuality whose property is replaced by the eventuality P(a ) in which the additional argument is the agent (or theme, or ...) of the verb the VP complements or modifies. When applied to the semantic translation of (22), this meaning postulate replaces the expression on the left of the arrow in (24) with the expression on the right of the arrow, thus ensuring the appropriate identification of the variable for the agent of sell and the unexpressed PRO subject of pay.10 (24) sell' (x, h, A}' pay' (y, .t)) -+ sell' (x, h, pay' (x, t) ) 10
We simplified somewhat Chierchia's account in irrelevant ways. The reader is referred to his article for details. Note that rationale clauses in his analysis are a kind of 'ad-verb' in the sense of McConnell-Ginet (1982) and thus augment the arity of any predicate they modify.
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The problem with Carpenter's semantic analysis is that the agent argument position of passive verbs in short passive sentences is satisfied within the VP and does not therefore have a lambda abstracted variable corresponding to its semantic translation. As a consequence, there is no possibility of identifying it with the variable x in (2o). The difficulty does not lie in Carpenter's particular proposal. In any (simple) Categorical Grammar style analysis, the representation of short passive verbs must follow the .8-reduction of the variable corresponding to the agent argument, thereby preventing the unreduced variable of the rationale clause to be identified with it. One possible solution to this difficulty is to adopt a 'property' analysis of control along· the lines of Chierchia (1989). In such an analysis, the determination of the controller of the PRO subject of infinitive complement VPS and rationale clauses is done through meaning postulates of the kind informally stated in (23):
.2..2.0 A-definites and the Discourse Status of Implicit Arguments
4
C OREFERENCE AND INFERENCE 4.1
Apparent counterexamples
We have until now considered two kinds of (broadly understood) anaphoric expressions: definite pronouns, for which neither explicit nor implicit a-defmites can serve as the antecedent, and reflexives and unexpressed PRO subjects for which both can. Our explanation for the differential behavior of a-definites with respect to these two kinds of anaphoric processes lies in the way the anaphor is resolved in each case. Only the former requires the presence of a discourse marker for the antecedent and thus excludes a-definites from their purview. But the literature provides apparent counterexamples to our general claim on the discourse role of a-definites which we must now address. Consider example (25) from Mauner (1996). (25) a. b. c. d.
The rebel priest was tortured for days. The rebel priest was tortured for days. The rebel priest was profoundly tormented for days. They wanted him to reveal where the insurgents were hiding out.
The indefinite use of the pronoun they (synonymous with someonet in sentence (25d) is anchored to the implicit agent of tormented or tortured in (25b) and (25a) respectively. The behavior of definite NPS parallels that of indefinite uses of they. They too seem to be able to refer back to an implicit 11 By 'indefinite use of tky', we refer to its non-anaphoric, unspecific use, which can be paraphrased by either so�o� or peopk. This use of tky is close. although not identical, to the a definites uses of on in French or Man in German. See Condoravdi (1989) for a discussion of the semantics of this non-anaphoric, indefinite use of they.
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By relying on a meaning postulate to identify the PRO subject of rationale clauses with the implicit a-definite argument of short passives, Chierchia's property approach to control eliminates the need for discourse markers equality clauses, just like the semantic detransitivization analysis of reflexive marking did. The upshot, then, is that a-defi.nites can enter into a grammatically induced relation of coreference, provided this relation is not mediated by discourse marker equality clauses (which would contra vene their a-definite semantics). Semantic reflexivization and property control are two such cases; in both, coreference does not rely on the existence. of available discourse markers, but rather, is the result of lexical semantic detransitivization or constraints on appropriate models (meaning postulates).
Jean-Pierre Koenig and Gail Mauner 221 argument, as argued by Carlson & Tanenhaus (1988) (see their example in (26)). (26)
a.
b. c.
Bill hurried to catch his plane. Bill hurried to unload his car. The suitcases were very heavy.
The definite NP the suitcases in (26c) is anchored to the implicit theme argument of unload, that is, what is being unloaded. These two classes of examples thus show that subsequent ipdefinite uses of they and definite NPs
(27) On
a tue le PDG de Renault. have.PR murder.PPT the CEO of Renault Les terroristes ont ete impitoyables. The terrorists have.PR be.PPT merciless INDEF
Renault's CEO was murdered. The terrorists were merciless. The NP les terroristes in the second sentence of (27) is easily interpreted as coreferential with the referent of the variable satisfying the agent argument position of tui in the first sentence, in apparent contradiction to what we suggested in section 2 (see Koenig 1999 for more examples). Such examples seem to contradict our hypothesis that implicit a-definites are inert. in discourse. In' fact, example (26) was used by Carlson & Tanenhaus to demonstrate the discourse relevance of implicit arguments. The intuitive felicity of examples such as {26c) and (25d) was confirmed experimentally in Mauner (1996). Mauner investigated how well readers process a target sentence containing an indefinite pronominal subject (e.g. an indefinite ·use of they or someone) which is the intentional agent of its own sentence (e.g. (2sd)), when it follows a short passive context sentence introducing either a volitional or nonvolitional implicit agent as a likely referent for the pronoun. In this experiment, participants read pairs of sentences one sentence at a time and judged whether target second 12 This statement constitutes an oversimplification of the data. & demonstrated by various authors, definite pronouns can refer back to referents that are not explicitly introduced in discourse. In particular, as pointed out to us by a reviewer and discussed ar length in Bosch (1983), 'anaphoric island' examples (Postal x¢9) or sentences that contain stressed definites pronouns contradict the claim that pronouns cannot refer back to antecedents which have not been explicitly introduced (see also Postal 1972 for additional examples). (33) a. John became a guitarist, because he thought it was a beautiful instrument. b. I saw Mr. Smith the other day; you know SHE died last year. (Wasow 1972) We address this issue at the end of this section.
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behave differently from definite pronouns: they can find their antecedents in implicit a-definites.12 What Mauner and Carlson & Tanenhaus observed for implicit a-definites holds of morphosyntactically realized a-definites too. Consider· the following example:
222 A-definites and the Discourse Status of Implicit Arguments
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sentences made sense given their context sentences. The coherence relation between the two sentences was kept constant throughout the experimental items (see Hobbs 1979, and Mann & Thompson 1988, among others, on the notion of coherence or rhetorical relation): the eventuality described by the second sentence was always the cause of the eventuality described by the first. Furthermore, the second sentence always described the referent of the indefinite pronoun's state of mind and strongly suggested that its referent intentionally brought about the event described by the first sentence as a result of this state of mind. For example, the desire felt by the referent of the second sentence's subject in (zsd) is most plausibly the reason why (s)he/they intended to torment or torture the rebel priest in (zsa)-(zsc). A coherent interpretation of the three simple discourses consisting of any of (25a) through (25c) followed by (zsd) thus relates the situations s and s' described by the two sentences via a causal relation, more precisely, via a reason relation: s' was the reason the second sentence's subject's referent causally initiated s. Furthermore, the establishment of a reason coherence relation between the two propositions expressed by the context and target sentences required the referent of the unspecific pronominal to be identi fied with the implicit agent of the preceding sentence. If s' is a reason for X to causally initiate s, then x must be the agent of s. Because the coherence relation was constant across conditions, the most natural interpretation of all experimental items had as the referent of they the implicit agent of the passive verb that occurred in the first sentence. That the referent of they was indeed equally likely to be taken as the implicit agent of the context sentences across all conditions was independently assessed through a prior questionnaire study. In this study, subjects were asked to rate on a s-point scale how likely the referent of they in a target sentence was whoever or whatever was responsible for the event described by the context sentence. There was essentially no difference across contexts in how probable participants thought the implicit agent was as the referent' of they. The results of this questionnaire study thus demonstrated that the materials were effective in equally inducing readers to treat they as coreferential with the implicit agent of the first clause. · Despite the equal likelihood that they be interpreted as coreferential with the implicit agent, the ease with which this coreference was established on line was predicted to differ across conditions. This is because two conditions included short passive context sentences whose main verb was either unambiguously eventive, e.g. (zsa), or biased toward an eventive interpreta tion, e.g. (25b), while a third included short passive context sentences whose main verb was biased toward a stative interpretation (25c). Context sentences from the first two conditions described situations that required a volitional agent. By contrast, the main verbs of state-biased context
Jean-Pierre Koenig and Gail Mauner 223 Table I Mean percentages of 'No' judgments, reading times and respective standard errors for target sentences following unambiguously eventive, event-biased, and state-biased short passive context sentences Sentence type
%
'No' jndgments (and standard errors)
Unambiguously eventive (c.£ (25a)) 17.0 (2.6) Event-biased (c.£ (25b)) 23.6 (J.I) State-biased (c.£ (25c)) 32 (3.5)
Reading times (ms) (and standard errors)
3 180 (183) 33 00 (203) 3484 (248)
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sentences describ ed situations that did not require a volitional agent. Now, the coherence relation (i.e. reason) that forms the b asis of the identification of the referent of they and the implicit agent of the context sentence requires the effect of the mental state describ ed b y the target sentences to b e an act. If I is the reason for s, then s must b e an act (under the sense of reason with which we are concerned here). State-biased context sentences do not therefore describ e a situation of the right sort to serve as an argument of this coherence relation. Of course, coercing this state situation into a sortally appropriate act is not difficult; one need only evoke the process from which the state resulted. But this coercion, Mauner predicted, would affect readers' judgments as to whether the second sentence made sense in the context of the first sentence. More precisely, the need to coerce the interpretation of state-b iased context sentences for readers to estab lish the coherence relation (and the identity of the implicit agent and the referent of they) would lead to increased reading times for the corres ponding target sentences and prompt some subjects to respond that these sentences did not make sense. Tab le I illustrates the percentages of 'No' judgments and reading times for target sentences following the three types of context sentences. As one can see, the experimental data support Mauner's predictions. Targets following state-b iased sentences which provided neither appropriate acts for the coherence relation nor volitional implicit antecedents for the unspecific pronouns elicited significantly more 'No' judgments and longer reading times (to sentences that readers continued to judge sensical) than either type of eventive context sentence. Targets following eventive sentences did not differ from each other in judgments or reading times. These results are plausib ly explained b y the hypothesis that the stative interpretation of profoundly tormented does not introduce an appropriately typed situation into the· Discourse Representation Structure to support the most plausible coherence relation among the situations describ ed b y the context and target sentences. The sequence consisting of (2 sc) and (2sd) requires additional inferencing to support the relation compared to the
224 A�definites and the Discourse Status of Implicit Arguments sequence consisting of (2 5b} and (2 5d). It therefore leads to an increase in total reading time for sentence (25d). For the purposes of this paper, Mauner's results are important in that they support empirically the claim that indefinite uses of they can refer back to implicit a-definites in on-line processing, as the following reasoning explicates: I.
2.
We conclude that examples (25), (26), and (27) and Mauner's experi mental results require us to qualify our main hypothesis that a-definites (be they implicit or morphosyntactically realized) are inert in discourse. Some NPs in subsequent sentences can target a�definites and be anchored to the same individuals in the universe of discourse. But under what conditions is such an anchoring possible? We turn to this issue in the next section.
4.2
Kinds of referential identification
·
As was the case for the unexpressed PRO subject of rationale clauses and intrasentential reflexives, we claim that anaphoric identification of the referents of indefinite pronouns and definite NPS to the referents of a�definite 'antecedents' differs from what typically happens in the case of definite pronouns (see (13a)). (This is an oversimplification; see fn. 1 2 for examples of atypical binding of definite pronouns.) Our hypothesis is that the discourse function of iinplicit arguments in the case of definite lexical NPS and indefinite pronouns (i.e. in (25), (26), or (27)) is mostly a matter of inference not coreference. More precisely, we suggest that the identification of referents follows from a process of accommodation (or bridging inference) of the kind described in Clark & Haviland (1 977} or of factoring. (Hereafter, we use the terms accommodation as the name of the phenomenon and bridging inferences for the mechanism underlying it.) For clarity, we discuss each case individually. We begin with definite NPS, then discuss indefinite pronouns and indefinite uses of they. We then
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3·
The degree to which the proposition expressed by the context sentence supported a reason relation was what was manipulated across conditions. The establishment of a reason coherence relation demanded the identification of the referent of they and the implicit agent of the context sentence. The presence of a difference in reading time between conditions suggests that speakers did perform the identification required by the coherence relation.
Jean-Pierre Koenig and Gail Mauner 2.2.5
bald (y)
X
y
king-of-France' (x)
king-of-France' (y)
I
Unresolved presuppositional
DRS
finally address atypical examples of binding of definite pronouns to which we alluded on several occasions. Our hypothesis that the apparent discourse relevance of a-definites results from a process of accommodation is independendy justified by recent work on presupposition in DRT, in particular, the work reported in Krahmer (1998) and Piwek & Krahmer (to appear). To present this corroborating analysis, we briefly digress on the treatment of definite NPS and their presuppositional behavior within DRT. As Beaver (1997) notes, the currendy most successful theory of presuppositional phenomena is the DRT based theory of presupposition-as-anaphor presented in van der Sandt (1992). According to this theory, presuppositions that are triggered by lexical or phrasal material are anaphors which need to find an accessible antecedent that binds them. Sentence (28) will serve as our illustration. Using a box representation for Discourse Representation Structures, the DRT representation of this sentence prior to the resolution of the presupposition carried by its consequent is as shown in Figure 1 (example from Krahmer
1998).13
(28) If France has a
king, the king of France is bald.
13 Variables at the top. of a box correspond to discourse markers. Material below these variables corresponds to predicative conditions. Thus, (J.j.a) corresponds to (34b) in the linear representation we used until now. X
(34)
a.
king-of-France' (x) b. x & king-of-France' (x)
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Figure
226 A-definites and the Discourse Status of Implicit Arguments
king-of-France' (x)
Figure 2 Resolved presuppositional
DRS
(29) If I remember correctly, the king of France is bald. •• Indefinite uses of th� of the kind we discussed in the context of (15� which constitute another class of cases that present difficulty for our hypothesis regarding a-defmites, are not anaphoric in the same sense as definite NPS and definite pronouns. Their anaphoricity is not grammatically induced and follows entirely from readers' striving for discourse coherence. We discuss the issue shortly.
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The unresolved presupposition associated with the conditional's con sequent is indicated by the most embedded box to the right of the arrow. This box encodes the proposition that there is a king of France, the presupposition traditionally assumed to be carried by the consequent (see Strawson I9So). Resolving the anaphoric dependency introduced by this presupposition amountS to binding the variable y to an accessible active discourse marker (x in the representation of the antecedent of the conditional in Figure I) and 'moving' the predicative conditions associated with the newly bound variable to the location of the variable binding it (into the box which contains the antecedent discourse marker x in Figure I). The result of the resolution of the anaphoric proposition that there is a king of France (after elimination of redundant conditions) is shown in Figure 2. This analysis of the presupposition , associated with (28) models the well-known fact that definite NPS carry (at least) an existential presuppo sition, which may or may not correspond to a presupposition of the sentence as a whole. The difficulty that definite NPS raise for our hypothesis regarding a-definites can now be more explicitly stated as follows. If presuppositions are anaphors and definite NPS are presupposi tional in nature, why does their behavior differ from that of definite pronouns?14 One property distinguishes definite NPS from defullte pronouns and presuppositions from non-presuppositional anaphora: the former's lexical content facilitates the process dubbed accommodation by Lewis (1 979). Lewis notes that presuppositions which are not satisfied are simply assumed by the hearer under some circumstances. Consider the following sentence:
Jean-Pierre Koenig and Gail Mauner 1.1.7 y CEO-ofRenaul( (y) kil( (x, y) merciless' (z)
terrorists' (z)
Figure 3 Unresolved presuppositional DRS for (27)
(3o) I bought a new car last week. The engine is already giving me trouble. The definite NP the engine is understood to be that of the speaker's new car because of a contextually available rule to the effect that each car has an engine. Piwek & Krahmer (to appear) casts their formal model of bridging inferences within a reformulation of DRT in terms of Constructive Type Theory. The details of their analysis are not our concern here. More crucial for our point is that it is compatible with our hypothesis that the anchoring of the referent of the terrorists to the referent of the a-definite subject clitic on in (27) does not require the presence of a discourse marker for the agent argument of tui. Simplying the representation for expository purposes, the representation of the discourse in (27), prior to the resolution of the existence presupposition associated with the subject of the second sentence is as shown in Figure 3·
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When this sentence initiates a discourse, the presupposition associated with the definite NP the king of France cannot be bound to a preceding discourse referent. Under certain circumstances, the presupposition is then accommodated, i.e. introduced within the ordinary (non-presuppositional) pan of the DRS. Of course, accommodation is a very powerful mechanism which should not be overused; otherwise, constraints on the context of utterance imposed by presuppositions would loose their force. Van der Sandt proposes a few conditions on accommodation. Our hypothesis is that the anaphoric nature of the definite NPS in (26) and (27) involves accommodation. We are not alone in proposing that bridging inferences in the sense of Clark & Haviland (1977) underlies accommodation. Piwek & Krahm.er (to appear) have recently suggested this is what is involved in the example from Hobbs, Stickel, Appelt, & Martin (1993) given in (3o).
228 A-ddinites and the Discourse Status of Implicit Arguments y, x CEO-ofReTUJU/( (y) lcilf (X. y) merr:iless' (x) terrorists' (x)
Figure
4
Resolved presuppositional DRS for (27)
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No appropriate antecedent for the discourse marker z is present in DRS. The resolution of the presupposition must therefore rely on accommodation and bridging inferences. It might, for example, rely on the knowledge that whenever a killing event occurs, an entity must be present which effects the killing and that terrorists 'often' kill government officials and top executives. Such knowledge can then be used to infer that z should be accommodated to the x agent variable of kill. The result of this accommodation via bridging inferences is shown in Figure 4· Although we view our hypothesis concerning presupposition resolution for discourses such as (27) as identical to Piwek and Krahmer's claims concerning presupposition accommodation. it should be noted that the representational mechanisms they propose to model accommodation are not rich enough to accommodate the difference between indefinites and a definites. Indeed, their Constructive Type Theory representation wrongly nullifies the distinction between the introducttoil of a discourse marker and the satisfaction of a predicate's argument position, which we argue is needed to model a-definites. We will not pursue the matter in this paper, though. since the issue of the exact mechanisms through which accom modation takes place is less important than the consensus that definite NPS can find their antecedent through accommodation and inferential processes. It is sufficient to show that (27) does not violate our contention that a definites are inert in discourse in the sense in which we introduced the term. Identification of referents through inferences applies to more than · definite NPs. As (25) indicates, indefinite uses of the pronoun they can also be resolved to the implicit agent of a preceding short passive verb. Again. we invoke inferential processes. In this particular case, the inference is not driven by a grammatically marked presupposition. It is not a bridging inference. Rather, it results from a hearer's drive to make the speaker's discourse coherent. As Hobbs and his colleagues argue, satisfying this maxim of interpretation requires (at least) establishing coherence or rhetorical relations between the eventualities described by discourse
this
Jean-Pierre Koenig and Gail Mauner
.2..2.9
Table .2. Anaphoric possibilities per NP type for (in)definites and a-definite antecedents
Reflexives and PRO subjects Definite lexical NPS Definite pronouns
Cross-sentential anaphors
{In)definites
(In)definites
A -definites
A-definites
lexical lexical identification identification (presuppositional) (presuppositional) binding binding through accommodation binding binding through accommodation factoring factoring through accommodation
segments and reducing the number of entities involved in the model for a stretch of discourse, what Hobbs et al. (1993) call 'factoring'.15 'Factoring' in Hobbs' terminology refers to all instances of identification of two predicates' arguments. For clarity, we reserve the term for instances in which the inferred identity of the values of the argument positions is not grammatically marked. We use the term identification for the general case. We use the term presuppositional binding to refer to the identification that results from the presupposition of 'availability' of antecedent discourse markers associated with definite NPS. Note that instances of both factoring and presuppositional binding can either target discourse markers explicitly introduced in the previous discourse or coerce the addressee or reader into accommodating the discourse with which she is presented and introduce an antecedent discourse marker after the fact, so to speak. We list in Table 2 the processes by which various kinds of NPS can find their antecedents that we have discussed so far. As indicated in the last column of the table, accommodation pervades non-lexical identification of anaphors to 15 Hobbs et aL derive these requirements from a general model of text interpretation dubbed interpretation-as-abduction. According to this model, readers strive to fmd the best explanation for what they have read or heard. In the case of the interpretation of an obsetved discourse, this abductive process amounts to inferring the best explanation for the validity or truth of the discourse with which they are faced. Since the fewet unproven assumptions a discourse comprises, the easier it is, all things being equal, to make it true, the presence of causal or other relations between eventualities, which reduces the number of unmotivated and unrelated events which the hearer must assume, should aid hearers' interpretive goals. The same reasoning explains why reducing the number of existing entities in the discoutse model helps the abductive process. Even though we fmd Hobbs' framework for discourse interpretation quite appealing, our point does not depend on its correcmess. It is sufficient for oUt purposes that discourse interpretation requires the establishment of coherence or rhetorical relations and the identification of participants even when no grammatical marking induces readers to do so. Both of these points seem uncontroversial
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Indefinite pronouns
Identification of argument positions
230
A-definites and the Discourse Status of Implicit Arguments
16
It is worth noting that non-grammatically induced 'factoring' inferences such as those necessary identify the referent of they to the implicit argument of tormented or tortu�tl can be drawn in the face of minimal lexical content (sec example (2 5)� This confirms that (prcsuppositional) binding and non-linguistically driven 'factoring' constitute distinct processes. to
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a-definites, whether that identification is grammatically marked or is the result of a linguistically unguided search for coherence. We also mention in the table instances of atypical binding of definite pronouns, to which we now turn. Now that we have placed cross-sentential anaphors within the larger context of identification of referents during the process of text interpreta tion, we must make good on our promissory note and discuss instances of atypical binding of definite pronouns. In the framework of presupposition as-anaphor, there is no difference in kind between definite pronouns like he and she and traditional presupposition-carrying definite NPs. Both involve an anaphor that must be resolved In both cases, the absence of a previously introduced discourse marker prevents the anaphor attached to the definite pronoun or definite lexical NP to be bound But, as already noted by van der Sandt (1992), whereas definite NPs often provide enough descriptive content for accommodation to take place (i.e. for the bridging inference to be supported by the speech participants' mutual knowledge), definite pronouns provide too little descriptive content (at most, the gender and number of the antecedent) to typically support accommodation. Because the anaphor cannot be resolved through either 'direct' binding or binding through accommodation, {8) and (13) are judged infelicitous.16 By contrast, the descriptive content of the lexical NPS in (26) and (27) lets accommodation rescue the presupposition whose 'direct' binding failed In other words, definite pronouns only differ from definite NPS in the degree to which they support bridging inferences when accommodation is required, not in kind. The fact that definite pronouns do not differ in kind from definite NPS explains the data presented in £D. 12. When the context sufficiently supports the identification of two arguments, definite pronouns, despite their impoverished descriptive content, can take implicit arguments as ante cedents. As Bosch (198 3) argues, one should distinguish between pronouns finding their antecedents and whether or not their antecedents are explicitly introduced by referring expressions. The antecedent might be introduced through means other than explicit morphosyntactic expression. In our terminology (excluding exophoric uses of pronouns for now), pronouns can coerce addressees and readers into introducing an additional discourse marker to serve as their antecedent. (See Bosch (1983 and 1988) for more details on how antecedents of pronouns may be 'implicitly' introduced.) But since the descriptive content of pronouns is lower than that of most definite
Jean-Pierre Koenig and Gail Mauner
231
3 A -definites can serve as antecedents for NPs only via accommodation (definite pronouns or definite lexical NPs) or foctoring (indefinite pronouns). Hypothesis
I. 2.
Lexical NPS make such inferences easier than pronouns because of their additional lexical content. Short passives make such inferences easier than middles because the presence ofa lexically encoded agent can be used in bridging.
This hypothesis leads to the following two predictions: I Sentences containing definite pronouns or definite lexical NPS should be harder to process when their only plausible antecedent is an a-definite than when it is an indefinite because this identification requires accommodation (i.e. additional inferencing) to take place.
Prediction
2 The increased processing difficulty should vary as a Junctio� of the lexical content of the anaphor and the availability of a lexically encoded argument position that can be used in bridging.
Prediction
These two predictiorls would rank the relative processing difficulty of the five sentences in (3 I) as their order of presentation suggests. The discourse in (J I a) should be less difficult to process than those in (3 I b) or (J ic), since the latter two require an additional bridging inference to
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NPS, more contextual clues are needed for this accommodation to be possible. This is what happens in the case of the examples cited in fn. 1 2 where the occurrence of lexical items which are morphologically related to the names ofthe discourse antecedent to be introduced and marked stress help readers in accommodating the otherwise descriptively poor definite pronouns. We conclude that_ pronouns, like definite and indefinite NPS, can prompt the accommodative introduction of their antecedent discourse markers: accommodation may target all anaphors. It simply does not target all anaphors to the same degree because the ease with which readers accommodatively introduce antecedent discourse markers is, in part, a function of the lexical content of the anaphor. For ease of reference, we call the process by which cross-sentential anaphors .whose lexical content is minimal find their antecedents binding simpliciter and r�erve the term presuppositional binding for anaphors whose lexical content is rich. ie. definite NPS. The (partial) classification of anaphoric process which we surveyed in this section embodies the following (revised) hypothesis regarding the discourse role of a-definites in the face of the data presented in section 4-1.
232
A-definites and the Discourse Status of Implicit Arguments
(3 I)
a.
b. c.
d. e.
The antique vase was sold to a wealthy woman by someone. The owner/He charged a lot of money for it. The antique vase was sold to a wealthy woman. The owner charged a lot of money for it. The antique vase was sold to a wealthy woman. He charged a lot of money for it. The antique vase had sold to a wealthy woman. The owner charged a lot of money for it. The antique vase had sold to a wealthy woman. He charged a lot of money for it.
5
C ONCLUSION
This paper has focused on the semantic representation of implicit arguments. Its main claim is that implicit arguments are a particular kind of non-quantificacional argument-what we call a-definites. As stressed throughout the paper, the category of a-definites is larger than that of implicit arguments. Morphosyn:tactically realized arguments can also be a definites, as the example of French subject clitic on demonstrates. What characterizes a-definites is their inability to serve as antecedents for future reference, i.e. their discourse inertness. The existence of a-definites is theoretically important because it confirms the underlying assumption of Discourse Representation Theory that semantic representations must include two kinds of information, a set of available discourse markers and a set of predicative conditions. By only contributing to the latter kind of information, a-definites confirm the need for the dissociation of these two types of semantic contributions. TJ:le discourse inertness of a-definites is also important within the narrower theoretical context of dynamic
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introduce a discourse marker which can serve as antecedent of the definite pronoun or lexical NP. The discourses in (J ib) and (31c) should be easier to process than either (J I d) and (J I e) since the latter do not lexically encode an agent argument to help the construction of a bridging inference. Finally, the discourses in (J ib) and (J id) should be easier to process than those in (J ic) and (J I e), respectively, because the lexical content of the former guides the bridging inference (the owner of the second sentence is likely to be the seller of _the first� Although it is difficult to compare (J ia) and (J ib), because of the slight pragmatic oddity of agentive passives whose agents are explicit indefinite pronouns, we think our predictions are borne out. In particular, it seems that {3Ib) < (3 Ic) < (3 1d) in processing difficulty. We are currently testing these predictions experimentally.
Jean-Pierre Koenig and Gail Mauner 233
JEAN-PIERRE KOENIG 609 Baldy Hall SUNY at Buffilo Buffilo, NY 14260-JOJO USA
[email protected]
Final
Received: I 5.06.99 version received: 16.02.00
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approaches to meaning. Their existence favors a DRT-style approach to the dynamic aspect of meaning over other approaches such as that proposed in Chierchia (1995a). As we showed, if the protracted availability of discourse markers for coreference reduces to a dynamic interpretation of existential ,quantifiers, the difference between indefinites and a-definites cannot be represented. The (relative) discourse inertness of a-definites does nqt entail their imperviousness to anaphoric processes. They can indeed serve as ante cedents of anaphors. We discussed two such cases in this paper: intra sentential reflexive binding and cross-sentential (in)de£nite NPS and pronouns. We showed that in both cases, independently motivated analyses of the phenomena existed that did not require abandoning the claim that a-definites do not introduce discourse markers. Following work by Bach & Partee (1980) and Chierchia (1989), we claim that intrasentential reflexives and PRO subjects can be bound to a-definites through a semantic detransitivizing process or meaning postulates, neither of which require the presence of a discourse marker corresponding to the a-definite's referent. Binding of definite NPs or pronouns to a-definites is the result of an accommodation process by which the presupposition attached to definite NPS or definite pronouns is resolved through bridging inferences even when no appropriate antecedent is available (as is the case when the antecedent is an a-definite, according to our hypothesis). Finally, identi fication of an indefinite pronoun to a previously introduced discourse marker is the result of hearers' or readers' drive to make the discourse coherent. Within this more detailed view of the nature of cross-sentential anaphoric processes, the difference between definite pronouns and definite lexical NPS reduces to the availability of enough lexical content to support bridging inferences. As we pointed out in the last section, if correct, this analysis of cross-sentential anaphoric processes leads to a set of testable empirical predictions regarding the relative difficulty of processing clauses containing anaphors that must be resolved to a-definites.
234 A-ddinites and the Discourse StatuS of Implicit Arguments
REFERENCES constraints on word classes in a hierarchical lexicon', Language, 76, 56-91. Dowty, D. (I98I), 'Quantification and the lexicon: a reply to Fodor and Fodor', in The &ope of Lexical Rules, Foris, Dordrecht, 79- I o6. Farkas, D. (1997), Towards a semantic typology of noun phrases', Paper pre sented at the 2nd Colloque de Syntaxe et 5emantique de Paris, Paris. Fodor, J. A. & Fodor, J. D. (I98o), 'Functional structure, quantifiers, and meaning postulates', Linguistic Inquiry, I I, 75 9-70. Garrod, S. & Sanford, A. (I 990), 'Referential processes in reading: focusing on roles and individuals', in D. A. Balota, G. F. d'Arcais, & K. Rayner (eds), Compre hension Processes in Reading, Lawrence Erlbaum, Hillsdale, NJ, 465-85. Giv6n, T. (I995), Functionalism and Gram mar, John Benjamins, Amsterdam. Groenendijk, J. & Stokho( M (I99I), 'Dynamic predicate logic', Linguistics and Philosophy, 14. 39-Ioo. Heim, L (I982), The semantics of definite and indefinite NP's', Ph. D. thesis, University of Massachusetts at Amherst. Hobbs, J. (I979), 'Coherence and corefer ence', Cognitive Science, 3, 67-90. Hobbs, J., Stickel, M, Appelt, D., & Martin, P. (I993), 'Interpretation as abduction', Artificial Intelligence, 63, 6g-I42. Kamp, H. (I981), 'A theory of truth and semantic representation', in . A. G. Groenendijk, T. M Janssen, & M B. Stockhof (eds), Formal Methods in the Study ofLanguage, Foris, Amsterdam, I -4 I. Kamp, H. & Reyle, U. (I993), From Discourse to Logic, Kluwer, Dordrecht. Koenig, J.-P. (I999), 'On a tue le president! The nature of passives and ultra indefinites', in B. Box, D. Jurafsky, & L Michaelis (eds), Cognition and Function in Language, CSLI Publications, Stanford, 256-72. .
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Bach, E. & Partee, B. (1980), 'Anaphora and semantic structure', in K. J. Kreisman & A. Ojeda (eds), Papers.from the Parasession on Pronouns and Anaphora, Vol 10, Chicago Linguistics Society, Chicago. Beaver, D. (1997), 'Presupposition', inJ. Van Benthem & A. Ter Meulen (eds), Handbook of Logic and Language, MIT Press, Cambridge, MA, 939-I008. Bosch, P. (I983), Agreement and Anaphora: A Study ofthe Role ofPronouns in Syntax and Discourse, Academic Press, London. Bosch, P. (1988), 'Representing and acces sing focussed referents', Language and Cognitive Processes, 3, 207-3 I. Carlson, G. & Tanenhaus, M. (1988), 'Thematic roles and language compre hension', in Syntax and Semantics, Vol. z1, Thematic Relations, Academic Press, New York, 263-9I . Carpenter, B. (I997), Type-Logical Semantics, MIT Press, Cambridge, MA. Chierchia, G. (I989), 'Structure meanings, thematic roles and control', in Properties, Types and Meaning, Vol. II: Semantic Issues, Kluwer, Dordrecht, 13 1-66. Chierchia, G. (1995a), The Dynamics of Meaning, Chicago University Press, Chicago. Chierchia, G. (1995b), 'The variability of impersonal subjects', in E. Bach, E. Jelinek, A. Kratzer, & B. H. Partee (eds), Quantification in Natural Languages, Kluwer, Dordrecht, 107-143· Chomsky, N. (198 I), Lectures on Government and Binding, Foris, Dordrecht. Clark, H. & Haviland, S. (I977� 'Compre hension and the given-new contract', in R Freedle (ed.), Discourse Production and Comprehension, Lawrence Erlbaum, Hillsdale, NJ, I-40. Condoravdi, C (I989), 'Indefinite and gen eric pronouns', in E. J. Fee & K. Hunt (eds), Proceedings of the 9th West Coast Conference on Formal Linguistics, CSLI Publications, Stanford University, 7I-84Davis, A. & Koenig, J.-P. (2ooo), 'Linking as
Jean-Pierre Koenig and Gail Mauner 23 5 Context, Kluwer Academic Publishers, Dordrecht. Pollard, C. & Sag, l (1992), 'Anaphors in English and the scope ofbinding theory', Linguistic Inquiry, 23, 261-303. Postal, P. (1969), 'Anaphoric island', Proceedings of the sth Regional Meeting of the Chicago Linguistic Society, Chicago Linguistic Society, Chicago, 205-39. Postal, P. (1972), 'Some further limitations of interpretive theories of anaphor, Linguistic Inquiry, 3, 349-71. Reinhart, T. & Reuland, E. (1993). 'Reflexivity', Linguistic Inquiry, 24. 657-720. Roeper, T. (1987), 'Implicit arguments and the head-complement relation', Linguistic Inquiry, 1 8, 267-310. Strawson. P. F. (1950), 'On referring', Mind, 5 9· 21-52. van der Sandt, R. {1992), 'Presupposition projection as anaphora resolution', journal ofSemantics, 9, 333-77· van Eijck, v. & Kamp, R {1997), 'Repre senting discourse in context', in J. van Benthem & A. Ter Meulen (eds), Handbook of Logic and Language, MIT Press, Cambridge, MA, 179-237. van Valin, R. & Lapolla, R (1997), Syntax: Form, Meaning, and Function, Cambridge University Press, Cambridge. Wasow, T. (1972), 'Anaphoric relations in English', Ph. D. thesis, MIT, Cambridge, MA.
Williams,
E. (1987), 'Implicit arguments, the binding theory, and control', Natural Language and Linguistic Theory, s. 1 51-80. Zeevat, H (1989). 'A compositional approach to discourse representation theory', Linguistics and Philosophy� u., 95-1 31.
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K.rahmer, E. (1998), Presupposition and Anaphora, CSLI Publications, Stanford. Lewis, D. (1979), 'Scorekeeping in a lan guage game', journal of Philosophical Logic, 8, 339-59· McCawley, J. (1988� The Syntactic Phenom ena of English, Chicago University Press, Chicago. McConnell-Ginet, S. (1982), 'Adverbs and logical form: a linguistically realistic theory', Language, 58, 144-84Mann, W. and Thompson. S. (1988), 'Rhetorical structure theory: toward a functional theory of text organization', Text, 8, 243-81. Manning, C. & Sag, L (1998), 'Dissociations between argument structure and grammatical relations', in G. Webelhuth, J.-P. Koenig, & A. Kathol (eds), Lexical and Constructional Aspects of Linguistic Explanation, CSLI Publications, Stanford. Manzini, M R. (1983), 'On control and control theory', Linguistic Inquiry, 14. 421-46. Mauner, G. (1996), 'The role of implicit arguments in sentence processing', Ph. D. thesis, University of Rochester. Mauner, G. & Koenig, J.-P. (1999), 'Lexical encoding of event participant informa tion', Brain and Language, 68, 178-84Mauner, G. & Koenig, J.-P. (to appear), 'Linguistic vs. conceptual sources of implicit agents in sentence compre hension',journal ofMemory and Language. Mauner, G., Tanenhaus, M., & Carlson. G. (1995), 'Implicit arguments in sentence processing', journal of Memory and Language, 34. 357-82. Piwek, P. & K.rahmer, E. (to appear). 'Pre suppositions in context: constructing bridges', in P. Brezillon & M Cavalcanti (eels), Formal and Linguistic Aspect of
Normal Objects, Normal Worlds and the Meaning of Generic Sentences REGINE ECKARDT
University ofKonrtanz
Abstract
I INTR ODUCTION The main aim of my paper is to propose a semantics for the genenc operator GEN, widely used in the semantic investigation of generic sentences, a semantic account which moreover does justice to certain well known empirical observations about generic quantification: (r.r) Generic sentences allow for exceptions: Dogs bite postmen Dogs don't bite postman Otto. ( 1.2) Generic sentences involve a modal component: Rose handles mail from Antarctica. (r.3) Generic sentences can be embedded in other modal constructions: If every postman got dog training, then dogs wouldn't bite postmen. ( r-4) Nested generic quantification: Dogs bite men wlio are afraid of dogs. For a more general overview over the discussion of genericity, the reader is referred to the comprehensive survey article ofKrifka, Pelletier, Carlson, ter
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It has sometimes been proposed that generic sentences make statements about prototypic members of a category. In this paper I will elaborate this view and develop an account where generic sentences express quantification about the normal exemplars in a category here and in counterfactual worlds sufficiendy similar to our own. Comparing the account to the currendy most widespread analysis which views generic sentences as universal quantifications in carefully chosen best-possible worlds, we find that an analysis that is based on the choice of normal objects does better justice to the data in question than an analysis that relies on a choice of normal worlds alone. A further conceptual advantage of an explicit separation of (a) a choice of best exemplars and (b) a modal component of generic quantification consists in the fact that it highlights that different generic sentences can rely on different kinds of choice of best exemplar. Comparing their logical behaviour, I will demonstrate that we should at least distinguish between nonnal-generic sentences and ideal-generic sentences. Finally, the paper proves that the account I propose is a modal variant of some recent purely extensional default logics, developed in AI.
23 8
The Meaning of Generic Sentences
Meulen, Chierchia and Link (Krifka et al. 1 995) as well as Pelletier and Asher (Pelletier & Asher 1 997). Note that in the light of these surveys, the enterprise of the present paper is restricted to a sub-question in the investigation of genericity, the question about the nature of the GEN operator. Even if I might occasionally talk about the 'investigation of generic sentences' in the following, this is always meant in that limited sense. The account I propose will be based on two main ingredients. First, I will introduce a family of operators that select for each domain p· the subset of normal individuals in P. Second, I will use a notion of accessible counterfactual worlds, the dispositional orbit of the world of evaluation, which will do justice to the modal nature of generic statements. Finally, the discussion will reveal the fact that we have to distinguish two types of generic sentences, normal-generic statements and ideal-generic statements. In proposing a new account, it is furthermore of interest to see in what respect this account can do better than previous theories. I will mainly concentrate on a comparison to a family of theories for GEN that I will call 'Best World Theories'. I will raise two main criticisms. First, these accounts only allow for one-dimensional quantification, which causes problems in certain cases. Second, Best World Theories try to capture several interacting factors in generic quantification by one un-analysed operator, and I will argue that this obscures the facts under consideration. The paper is orgaruzed in the following way. In sections 2 and 3, Best World Theories of generic sentences and the Normality Based Theory of generic sentences will be introduced. Section 4 will examine the treatment of exceptions in either account. It will tum out that Best World Theories are forced to follow a strategy of 'stepwise quantification', which will be shown to be inappropriate upon closer investigation in section S· The NorMality Based Theory can face exceptions without problems. Section 6 investigates genericity in complex modal constructions. The remainder of the paper will argue in favour of a distinction between normal-generic sentences and ideal-generic sentences. In section 7, I will take the example of the 1oo-year-old turtle as an intuitive starting point to introduce this distinction. Section 8 will elaborate on this point, arguing that different kinds of axiomatic restrictions are appropriate for either kind of statement. While 'normality' should be more statistical in spirit, 'ideality' refers more to a conceptual ideal. Section 9 addresses the question of a sound conceptual basis for theories of GEN. An Appendix relates the approach to theories of default reasoning in AI literature. It turns out that the account can integrate widely used systems of nonmonotonic reasoning in a modal framework-a combination that already bore fruit in section 6. ·
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·
·
Regine Eckardt
2
239
BEST WORLD THE ORIES
(2.1) (2.2)
Rose handles mail from Antarctica. This machine peels oranges.
might be true even if no letter from Antarctica has ever reached our office, and (2.2) can felicitously describe a machine that is brand new and has never before been set to work, and would still be true even if the machine got destroyed by an accident before it had seen its first orange.1 The overview article by Krifka et al. (1995) sketches a modal treatment of the GEN operator in Section 1.2.6, which is based on the rich and elaborate modal accounts developed in Kratzer (1978) and Heim (1982). Apart from that proposal, Kri£ka et al. (1995) list a number of modal treatments of the GEN operator that I will refer to with the cover term 'Best World Theory'. For concreteness, I compare my approach with the recent account proposed by Pelletier and Asher (Pelletier & Asher 1997) which, however, for the issues under discussion can be taken to represent a much larger class of similar approaches, some of them listed below. Best World accounts of a generic statement of the form GEN[xll . I Xn](�. w) elaborate the following paraphrase: take any a, go to those possible worlds which are most normal, or most undisturbed, or most ideal, with respect to matters in question-that is, the proposition �(a)-and see whether W holds true. If yes (for all a, for all normal-�(a)worlds), the generic statement is true. If no, it isn't. . Formally, Pelletier & Asher assume a function * which maps pairs of worlds and propositions onto the set of those worlds where the proposition holds true in the most normal way:
(2.1)
.
(2.3) *:
W
.
X
P(W)
�
P(W)
1 The sentences can also become false under such circumstances. This is the more striking observation, as generic sentences tend to be regarded as universal quantifications of some kind.
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In this section, some reaSons to prefer a modal account for GEN over other proposals will bqefly be reviewed, and one most recent such account, the theory of Pelletier & Asher (Pelletier & Asher 1997), will be introduced. This theory will provide a concrete object of comparison to the account to be developed in this paper. It has often been observed that generic sentences cannot be. analysed in a purely extensional manner. The most striking evidence in favour of this observation is generic sentences that express regularities that, in fact, have never so far been instantiated. Examples (2.1) and (2.2), taken from Krifka et al. {1995), illustrate this point:
.2.40
The Meaning of Generic Sentences
*(w,p) p' where p' = all the normal p worlds according to w, that is, all the worlds in which p, along with the typical consequences ofp holds true. =
Next, a conditional operator > is defined on basis of the * function in the following way: (24) M , w , g I= � > w iff *(w, [�]M,g) � [�]M,g, where [�]M,g = {w e W: M, w, g f= �} We can now translate the logical form of generic sentences, something of the form into the logical expression . . . , Xn (� > w) and will thus get exactly the semantics for (2.5) that was given in the paraphrase above. Further restrictions of * are discussed in Pelletier & Asher (I 997), like FACTICITY and OR. I will give FACTICITY for illustration, an axiom that ensures that worlds that are normal for p also support p: (2.6)
Vxn
(2.7)
FACTICITY:
*(w,p) � p
Although such axioms, of course, influence the resulting logic of GEN, they will not be of primary importance in my criticisms of best world theories. I therefore refer the reader to the original source for more details, as well as a treatment of defeasible inferences on the basis of this logic. With respect to the issues to be discussed below, the theory of Pelletier& Asher can be seen as standing for a range of related accounts, ranging from Morreau (1996), Asher & Morreau (1995), back to classical papers like Delgrande (1987, 1988). 3
NORMAL OBJE CTS
section serves to introduce the basic notions of the account I want to propose as an alternative to Best World Theories. The most important ingredient will be the idea of distinguishing normal from not-so-normal objects in a category. If one starts out from a world w to look for those worlds ul in *(w,p) which are normal for p. one will in many cases immediately leave w. Formally speaking, w ¢ *(w,p) for most propositions p and worlds w, and this should be so, because otherwise default implication would come too This
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(2.5) GEN[x J, . . . , Xn](�. w)
Regine Eckardt 2.41
close to classical implication, as demonstrated in Pelletier& Asher (I 997), or Morreau ( I 996). In face of this (quite informal) observation, one might ask how speakers can acquire this kind of sophisticated knowledge about counterfactual worlds. All they can look at is the real world around them. Generic beliefs should be the result of '[speakers'] desire to understand and characterise the world immediately surrounding them' (Pelletier & Asher I997: I I29), yet the 'normal case in the world immediately surrounding one' seems to be deeply hidden in Best World Theories. One could propose to define the set of objects a which are normal Ps, for every property P, in world w:
Evidently, the notion of 'real, normal Ps' is an indirect part of the Best World Theory. Yet it remains unclear how the global function * builds on such real cases which are intuitively fundamental for our generic beliefs about P. We could also turn things upside-down and introduce an explicit notion of 'normal object in P' in order to analyse generic sentences.2 The basic idea is to distinguish between all Ps and normal Ps. In order to do that, we introduce a family of functors (3.2) Nn:
W
X
(De)" --+ W X (De)"
These functors will map all n-ary properties P on to their normal parts N(P). For all worlds w, Nn(P)(w) is the set of all those tuples alt . . . , an which are normal Ps in w. Evidently, these a,, . . . , an should also be P in w, such that we sensibly require (3.3 ) to hold true.3 Assumption (3.3) will be repeated as (N I ) in section 8, where more �omatic restrictions will be discussed. (3.3) For
all
w:
N(P) (w) � P(w)
On the basis of these operators, we can present a first try at analysing generic sentences. (3.4) FIRST VERSION: GENx, , . . . • Xn (� ; w) iff 'Vxlt . . . , Xn(Nn (>.s>.x,, . . . , Xn�)(x, , . . . , xn) --+ llt) z In the survey part of Pelletier & Asher (1997� this strategy is discussed in section 2.3 on 'Prototypes'. The theory to be developed here can actual.ly be seen as a spell-out of that sketch. A similar sketch can also be found in Kri£ka et al. (1995� 3 I will sometimes omit the ariry index ofN in the following when this is possible without causing misunderstandings.
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(3.I) a is normal with respect to P in w iff w E *(w, "P(a) )
242
The Meaning of Generic Sentences
{34) reads as follows: 'Form the n-ary property cp, and use it as an argument of the N functor. That will give you the normal cps. If all of them show W , the generic statement holds true.' The analysis proposed in (34) captures the intuition nicely that generic statements are just talking about normal objects. However, it can easily be argued that the treatment in (34) is still too extensional. Many such universal sentences come out true by accident, or because of the finiteness of our world. Consider the following scenario where no choice, however sophisticated, of normal Ps can rescue the case. (3.5)
•
•
•
Similar examples arise whenever the set of Ps in a world w is small enough to fail to exemplify some property that, in and of itself, would be quite natural for some P to have. And interestingly, speakers can have quite clear feelings about whether some property isn't exemplified by accident or by principle. (Compare the generic statement: 'Popes aren't called Goofy'.) Another well-known problem for a purely extensional analysis are generic sentences about something that has not occurred yet. Let me repeat example (2.1) for illustration. (3.6) Rose handles letters from Antarctica. According to the translation suggested in (3.4), we could only say that for all normal letters x from Antarctica (arriving in our office), Rose handles x. In the (likely) case that no letter from Antarctica has ever reached us, we would quantify over an empty set, which does no harm in making the universal true-however, more things will spuriously be true, due to the fact that there are no letters form Antarctica: All generics in (3.7), for example, would be predicted to be true: (3.7) Rose eats letters from Antarctica. Rose answers letters from Antarctica rudely. Joe handles letters from Antarctica. ·
This is, of course, unwelcome. We can avoid these kinds of accidental universal truths by evaluating (9) not only in ours, but also in a range of other possible counterfactual worlds. Thus, we arrive at an analysis like in . (3.8):
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•
It is true that no pope ever had the name 'Bartholomew'. Normal popes are a subset of the actual popes. Thus, (3.4) renders the generic sentence 'Popes aren't called Bartholomew' true. Actually, the sentence is-false.
Regine Eckardt 243
(3.8) SECOND VERSION: w I= GENxu . . . , x,. (� ; 'Ill ) iff 'Vul (w � ul --+ 'VxJ , . . . , x,.(N,.(>u>.xJ , . . . , x,.�)(xJ , . . . , x,.)(ul) --+ 'lt(ul) )
A paraphrase would be 'Go to any world ul that is related by � to w. Now, check for all objects a1, , � that are normal � in ul whether they are also 'Ill in u/. If this works, for all a1, , 3u , in all u/, the generic Statement is true.' Definition (3.8) makes use of the new binary relation � which singles out those counterfactual worlds that are relevant in the evaluation of the generic sentence. Formally, the relation � is an accessibility relation between possible worlds. Coming from some world w, all those other worlds are � -accessible that are like w with respect to causal and statistical dependencies and regularities, but may differ from w in isolated acciden� facts. (3 .9) is meant to capture the intuitive content of � : • • •
• • •
facts.
We will call { w' l w � w' } the dispositional orbit or briefly DO of w. Note that there is a crucial difference between the underlying content of world selection in Best World Theory, coded as * function, and accessibility � in (3.8). While * selects ·better worlds than ours, � accesses those worlds which behave like our own. These need not be more normal in any way, but may differ with regard to facts that are relevant for our generic beliefs. In example (3.5) above we might reach worlds w' where some normal pope is called Bartholomew, where some unnormal pope has the name Goofy, where no normal pope is called John, but none where a normal pope is called Goofy. Having strong faith in normality operators N,., one .might even think about not restricting the universal quantification at all. As long as N,. picks out the proper subsets in all words, might not the universal in (3.8) hold in all worlds w', not only in some carefully selected ones? It will become clear in sections 6 and 9 that dispositional orbits are indispensable, and their role will become more colourfuL Yet we will first look at some quite un-intensional cases in order to see the approach at work.
4
NORMAL OBJECTS AN D EXCEPTIONS
In this section, I will show that Best World Theories are not very robust in dealing with exceptions. The orily way in which they can account for them will lead to further trouble in section S· The Normality Based Theory does not face comparable difficulties.
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(3.9) w � ul stands for: ul is like w with respect to all dispositions, causal and statistical dependencies and regularities but may differ in other
244
The Meaning of Generic Sentences
Consider the two sentences in (4-1): (4-1)
Dogs bite postmen. b. Dogs don't bite the postmail Otto. a.
The two sentences are not contradictory. In fact, they perfectly illustrate that generic sentences allow for exceptions. Surprisingly, Best World Theories face some annoying difficulties when treating exceptions. A first natural representation of (4-1a) in the Best World Theory introduced in section 2 could look like this, where the reasonable reference to occasions s when postmen and dogs meet is added:
of normal 'a is a dog and h is a postman and they meet in s'-worlds is a subset of the set of worlds where a bites h in s. {4-3) *(wo, DOG{a) & POSTMAN(b) & MEET(a, b, s)) � BITE(a, b, s) In particular, this should also hold true if h happens to be otto the brave:
(44) *(w., DOG(a) & POSTMAN(otto) & MEET(a, otto, s)) �
BITE(a, otto, s)
Sentence (4-1b), on the other hand, will be represented as in (4.5): (4-S) 'v'x(DOG(x) & POSTMAN(OTTO) & MEET(x,OTTO,s) > ..., BITE(x, OTTO, s)) Given that the constant name OTTO is interpreted as the individual otto,
this amounts to the condition in (4-6):
(4-6) For all a, s: *(w0, DOG(a) & POSTMAN(otto) & MEET(a, b, s)) � BITE(a, otto, s) ...,
We may faithfully assume that the set ofworlds where a bites otto in s and where a does not bite otto in s are disjoint. Therefore, the conditions in (44) and (4.6) contradict each other. This is an unwelcome result,' because the consistency of (4-1) is a paradigm issue in genericity. The shortcoming was pointed out, among others, by Lehman (1989). Pelletier & Asher answer it by offering a different analysis of (4-1a): {4-7) 'v'x (DOG(x)
>
'v'y {POSTMAN{y)
>
'v's (MEET(x,y,s)
>
BITE(x,y,s))))
Matters become quite intricate at this point, which is why we will take the time to spell out (4-7) in detail:
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(4-2) 'v'x'v'y'v's (DOG(x) & POSTMAN(y) & MEET(x,y,s) > BITE(x,y,s)) Formula (4-2) is true in w0 iff for all instantiations a, b, s for x, y and s the set
Regine Eckardt 2.4 5
(4-8) For all a: *(w0, DOG(a)} � {w 'l for all b: *(w', POSTMAN(b)) � { w'l for all s: *(w", MEET(a , b , s) ) � BITE(a , b , s } ) }}
This reads as follows: we pick an a. We proceed, starting from W0, to all those worlds where a is a normal dog. There we are. Now we pick up some b. We proceed, starting from ul, to those further worlds ul' where b is a normal postman. In ul', we pick some s and proceed to those worlds ul" where s is a normal meeting of a and b. Now, a should bite b in s. (4-1b) receives the corresponding representation given in (4-9):
(4-9) \fx(DOG(x) > V's(POSTMAN{OTIO} & MEET(x, OTIO, s} ..., BITE(x,OTIO,s) }) )
>
In this case, we choose an a and proceed from W0 first to all worlds which are normal for a being a dog. In those, we c�oose some s and go further to worlds ul' which are normal for s being a meeting of a and the postman otto. As the world need not be normal for Otto being a postman, Pelletier & Asher would argue,4 the meeting can be such that otto does not get bitten, although he would be if we were in one of those worlds where he is a normal postman. So far, so good. We see· that if the Best World Theory wants to account for examples like (4-1), it needs to adopt the strategy of 'stepwise quantification' where only one variable is bound at a time. For instance in (4-7), we have one default implication for variable x, one for y, and one for z. Treating them all in one step, like in formula (4-2), was fatal. At this place, it is necessary to remember what all that walking around in possible worlds was good for. When we had picked some individual a, in (+8), we moved to worlds ul where a was a normal DOG. Next, we picked some b and went from ul to somewhere else, ul', where b was a normal POSTMAN. The problem seems to be that a might cease to be a normal DOG (or worse, cease to be a dog at all) once we have reached ul'. After all, the global function * in ul cannot take care of some odd chosen dogs when looking for further worlds where b conforms to the typical POSTMAN. Let me put it even more dramatically: if * indeed could care for such things, • Actually, I am not sure whether they would defend this cl.aim. In the example they present, they can afford to keep their twin of posanan Otto, the zookeeper Joe, out of all antecedents of default
implications. We cannot represent Otto in the consequent of >, because Otto is part of the event description in the anteeedc:nt. Thus, it might be that Pelletier & Asher fail at this example, but let us act as if they didn't.
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Example
.2.46 The Meaning of Generic Sentences
(4-1 0)
(4.1 r)
'r/ul (wo � ul --+ 'rlxys(NAAWAXYS.DOGw(X) & POSTMANw(Y) & MEETw(X, Y,S)))(x, y, s)(ul) --+ BITE(x, y, s)(ul)) 'r/u/ (wo � ul --+ 'r/xs(Nz(AWAXS.(DOGw(X) & POSTMANw(otto) & MEETw(X.otto,S)»(x,s)(ul) --+ ..., BITE(x,otto, s)(ul) ) ·
Some remarks about notation: in order to make things more readable, I have underlined the arguments ofN3 and Nr The possible world parameter of properties is notated, somewhat inconsistently, sometimes as an subscript of the predicate in question (DOGw(x) ), sometimes as an extra argument (DOG(x, w) ). I hope that this is not too disturbing. I moreover adopt the convention to use capital letters for the lambda-bound variables X and >.X in the argument AX�(X) of N in order to facilitate reading. Formula (4- r o) says that in all ul in the dispositional orbit ofW0, we find that all triples of normal dogs, postmen and their meeting are such that the dog will bite the postman. In contrast, formula (4-1 1) says that in all worlds in the dispositional orbit of W0, we find that the normal meetings of normal dogs with postman Otto are such that Otto doesn't get bitten. In particular, Otto can stay an unnormal postman throughout all of the DO of W00 This is, as we shall see in section 6, one of the explicit functions of the dispositional orbit.
s
N ORMAL PAIRS AND PAIRS O F NORMALS
In this section, it will be shown that Best World Theories cannot account for situations where normal pairs over A and B are not the same
as
pairs of
.
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then the first step from W0 to ul where a is a normal dog must have already preserved, without us noticing, all those normal cats, normal mice, normal sausages etc. which we might want to use in the next sentence about the normal dog a. Evidendy, the star operator * has to perform a much more tedious task than authors in Best World Theories acknowledge. A normal-operator analysis of (+r ) is unproblematic, because we have the explicit means of specifying which variables refer to normal individuals, and which do not. We chose which things should be normal, we form the property with respect to which they should be normal, we can now apply the normality operator of an appropriate arity, and get the antecedent for the implication as a result. The representations for (4-ra) and (4-rb) are given in (4-ro) and (+I I), respectively:
Regine Eckardt 247 normal A and normal B. Moreover, other perspectival issues in generic statements will be addressed. The strategy of stepwise quantification becomes more problematic in cases where normal encounters of A and B are usually not encounters of normal A and normal B. I will use the Wolves-and-Men example for illustration. The following generic sentence reflects traditional wisdom about wolves and men. Picturesque tales to that end can already be found in general survey articles on lupus lupus as in Brehm ( r 876) and, more recently, Grzimek (1987).
( s .r) Wolves kill men.
(5.2) 'v'x(WOLF(x)
>
'v'y (MAN(y)
>
'v's (MEET(x,y, s)
>
KILL(x, y, s)) ) )
As in previous examples, we select an object a and go where a is a normal wolf, we pick a man b and make it a normal man, and now we look for normal encounters of these. Either we shall find that such encounters usually end with the man successfully chasing away the wolf, and will deny the truth of (5.2) on these grounds. Or we can claim that none of these encounters is like what we had in mind (our expectations being shaped by the things that immediately surround us), such that there are none. In this case, (5.2) becomes true, as do many other generic sentences about wolves and men, because vacuous universal quantification is a powerful truth maker.
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However, modem descriptions of wolves tend to stress the fact that the generic sentence (s .r) is not, strictly speaking, true. In fact, normal wolves tend to avoid encounters with normal (healthy, adult) humans altogether. In the rare case that a normal wolf meets a normal man, so these treatises report, it is usually easy for the human to frighten the wolf away. Yet these facts are compatible with the assumption that the generaliza tion expressed in (s .r) is talking about what normally happens under those circumstances when wolf and man meet at all: Such meetings most probably take place at times when the wolves are unusually hungry, and where the human is of a special kind, namely looking helpless and easy-to attack: a sick or wounded person or a child. Although the generic sentence does not talk about normal wolf and normal man, its content is still important enough for humans. We are equally interested in sick and healthy, young and old, when it comes to the question who should be killed and eaten by wolves: nobody should. Therefore, even regularities about normal encounters of a rare kind are worth reporting, if they have such far reaching consequences. Stepwise quantification cannot handle such examples properly. Sentence (s.r) will acquire a representation as in (5.2):
248 The Meaning of Generic Sentences
We could try and make all three, wolf a, man b and encounter s normal at once: a world that is most normal for the proposition 'a is a wolf and b is a man and s is a meeting of a and b' need not support the normality of either conjunct in isolation. Hungry wolves can meet weak men and kill them in such worlds, without changing our standards of a normal man.
(5.3) VxVyVs(WOLF(x) & MAN(y) & MEET(x�y,s}
>
KILL(x,y,s}}
(54) Waldo the Wolf doesn't kill men. analysis in terms of normality operators will properly distinguish between (5.1 ) where we quantify over wolves, men and their meetings, and make a statement about the normal triples of that kind, and example (54) where we only quantify over normal encounters of Waldo with men.
An
( s.s) Vu/ (wo � ul Vxys(N3(..\WAXYS(MANw(X) & WOLFw(Y) & MEETw(X. Y,S)»(x, y, s}(ul) KILL(x, y, s)(ul) ) (s.6} Vul (wo � ul Vxs(Nz (..\WAXS(MANw(X) & WOLFw(waldo) & ---+
---+
---+
MEETw(X.waldo,S)»(x,s)(ul)
---+ ....,
KILL(x,waldo,s)(u/))
Let me briefly discuss a further constellation in which normal encounters with A are not the same as encounters with normal A. These highlight the fact that 'Normality' is a subjectiv� notion.
(5.7) Bees are busy. The generic statement in (5.7) is a variant of the proverbial 'Bienenfleill' ('busy as a bee'). However, on closer investigation it was found that the normal bee most of the time is dozing hidden in the hive, and that ( 5.7) is but an unimportant generalization about the normal-bee-we-see and what it does on normal-occasions-when-we-see-it. When we see a bee, we see it working, but this does not mean that most bees work most of the time. (It means that there are many bees in hives, though.) Example (5.7) is thus based on some selection of occasions wheri we meet bees. Similar generalizations arise when we meet only a special subset of a set of A (what would we think about bees if all we knew were the drones?).
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However, the structure of this formula corresponds to (4-2). The same argument as above will show that sentence (5.1), in the representation suggested in (5.3 ), will contradict sentence (5.4) which states that Waldo the Wolf is an exceptional case. {Waldo might be a tame wol£) Again, this prediction does not fit in with our intuitions:
Regine Eckardt 249
Real world examples of that kind. howeve� already have the nasty smell of a prejudice, which becomes even more stinging if the generic statement reflects what some special person x thinks is true about members of a class A. Yet one should be aware of the fact that 'being a normal A' in many cases amounts to a characterization of 'the kind of A I meet with highest frequency'. If those A that I meet are the same as those everyone else meets, the arising subjective notion of 'normal A' is accepted by the whole community. If we can be sure, moreover, that the 'normal A we meet' is also the most common A there is, our generalizations about 'the normal A' become trustworthy observations of reality. One can capture these three stages of subjectivity by adding a further hidden_parameter to the normality operators: N(x,P) yields normal Ps seen from the perspective of x. N(A, P) yields normal Ps seen from the perspective of the community A, something like a weighted generalization of the single subjective normalities N(a, P) of members a of A. In fact, such N(A, . ) is all we ever can get, taking larger and larger communities A, but in some cases one might want to express that we have good reason to expect that these subjective normalities will not be shaken by any further community, in which case we could use an unparametrized notion N(P). Note that a subjective notion of normality can not help us to safe example (5.1). It is not that we only see a special selection of meetings between wolves and men. And even the wolf might realize that the men he would be apt to meet (in order to kill) are not those that are most common. Sentence (5.1) is a generalization about normal encounters of wolves and men, and not a statement about normal men and normal wolves, from whatever perspective. •.
This section discusses the normality based theory and its predictions in modal embeddings. Let us come back to example (4-1)a, repeated here as. (6.1). The corresponding semantic representation is given in (6.2). (6.1) Dogs bite postmen. (6.2) Vu/ (w0 � ul Vxys(NAAWAXYS(DOGw(X) & POSTMANw(Y) & MEETw(X. Y,§1))(x,y, s)(ul) BITE(x,y,s)(ul)) -+
-+
What should worlds look like which we look at in the evaluation of (6.2)? Certainly both; dogs and postmen should generally look like they do in our
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6 NORMAL HERE AND NORMAL ELSEWHERE
·
2.50 The Meaning of Generic Sentences
(6.3) If every postman got dog training, then dogs would not bite postmen (64) W0 f: .Xw('v'x (POSTMAN., (x) -+ DOGTRAINED., (x)) D-+ .Xul(ul :::= w -+ 'v'yVz(N(.XW.XYAZ.DOGw(Y) & POSTMANw(Z) )(y, z, ul) BITE(y, z, ul) )) -+
..,
Formula (6.4) can be paraphrased as: 'For all "nearby" worlds w where all postmen get dog training, it holds true that in all worlds ul in the dispositional orbit of w, all pairs of normal dogs and postmen you find are such that the dog will not bite the postman in ul. J6 The closest worlds where the antecedent is true are beyond the DO of the actual world. In the DOs reached from that remote starting point, all postmen get dog training and dogs remain sane enough not to bite them. That is, the notion of closest worlds in the sense of Lewis is wed in counterfactual reasoning with generic sentences to ensure that everything else stays as normal as it was in 5 I will suppress reference to meeriligs between postmen and dogs in (6-4). The arguments in the following will mainly be concerned with the looks of normal dogs and postmen and not with the exact nature of their regular meetings. I use explicit quantification over worlds here, following the analysis of counterfactual statements proposed by Lewis (Lewis 1973a.b). The box-arrow is to be read as qwlifying the universal quantification over possible worlds in the sense defined by Lewis. 6 The shortcut 'nearby' in the paraphrase abbreviates the Lewisian quantification in counterfactual implication: Wo
F 11t/J D-+ 111/J iff
for all worlds w where tf> is true, there is a world w' which is equally or more similar to W0 such that for all further worlds w'' of equal or greater similarity to W0 where t/J holds true, TjJ will be true also.
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world. For example, we should not reach a world ul' where postmen generally get special dog training in order to impress dogs. If that was the case, the normal postman in such a world would probably not be bitten by dogs, and thw the universal in (6.2) would become false. Therefore, such a ul' should not be element in the dispositional orbit DO of woHowever, it makes sense to argue that, if we look for triples of a dog, a postman and a meeting of these which are like those that we see in our everyday surroundings, then the postman of the triple should be like the postmen here in such triples, and in particular he should not have special dog training. The worlds in DO are there to provide counterexamples to accidental universal truths in our world, and to give examples for concepts which happen to be empty, but they are not there to change dispositions. Things are different in counterfactual embeddings of generic sentences. Sentence (6.3) provides an example where the antecedent of the counter factual already carries w into counterfactual worlds which have disposi tiona! orbits quite different from ours. In such a world, different generic statements can hold true. The representation is given in (6.4).5
Regine Eckardt
:Z.SI
the world we started from. In particular, even in worlds where postmen get dog training, normal dogs should remain as they are in the actual world. Dogs should not, for example, change their character and all act like pitbull terriers who would fiercely attack even the best-trained postman. In fact, all schemes of improving the chances of postmen against dogs are based on the expectation that dogs will remain like they are while postmen get better equipped for their job. Let us also have a look at nested generic quantifications, which turn out to be another case of modality in genericity, for example in relative clause constructions as in (6. s ):
·
The representation in (6.6 d) explicates that we make a generic statement about people with certain dispositions, that is for whom other generic statements are true. I give some intermediate steps of the derivation in (6.6a-c). The parts in boldface are those that are new, respectively. (6.6)
>.w>.y'Vw' (w' ::::: w FEAR(y x, w'))
a.
,
(6.6) b.
�
'Vx(N(>.W>.x.DOGw(X) )(x, w')
�
= the property of being someone who is afraid of dogs >.w>.y (MAN,(y) &
'Vw'(w' ::::: w -+ 'Vx(N(>.WAXDOGw(X))(x, w') -+ FEAR(y, x, w')) )
= be a man who is afraid of dogs (6.6) c. N[>.W>.x>.Y{DOGw(X) &
MANw{Y) & Yul(ul :=::: s -+
(6.6) d.
'Vx(N(>.W'>.X'DOGw(x'))(x, w') -+ FEAR(y, x, w'))))]
. = property of being a normal dog-man-pair such that the man is afraid of dogs
'Vw"' (wo :=::: w"' -+ 'Vx'Vy(N(>.WAX>. Y(DOGw(X) & (MANw(Y) & 'Vul(w' ::::: W-+ 'Vx(N(>.W'>.X'DOGw(x'))(x, w') BITE(x,y,wj ) -+ FEAR(y x, w')) ))](x,y,w*) ,
-+
In (6.6)b, we collect all pairs of men and worlds such that 'man a is afraid of
normal dogs, in w'. This tnight be the case because a is a shy person in w and has had bad experience with dogs, which behave like our dogs do otherwise. However, it also might be the case that normal dogs in w are very fierce animals; they might be 2 metres high, very aggressive, and a needs no special characteristics in order to be afraid of them. It even might be that a is a lunatic and dogs are all very small, peaceful and friendly animals but he fears them nevertheless. (It might even be that all men show this kind oflunacy in some world, such that even the normal ones are afraid of friendly dogs.) The operator N in (6.6c) takes an argument that is the
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(6.s) Dogs bite men who are afraid of dogs.
252. The Meaning of Generic Sentences
conjunction of DOGS and MEN who are afi-aid of dogs in any sense. Only in (6.6d) do we restrict our attention to those worlds that are in the dispositional orbit of w the world of evaluation. In doing this, we get rid of worlds where a normal peaceful, small, friendly dog meets the normal lunatic who is afraid of dogs-and does not bite him. Formula (6.6d) states that in all worlds that are dispositionally like our own, a normal pair comprising a man who is afi-aid of a normal dog (in our sense) is such that the dog will bite the man. Ot
I propose to distinguish between normal -generic statements and ideal generic statements. In this section, the distinction will be motivated in terms of plausibility arguments. The following (true) generic sentence is known from the literature as a puzzling case:7 (7.1) Turtles live to be a hundred years or more old. Sentences like (7.1) were used in the first place to argue that generic quantification does not amount to a quantifier like 'most' or 'the most common'. We know that the vast majority of turtles die very young and only the strongest, happiest exemplars live to a biblical age. It is simply false to claim that (7.1) is true because most turtles live to 100 years. However, sentence (7.1) is problematic even for more sophisticated theories, as Pelletier & Asher (1997) demonstrate. Those Best World Theories that imply the existence of absolutely best worlds (e.g. Delgrande 1987) will have to assume a world where in fact all turtles do live for 100 years. It is easy to imagine that such a world would not be the absolutely best world in many ecological respects. Pelletier & Asher represent sentence (7.1) as in (7.2): (7.2) 'v'x (TURTLE(x)
>
LIVE-T0 -1oo-YEARS(x))
Thus, we take any b, move to a world that is normal for b being a turtle, and see b become very old. As we can check for each turtle separately, Pelletier & Asher would argue, there is no need for worlds where all turtles get old at once, and ecological disasters are avoided. 8
7 Quoted from Pelletier & Asher (1997: 1164).
8 Remember, however, our observation in section 4 that stepwise quantification requires that objects once made normal remain normal in the next step. Therefore the question whether Pelletier & Asher can avoid such ecological disasters is not finally settled yet.
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7 NORM AND I DEAL: THE 1 00 -YEAR- OLD TURTLE
Regin,e Eckardt 253 In principle, we can easily mimic the analysis in (7.2) by the representa tion in (7.3). Yet I will argue that (7.3) offers a hint that there is more to the roo-year-old turtle than that captured in (7.2)/(7.3). (7.3)
'v'w(w � W0 -+ 'v'x(N(AWAX.TUR1LEw(X)) (x, w) LIVE-TO- roo-YEARS(x, w))
-+
Taking (7.3) seriously, we have to acknowledge that not many turtles are
· normal in our world. Even ifwe ignore the many little baby turtles that get
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eaten soon after birth and argue on the basis of turtles-we-meet (compare section 5), we have to face the fact that (7.3) requires that the turtle we normally meet is not a normal turtle. In the extreme, it would even be possible that there was but one known turtle that was observed to be that old. Given the biological background we have, we would still accept (7.1) to be true. This amounts to the claim that there was but one known normal turtle. It is certainly obscure to claim that all the generic knowledge we have about turtles has come about from the observation of this one single exemplar. The Methuselah Turtle might have many accidental features. We perhaps acquired our knowledge about interbreeding behaviour from quite different turtles, and gourmets' expectations about the taste of turtle meat were certainly shaped by younger exemplars. The standard reply to that kind of worry would be that generic generalizations are something more sophisticated than talking about 'the average Q': generic sentences need not always talk about the average Q. Yet there is a certain tension between the standard reply and the equally plausible claim that 'people notice regularities in nature, and form . . . folk laws to codify these regularities and to predict what the future will bring' (Pelletier & Asher 1 997: I 129). The quotation offers a sane conceptual basis for generic quantification. The position underlying the standard reply, on the other hand, can only say that there might be some basic cases in which 'generic' means 'normal' but that, in an undescribed process of holistic integration of all kinds of bits of folk-law, the * operator, or N functor, emerges which selects worlds/objects due to laws quite different from those of statistics. I propose that we should treat (7.1) as a different kind of generic sentence, and we should do that because it actually is different from statements like (6.r). (7.1) states something about the potential age of turtles, but matters are such that the normal course of the world rather inhibits turtles from exemplifying this potential. A turtle that is 'normal' in the sense of (7.1) is one where, quite un-normally, all those incidents that cause an early end for most turtles did not take place. The 'normal' turtle in (7.1)
254 The Meaning of Generic Sentences and the 'normal' course of events are in conflict. Therefore, I propose to analyse (7.1) in terms of ideal rather than normal turtles. Generalizations like (6.1), on the other hand, are such that the normal Q (dogs, postmen, etc.) talked about are those Q which are the product of the normal course of the world. A normal world does not prevent dogs from developing an infelicitous sportive interest in postmen's trousers. And it is not the case that only those postmen who are spared the experience of the everyday postman show dog-incensing timidity: normal postmen are those we find in the normal world. We thus need to distinguish nonnal from ideal generic sentences:
•
•
Normal Qs are those produced by the normal course of events. Ideal Qs might be rare, because the normal course of events inhibits them. Thus, we select normal and ideal Qs by different criteria. Generic sentences about normal Qs can be falsified by pointing out a large number of counterexamples. Generic sentences about ideal Qs are immune to such an argument. Many properties that we observe with 'normal Qs', and that are not supported by any kind of theory, are-perhaps accidentally-never exhibited by one of the rare ideal Qs. If our generic beliefs about Qs were based on ideal exemplars, the corresponding generic ·sentences should be judged false. This is an intuitively wrong prediction.
Let me give another example. It is well known that books on mushroom with drawn pictures are more reliable guides for the mushroom-hunting gourmet than books with photographs. The reason is that it seems to be almost impossible to take a picture of an exemplar that has escaped all damage to its ideal shape. The painter can develop a picture of an ideal exemplar, relying on other, nonna/-generic sentences: 'Normally, we find a bite like that when a snail has eaten from a mushroom. Thus, a snail has damaged this mushroom. If the mushroom had escaped this damage, the bite would be missing.' We know what an ideal undamaged mushroom looks like because we know what features are the result of damage. And, even if there was not a single undamaged toadstool in the world, we would still be sure that the undamaged ideal fly agaric was poisonous, because we know that all normal ones are. In formal terms, we will assume a further family of functors (In)n N of the same type as the 'normal' functors we have used so far. We can replace N by I throughout all definitions and will get a semantic account for ideal,. generic sentences analogous to the account for normal-generic sentences. For now, the difference between a representation in terms of N and a representation in terms of I is barely a conceptual one. The points at which
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•
Regine Eckardt 255
this conceptual difference yields quite concrete consequences will be pointed out in the next section. Clearly, Best World Theory would have the parallel option to add a second * operator which selects for <1>-ideal rather than
A more far-reaching observation in favour of separating ideal-generic and normal-generic sentences concerns the desirable link to theories of speakers' reasoning about new objects introduced in discourse. Formal investigations of speakers' understanding of longer texts have revealed that, in order to 'make sense' of a given piece of text, they have to rely substantially on defoult inferences in order to conjecture plausible anaphoric links, interpret definite descriptions and to form hypotheses about temporal and causal relations between the facts and events reported. It is natural to assume that the contents of generic sentences form, so to speak, the database in such everyday reasoning and that nonmonotonic logic investigates the inference patterns that speakers apply. Finally, speakers can apply generic laws to the special case if they assume that, unless told otherwise, the individuals they talk about are as normal as can be in the categories as members of which they were introduced in the discourse. This is indeed the intuitive core of several attempts to offer a formal spell-out of this link (see Delgrande I988; Asher & Morreau I995; Pelletier & Asher I997). Once more it turns out that normal-generic sentences play a different role in this kind of reasoning from ideal-generic sentences. Assume that we are engaged in a discourse about the turtle Agatha. While we can safely hypothesize that Agatha is a normal turtle who lives under water, moves slowly, is dark-greenish, etc., we will be more hesitant with respect to the assumption that Agatha will live to be roo years old. {One might compare one's own expectations about the age one will reach with the generic statement that humans live to be 90 years or more.) We have argued that the ideal is not usually what· we meet every day. In a normal world, Agatha will most probably not be an ideal turtle. Accounts of generic sentences that do not make this distinction will, however, predict that we expect Agatha to become a Methuselah Turtle as naturally as we expect her to live in water.
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(7.4)
256 The Meaning of Generic Sentences
8 NORM AND IDEAL: SOME L OGICAL CONSIDERATIONS
(8.I) (8.Ia) (8.2) (8.2a)
(N I): V'x(N (.-\ WAX.\Pw(X))(x, w) � �(x, w)) N(A) C A (I I ): V'x(I (.-\WAX.�w(X) )(x, w) � \P(x, w)) I (A) C A
If something is, for example, a normal cup, it must in particular be a cup, or if someone is an ideal postman, then he must also be a postman. Not only
are these assumptions intuitively plausible, but similar requirements are also made in Best World Theories, under the label of FACTICITY (see section 2). The next point is already more problematic. If there are Ps, should there also be normal/ideal Ps? Intuitions diverge at this point. With respect to normal Ps, it can reasonably be argued that no natural class of objects consists entirely of exceptions. Perhaps the class falls into objects of varied shapes and properties which we all encounter with equal frequency, but then this rather means that all of them are equally normal, and not that all of them are equally abnormal. Elaborating this thought (N(P) = P) amounts to the prediction that normal-generic statements about the class in question amount to, at least in the world under discussion, universal statements about P. This seems a fairly innocent conclusion. Matters are evidently different for ideal objects. In stressing that ideal objects in P might arise only under quite non-normal circumstances, we
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In this section, the distinction between normal-generic and ideal-generic sentences will be further defended looking at the different logical consequences speakers draw from either kind of statement. The present section discusses some axiomatic restrictions on the behaviour of normality and ideality operators. In testing our intuitions about appropriate axiomatic restrictions and their logical consequences, we find further support for the distinction between ideal-generic statements and normal-generic statements. While I will propose axioms for the normality operators N that turn normality into a quasi-statistical notion, the ideality operators I behave differently. The relation between N and certain similar proposals made in the AI literature (which elaborate the statistical origin of the definitions in more depth than I can do here) will be discussed in the Appendix. The first requirement that is reasonable for both normality and the ideal is that normal �·s milst be \P's, and equally, ideal \P's must be \P's. This is reflected in (8. I ) and (8.2). The appropriate first-order axioms are given together with the respective set-theoretic clauses in order to facilitate reading.
Regine Eckardt 2 S 7 already allowed for the possibility that some quite normal worlds (among them the real one) could be without any ideal P objects, for some categories P. Even if there are Ps, I(P) might be empty. Therefore, I propose the following non-emptiness condition only for N, but not for 1.9 (8. 3) (N 2): �(x, w) --+ 3xN(.XWAX.�w(X) )(x, w) (8. 3 a) A :/: 0 --+ N(A} # 0
CONTRAPOSmON.
(84)
(84a) (8.s)
(8.sa)
10
3x(N(.X WAX.�w(X})(x, w) 1\ w(x, w)) --+ V'x(N(.XWAX.�w(X) 1\ 'lll w(X))(x, w) N(.XW.XX. � w(X) )(x, w) 1\w(x, w)} N(A) n B N(A) n B # 0 --+ N{A n B) (I 3 ): 3x(I(.XWAX.�w(X))(x, w) 1\ w(x, w)) --+ V'x(I(.XWAX. �w(X) 1\ Ws(X} )(x, w) --+ I(.XWAX. �w(X} )(x, w) 1\ w(x, w)) I(A) n B # 0 --+ I(A n B) C l(A) n B (N 3):
�
=
(84) requires that if there are normal Ps that are Q at all, then the normal . P-and-Q's should be those normal Ps which are Q. It predicts, for example, that if we find normal postmen who have a beard then the normal postmen-who-wear-beards are those normal postmen who wear beards anyway. No different standards of normality should apply as a result of further known circumstances. In the case of idealness, the weaker requirement (8.s) can be paraphrased as 'if some ideal Ps are Q then the ideal P-and-Q's should at least be a . 9 We will see reasons that allow I to be a partial function. We seem to have stlltistical folk knuwkdgt about any kind of category, but not all categories need to have something like a 'prototype'. 10 The list docs not exhaust the range of'prominent' patterns in nonmonotonic reasoning but will
serve to exemplify the usefulness of (8-4)/(8.;). For a fuller discussion of the status o£; and logical relations among such inference patterns, see Chellas ( I 974), Ginsberg (1994), Kraus/Lchmann/Magidor (1990), Rott (1996)/(to appear). to name but a few. The Appendix shows how our approach tics in with this kind of literature.
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The lack of an analogous axiom (I 2) will lead to different predictions about the logical strength of normal-generic and ideal-generic sentences which are empirically justified. We will come to these in a moment. The final restriction will relate normal/ideal Ps to normal/ideal Ps that are Q. Axiom (84}/(8.5} are motivated by more intricate considerations than (8.r}-(8. 3). Basically, they will allow one to maintain the global perspective that everyday default reasoning is reasoning in terms of generic sentences. (8.4) is the weakest way of restricting the range of possible N functors in a way such that N will support the default inference patterns discussed below: RATIONAL MONOTONICITY, CAUTIOUS MONOTONICITY and WEAK
258 The Meaning of Generic Sentences
subset of the ideal Ps that are Q.' Once more, this reflects the fact that the logic of ideal objects is different. The ideal objects in a large and undifferentiated class can be less specific than the ideal of a more narrow class. Consider the following example: (8.6) Cups are (ideally) made of porcelain Porcelain cups (ideally) have a gold rim But: Do cups indeed (ideally) have a gold rim?
(8.7) Dogs (normally/ideally) don't suffer from cancer. Dogs who suffer from cancer may have quite different properties from dogs in general. Before moving on to such cases where statements about atypical exemplars are made, I shall list some simple consequences of (Nr)-(N3) and {I r ) and {I 3 ). Unless stated otherwise, the laws hold for both, normal-generic and ideal-generic statement. RATIONAL MONOTONICriY:
IfAs are 4 and it is not the case that As are non-B, then As that are B are C. Proof. Assume N(A) � C, and not N(A) � -, B. Then N{A) n B "# 0. and therefore N(A n B) = N(A) n B. Thus N(A n B) C N(A) � C. This holds in all possible worlds. For I, we must argue in some more detail. If I(A) =f 0. the conclusion follows as in the above case. However, we might be in a world w where I(A) 0. Yet there must be worlds in the DO of w where I{A) f; 0. as otherwise the second, negated ideal-generic sentence would not be true. (If we had no ideal As anywhere, anything could be claimed to hold true about ideal As.) In all those worlds, I(A) =f0 and moreover I(A) � C, as otherwise the first ideal-generic sentence would not hold true. As before, we can deduce that I(A n B) � C in all these worlds. RATIONAL MONOTONICriY is one of the most widely accepted requirements to generic implication, and default reasoning in general (see Pelletier & =
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Although the notion of a cup evokes the material 'porcelain', the idea we have of a spitting image of a porcelain cup is even more specific than the ordinary cup-of-porcelain. This is captured by (8.5) while retaining the intuition that even ideal P-that-are-Q should not lie far beyond the ideal P, as long as Q in and of itself is not an entirely un-ideal property of P. Note that if the latter should actually be the case, that is, if Q is a somehow odd property, normal/ideal P-and-Q are free to have any appearance. This is illustrated by the following example:
Regine Eckardt 259
Asher 1997, as well as the sources quoted therein). below:
An
example is given
(8.8) If dogs have hair, and it is not the case that dogs are generally not brown, then brown dogs have hair, too. CAUTIOUS MONOTONICITY:
IfAs are normally B, and As are normally G then As that are B should normally be C.
Proof Assume that N(A) � B, and N(A) � C. As N{A) n B is thus nonempty. N(A n B) = N(A) n B, and therefore is a subset of N(A). Therefore N(A n B) � C. In the case of ideal-generic statements, we have to make the assumption that there are worlds in the DO of ours such that these contain ideal As. As before, the conclusion follows in these worlds. in the same way as in the proof for the normality case. Given that generic sentences make necessity claims about all worlds in DO, the claim about ideal-generic statements follows. (8.9) If dogs have four legs, and dogs love sausage, then dogs that have four legs love sausage. Note that Pelletier & Asher doubt the validity of inferences like (8.9). I propose that this is the case because we tend to give causal force to the more narrow description of dogs in the consequent of (8.9): not only do dogs that have four legs love sausage, but they do so somehow because they have four legs. The sentence 'dogs love sausage', in contrast. suggests that dogs m general love sausage, not only when or because they have four legs. However, if we strip off these further implicit assumptions of the sentence, the result is a convincing deduction. (8.Io) Dogs have four legs. Dogs love sausage. Dogs that have four legs love sausage. A final property that can be proved for normal-generic sentences, yet not for ideal-generic sentences, is the principle of WEAK coNTRAPosmoN: WEAK CONTRAPOSmON:
If normal As are B, then ..,
(i) either normal ..., B's are A, (ii) or else .., B is an un-normal case altogether.
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Suppose that there are ideal As. Then, ifAs are ideally B, and As are ideally G then ideal As that are B should ideally be G too.
.2.6o
The Meaning of Generic Sentences
I will first give examples for both cases of the disjunct Example (8.I I) looks like a case where something like a principle of contraposition would be quite welcome. Example (8.12), in contrast, illustrates a case where general contraposition would yield absurd consequences, and which exactly fits the second case of the above disjunction. This discussion is adopted from Ginsberg (I 994). 1 1
(8.I I) Koala bears live on eucalyptus. Animals who don't live on eucalyptus are normally not Koala bears. (8.12) Men (normally) do not suffer from diabetes. ·f+ People who suffer from diabetes are normally female. --+
11 Note that we, following Ginsberg and similar discussions by von Fintel (1998), talk enrl!e universe were, for these marten, restricted to animals or humans, respectively.
as
if the
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Formally, the relation between both pairs of sentences in (8.I I) and (8.12) conforms to the pattern known as contraposition. The implication in (8.I I) looks sound. Something like it might actqally be in use when a biologist, knowing not much more about Koala bears than (8.I I), looks out for Koala bears in the jungle: If the putative Koala bear is seen regularly eating coconuts, the biologist will adopt the hypothesis that the animal is probably not a Koala bear after all. Yet example (8.12) would clearly be an undesired implication. The reason seems to be that suffering from diabetes is uncommon both for men and their complement (in context), women. Therefore, if normal men do not suffer from that disease, normal women may not suffer from it either. These two cases are exacdy what is allowed by WEAK CONTRAPosmoN. The proof that it holds true for N is not so simple as the properties we checked so far. In particular, it relies on (NI) to (N3) in full strength and therefore will not carry over to ideal-generic sentences. However, this might even be desirable. We argued in section 7 that normal-generic sentences and ideal-generic sentences should be kept separate. Ideal-generic statements say something about how a· member in P will look like under maximally undisturbed circumstances, while . normal-generic statements say something about members of P under 'normally disturbed' circumstances. Being confronted with some new, so far unknown, P object in conversation, one will assume that it most likely is a normal P, but probably not an ideal P. Therefore, default conclusions on the basis of normal-generic sentences, but not ideal generic sentences, will be drawn. Another facet of the same observation is that contrapositions of ideal-generic sentences like the one in (8.13) are
Regine Eckardt 261 intuitively wrong. (8.1 4) offers the general pattern which discussion below.
is used
in the
(8.1 3) Turtles live to be Ioo years old. f+ Animals who don•t get to be 100 years old are not turtles. (8.1 4) As are ideally B. f+ non-B•s are non-A. (in whatever reading)
In (8.1 3), it is not the case that the negated consequent ..., B, 'not living to be
·
12 An equally satisfying assumption, conceptually, might be that I(-. B) = 0 in such cases. However, this would lead to evident complications, because the respective generic sentences would then be predicted to be, first of all, true (empty universal quantification), and funher assumptions would be necessary to explain why that makes them unacceptable.
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IOO years old•, is generally a marked (non-normal or non-ideal) property in the animal kingdom, like 'suffering from diabetes• was for humans. Therefore, the failure of (8.1 3) would not be captured by something like case (ii) of weak contraposition. The deeper reason is that no version of contraposition makes any sense for ideal-generic sentences. Knowing that the ideal A has property B, we will not conclude anything about -, B•s, normal or ideal, and for various reasons. If we read (8.14) as talking about normal ..., B, we have to take into account that we were talking about ideal As and may therefore not be surprised if quite normal, but simply un-ideal As already show ..., B. Therefore, ..., B in the normal case should not lead us to conclude anything about being A. (If an animal dies at the age of So, it may still be a turtle without being anything extraordinary otherwise.) If we read (8.14) as taking about ideal ..., B, we will very often make the observation that the complement of a category, which is such that we have a notion of an ideal exemplar in that category, is not such that a similar 'prototypical -, B• would exist (see fn. 9). Consider the above example: what should an ideal animal-which-doesn•t-reach-Ioo-years look like? Would it rather be furry or skinny? Does it have four legs or more? These questions seem hard to answer. Thus, the implication in (8.14) in such cases does not hold true for reasons beyond those captured in weak contraposition: the second sentence in (8.14) is, first of all, a generic sentence. If we read it as a normal-generic sentence, the implication becomes false because normal ..., Bs may be A in spite of ideal As being ..., B. If we read it as an ideal-generic sentence, it is probably not even interpretable because the antecedent 'animals of age less than IOO years• is not an appropriate argument of the I functor. We will capture this by assuming that I are partial functors.12 However, sometimes matters are such that the complement ..., B in context is a category that is an eligible argument for I (that is, is the kind of
262 The Meaning of Generic Sentences
category where it makes sense to ask about ideal exemplars). One of the most prominent constellations is, of course, that we are in a context where our attention is restricted to humans, such that B and ..., B correspond to males and females. I am aware of the difficulty of producing any inoffensive generic statements involving gender, but let me, for this one occasion, nevertheless try. 13 (8.1 5) Ideal gynaecologists are women. Ideal men are not gynaecologists.
·
e w
(i) R, maps D" into Dn. (ii) for each subset A of D" which is definable in the logical language L we use N,.(A) = {a E A j R,.(a) is minimal in A} in w. Intuitively, the mapping R, transfers the ordering of Dn on to D", the set of all n-tuples over the univers� D. The ordering of D" is to be read as a ranking according to degrees of abnormality, such that the elements of lowest rank are the most normal ones while the ones with higher ranks are the more and more unnormal ones.14 Figure 1 illustrates this correlation. The proof that this is the case will be given in the Appendix. The use of such ranking functions, based on ordinal numbers n, was proposed by Spohn (1988). Its generalization to sets of individuals was first suggested independently by Brafman (1996) and Weydert (1997). The latter two approaches, however, are explicitly set up in purely extensional terms, and it is not· immediately evident how a modal dimension is to be 11 I can adopt at least the first sentence among my personal ideals-but remember that 'ide2l', like 'normal', in the end is a notion that reflects a personal perspective on the world . . . •• The direction of this ordering is a heritage of the statistical origins of this kind of modelling. I keep it in order to facilitate the formal comparisons to be made in section 9· I apologize for the extra effort'thereby caused for the reader. It takes some time to get accustomed to the inverse correlation that the more normal an object. the lo� its rank.
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Intuitively, the two sentences are logically independent. Or, more bluntly, why should our image of the ideal gynaecologist have any influence on our ideas about ideal men? To summarize, intuitions with respect to contraposition provide further evidence in favour of a distinction between normal-generic and ideal generic sentences. It remains to be shown that weak contraposition for normal generic-sentences follows from (N1)-(N3). This will require some more technical considerations. It can be shown that axioms (N1)-{N3) induce a global ordering of all objects in the universe of the model, according to their degree of normality. Formally, we can derive for each possible world w a family of functions (R,)n and ordered sets Dn such that
Regine Eckardt 263 �-----
very normal
rather unusual
fairly normal I
introduced. We have argued at various places (and follow the general discussion on generic sentences in that respect) that a purely extensional treatment of generic sentences is inadequate. In this sense, our treatment extends these approaches in a way necessary for reasonable application in natural language semantics, even if (as will be shown) our treatment of normality, extensionally speaking, is equivalent to the proposals by Brafman and Weydert. With this ordering at hand, we can now prove WEAK coNTRAPosmoN for N,. Let me repeat the claim in a more formal manner: WEAK coNTRAPosmoN: Let M be a model of a language L that includes a family of normality operators (NJn e w as introduced above. Assume that (N 1) to (N 3) hold true in M. Let A and B be £-definable subsets of D" such that N(A} � B. Then (i} N(..., B} � -, A or N(B U .:., B) � B
(ii)
Proof. Assume that (i) does not hold true. This means that N(..., B) n A # 0, and we now have to show that N(D} c; B. If N(..., B) n A # 0. then we know that an element x in the intersection must have higher rank R, (x), that is, be less normal, than all elements in N(A), because the latter-most normal ones in A-are all in B. The normal elements b in B in turn are either in part among those in N{A}, or else are all outside A and have thus even lower rank (= are even more normal). In any case, R, (x) � R, (b), which means that the most normal objects in D are in B. This is exactly what (ii) states. Before concluding this section, I will offer some more sample constella tions of global ordering according to normality. The first constellation, depicted in Figure 2 is one where we have A and B intersecting, and where some, but not all normal A are normal B.
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Figure
set A
.
264 The Meaning of Generic Sentences (females)
A = males
------- .·-··--······-····-··...···········-··
B
= parents
···············-·····························
N(A)
We could, for example, instantiate A as male and B as parent. The above picture would then say that some, but not all, normal males have children. (Or, in other words, neither sentence 'men have children', nor 'men don't have children', is a true normal-generic sentence in our world-which I take is a valid assumption.) Given this, the global notion of normality encoded by (NI )-(N3) will predict that the normal male parent is simply a normal male who happens to have children. In particular, this has the consequence that all normal-generic sentences about males carry over to males with children, which is a reasonable prediction. The diagram in Figure 3 offers a graphical summary of the diabetes example. Normal males do not suffer from diabetes. Nor do normal females. Suffering from diabetes is just an extraordinary property. Finally, Figure 4 illustrates the widely discussed Quaker/Republican example. We can have two properties A and B where N(A) does not intersect with B, and N(B) does not intersect with A either. No further implications follow. This is exemplified (quoting generic statements about B=
A = males
C = diabetes
Figure 3
females
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Figure z
Regine Eckardt 265 A == republican
G
G B
=
quaker
Figure 4
9
SOME CONSIDERATIONS AB OUT THEORY ARCHITECTURE
In this section, I will discuss potential circularities luring in Best World Theory and Normality Based Theory. I will argue that the distinction between normal-generic and ideal-generic statements is a first step towards avoiding such circularities. How do generic sentences relate to the more basic facts in the world? It seems reasonable to assume that generic facts, or at least some of them, are something like rough generalizations over simple facts (i.e. facts about single objects and their properties}. We have already seen some statements to this end in the literature on Best World Theory.
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political interests from the literature) by the observation that Republicans normally are not Quakers, and Quakers normally are not Republicans. Logically speaking, this leaves all options open for those persons who are both a Quaker and a Republican. They are just not cove�ed by the normal generic knowledge expressed in the sentences 'Quakers normally are not Republicans' and 'Republicans normally are not Quakers;. To summarize, the proposed set of three axioms turn out to restrict the logic of N, the normality operators, in a strong way. I will argue in the next section that such a strong version of normality is necessary for general reasons. Yet notice that one advantage of the account is its flexibility. The precise nature of axiomatic restrictions we want to adopt for normal generic, or for ideal-generic statements, can easily be adjusted according to further considerations, while the underlying intended content of the operator remains stable. This distinguishes our account from proof theoretic approaches to default reasoning, where new sets of derivation rules lead to entirely new logical systems without making it clear in what sense all are competing spell-outs of a uniform underlying idea.
266 The Meaning of Generic Sentences
(9.1):
(9.1}. Let
Ll be the set of all desirably true generic sentences GENx (J ; about 'Y in w,. A world w is normal for proposition 'Y(a) (that is,. w E *(w0, 'Y(a)) } if and only if w f= 8(a) for all 8 such that GENx {'Y ; 8 ) is in Ll.
8)
What (9.1) states is this: you want to know what generically holds true for 7's? Look into all those worlds that are normal for 'Y· How do I know what worlds are normal for 7? First of all, they have to fulfil all generic implications for 'Y . . . . 15 Now, (9.1) would be innocent if any independent criteria were in sight to characterize *(w0, 'Y)· Yet we have seen that the shape of object a is not enough for a world to be normal for 'Y(a), that even the idea that generally everything should be as normal as possible for 'Y was not enough, and that in the end *(w0, 'Y) was the holistic integration of various complex steps of generalisation. Therefore, (9.1) for now is all Best World Theory can offer 16 when it comes to characterising the star operator *. Naturally, a different set-up does not automatically prevent such obscurities. An uncharitable characterization (9.2) of normality N can make the same point against Normality Based Theory, as (9.1) does for Best World Theory: ·
(9.2) An object a is normal in 'Y· a E N('Y), only if for all generic implications 8 we want to hold true for 7's, 8(a) holds true. Yet, as normality N is something we first and most simply evaluate in our own real world, it can also be understood differently. We have consistently 15 I omitted universal quantification and instantiation in the paraphrase, for the sake of brevity. It might not be an accident that Pelletier & Asher even use a similar formulation in their definition that was quoted here in (2.3� They do not seem to find this a problem. 16
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However, Best World Theory ultimately does not reflect much of this insight. In section 3, it was shown that 'normal objects in P can be defined only indirectly, and that the *-operator does not systematically build on 'everyday objects in our world'. In section 4, we saw that the choice of worlds in *(w,p) has to respect the properties of previously chosen individuals a which were ideal in w for some property q,. (Of course, the more choices have to be made in stepwise quantification, the more individuals the *-operator will have to keep in mind.) In section 7 I showed that worlds w in *(w,p) are sometimes selected rather for idealness than the normality of the matters in question. Thus, it is not clear whether the 'meaning' of * can be characterized in independent terms, and for now the only safe thing that can be said about * is the somewhat uncharitable
Regine Eckardt 267 presented normality as a kind of folk-statistical notion. This led to the more specific observations that •
• • •
Normal encounters of participants of different kinds need not be the same asencounters of normal participants of these kinds Normality involves a perspective Normality is not the same as the ideal Normality is restricted by axioms {Nr)-{N3)
17 Again. under generally unchanged circumstances, which probably include something like the expectation that not much mail from Antarctica will ever arrive. Otherwise the office might decide to employ a further secretary in charge of the fans at Antarctica.
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Therefore the first important building block of the theory, normality, can be taken to be an independent notion. The second building block in the interpretation of (normal-)generic statements was the choice of the dispositional orbit DO of the world of evaluation. What is the status of this accessibility relation? In order to answer this question, it is helpful to remember why the dispositional orbit was introduced. The worlds in DO are needed to exemplify cases that by accident are not instantiated in the real world: the pope called 'Bartholomew', or the first letter from Antarctica. They should not provide counterexamples to generalizations that we actually want to become true Qike the normal pope called 'Goofy'). On the one hand, such worlds can exemplify cases where a real accident seems · to have played a role, and I assume freely at this place that we have a notion of 'accident' available. Yet it might well be that such cases are not of overwhelming importance because upon closer investigation it turns out that the notion of an 'accident' is itself based on knowledge of many similar cases which ended differently, and thus knowledge about. the general limits to the possibilities available. This leads us to a second, and perhaps more important, function of the DO. There are cases where the DO actually does reflect modal knowledge that lies at the basis of generic sentences. Consider one last time the example 'Rose handles mail from Antarctica'. This scenario certainly does not say anything about the normal case in a statistical sense, because there is no meaningful statistics over the empty set of events. In contrast, it is natural to assume that the sentence reports some kind of intentional (and thereby intensional) planning in Rose's office. Discussing various counter factual scenarios of what might happen, the employees of that office decide that the counterfactual scenario of mail arriving from Antarctica should end so that Rose is the person to take care of it.17 DO is shaped by our plans and intentions. This second function of DO is in concord with a distinction drawn by Carlson (1995: 233£), the distinction between generic sentences
2.68 The Meaning of Generic Sentences
Acknowledgements The research underlying this paper was conducted in project AI, SFB 47I Konstanz, funded by the DFG (Deutsche Forschungsgemeinschaft). I would like to thank Urs Egli. Manfred Kupffer, Hans Rott, Wolfgang Spohn and Emil Weydert for many fruitful discussions. Miriam Butt mercifully helped me at several stages to draw the language closer towards Standard English. All remaining errors (in content and form) are my own. REGINE ECKARDT University ofKonstanz Fachbereich Sprachwissenschaft/SFB 47 J 78457 Konstanz Germany
[email protected]
18
Received: 12..02..I998 Final version received: I 8.04-2.000
It would probably surprise Carlson to see that an account called 'Normality Based Theory' is one where this distinction can be discussed cleanly. He expresses the hope that an analysis for the 'rules-and-regulations' examples might lie more in the direction of proof theoretic accounts for default reasoning. Pelletier & Asher (1997), however, argue nicely that these accounts are not satisfying on semantic grounds. Another direction for analysing the 'rules-and-regulations' examples, which was pointed out by Carlson, is the Best World Theory proposed in Asher & Morreau (1995). For a more recent proposal, see Cohen {1999).
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which have an inductive flavour and generic sentences which seem to call for a 'rules and regulations' approach.18 One function of the DO is to take care of the 'rules and regulations' aspect of genericity. Yet the DO is combined with the (more) inductive notion of normality, such that the overall theory allows to capture mixed cases, like the pope example: certainly, there are no explicit laws about names for popes. but there are some general expectations and guidelines about how to behave when you are to be God's representative on earth. These will exclude worlds in the DO where popes eagerly chose the name 'Goofy', while other papal names are allowed even without an explicit list of eligible names hidden in the safe of the Vatican. In summary, the analysis of normal-generic sentences is built on the independent notions of normality and dispositional orbit. This is, however, only a partial answer to the question whether the overall theory is free from circular definitions, because I cannot, at present, offer a similar justification for ideal-generic sentences. My hope would be that, as sketched in section 7, at least part of our notion of an ideal is rooted in our knowlc::dge about what usually happens when all disturbing factors can be excluded. It is likely, however, that for instance perceptual notions of ideal Gestalt involve cognitive processes that are beyond the reach of this paper.
Regine Eckardt .269
APPENDIX: RELATIONS TO SOME OTHER NORMALITY -BASED ACCOUNTS In this appendix, I will prove the fact that (NI}-(N3)
tum normality N into a global notion. Our treatment of normal-generic statements on the basis of N, restricted by these axioms, is equivalent to certain statistically conunitted default logics developed in AI. I first give some definitions on which the following results are based. Note that, throughout this section. I adopt the convention that boldface variables abbreviate tuples of variables: � stands for x1,Xzo , x,.. ln the same way, boldface letters m, n . . . will stand for tuples of elemep.ts of the model domains under discussion. • • •
Operator-based normality
(NI) For all terms P of type ( en, t}, 'V� (N. (P)(x) -+ P (x)) (N.2) For all terms P of type (e", t} 3xP(x) -+ 3xNn(P)(x) (N I� Where P is a variable ranging over properties of arity n, we have 'VP'Vx(Nn (P)(x) -+ P(x)) (N.2� Where P is a variable ranging over properties of arity n, we have 'VP(3xP(x) -+ 3xN.(P)(x)) The difference between the first-order schemata and the second-order axioms is that the latter restrict the behaviour of N; with respect to all arguments while the former only make claims about definable subsets. Normality based on ranking
The logic languages that are at the basis of normality with ranking do not refer explicidy to any normal objects. Instead, they provide universal quantification that is restricted to the normal subpart to the respe<;tive domain of quantification. in a way much similar to the default implication used in Best World Theories. The following definitions are based on work by Weydert ( 1997) and Brafman (I 996), which is rooted in research in nonmonotonic logic (see Kraus et aL I990), and also on the work on ordinal conditional functions by Spohn (I988).
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Let L be a logical language with .>.-abstraction for variables that range over individuals. We can thus, even without introducing an entire hierarchy of types, define what it means for a term to be of type ( e" , t}. Moreover, we will assume that L contains a family of functor symbols (N;);e .., where for each term ¢ of type (e', t) the term N; (¢) is also of type (e', t} The functor symbols will be interpreted by functions that map the power set over D'; into itsel£ They are to be thought of as functions that map each relation on to its normal subpart. The interpretation of N; is restricted by axiom scheme (NI) that states that a normal P should always also be a P, and by (N.2) which requires that we should always be able to find some normal P. These axioms cortespond to (N I} and (N .2) in the previous section. The restrictions will come in different shape, depending on whether the language L provides quantification over relations or not. The second-order analogues are listed in (NI� and (N-2�. (Second-order versions will generally be marked with a superscript .2 in the following.)
.2.
70 The Meaning of Generic Sentences
Definition: Let L be a first order logical language which is augmented by the following kind of formulae: for all formulae ¢, t/J in L, and variables x,, , x1r the following is also a formula in L: •
•
•
The language L is interpreted in structures (M, R) where M is an L-model in the usual sense and R is a family of ranking functions of the following shape: {i) R = {Rn)n e w (ii) For each n E w, Rn is a function of IY; into an ordered set 0. (iii) For all formulae ¢ and t/J (M,R) I= .¢
-,., . . . , xi
n
E
E
{k I (M, R) I= ¢{k)} such that for all
{k I (M, R) I= ¢ {k)} with R; (n) :5 R; (m) we find that {M, R) I= t/J(n).
Further possible restrictions on R are discussed both in Brafman and Weydert. While Weydert attempts to fix the ranking so as to come close to a probability measure on the domain D, Brafinan keeps his models more flexible. Both authors give a sound and complete axiomatization for their respective versions of L The following axiom (WI) was suggested by Weydert (p.c.) in order to ensure that each L -definable subset A � U: contains one or more elements of minimal rank. These models are called 'smooth models' in the literature. where y does not share any variable with x. Intuitively, (WI) asserts that for all sets A, there are a E A such that for all c E A, R(c) �R(a). An analogous effect would be achieved by requiring that the ordered set 0 in fact has to be an ordinal (this approach is explored in Spohn I988). This even ensures that each set contains elements of minimal rank, not only definable subsets. However, the resulting models can no longer be characterized by a recursive set of axioms, which is why I refrain from this move. Ranked models can be mimicked with normality operators
Let (M,R) be an L-model with ranking. Let us further assume that (M,R) validates axiom (WI), such that all L -definable subsets in U: have minimal elements. We can now construct an L,..,""-model M which simulates (M,R): Let L,.,"" be the language with normality functors which equals L in its standard part. We will augment M to yield an L,..,rm-model M by the following definition: Nn (A) := {m e A j 'v'n e A (R. (m) :5 R. (n)) }
for all definable subsets A of U: and N.(A)
:=
A otherwise.
Now define a mapping * of Lran.l!"-formulae into Lnorm-formulae in the obvious way: For all atomic formulae i/J, let rp*
:=
rp.
For ¢ = ..., t/J, t/J 1\ 8, 1/J V 8 let ¢"' = ..., '1/J� 1/J* 1\ 8� '1/J* V 8".
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there is an m
t/J iff
Regine Eckardt 271 For rp
=
3x-rj;, let rp* := 3x-rf;*
For rp = '1/J -+" 8, let q,• := Vx(N(>.x.'I/J.,(x) -+ 8j Using this definition, we can prove the following equivalence to hold
true:
(M, R) F tP ¢} M' F q,• Proof. By induction on the complexity of formulae. The crucial step
(M, R) f: '1/J -+" 8
{:}
is
to show that:
M' f: Vx(N(Ax.'I/Jj(x) -+ 8 j
WEAK CONTRAPOSmON.
Certain
normality models can be mimicked by ranked models
The central property of ranked models is that intersections of sets inherit their normal parts from the respective supersets if the intersections are nonempty. While we get a strong equivalence between ranking and normality functors if we allow ourselves second-order quantification, we can at least translate the default implication part of the L,.,,-model with ranking. without second-order quantification. The latter is expressed more precisely in Theorem 1 1, while the former is formulated in Theorem 1 2•
Theorem I1: Let L110,. be a klnguage with normality operators and Lranlt be the corresponding klnguage with defoult implication, as defined above. (That is, both klnguages share the same classical core: the same constant symbols, relation symbols, Junctions.) Let M be an L,rm·model in which satisfies {N1) and (Nz) above, and where moreover (NJ) holds true,for all terms A, B of type (e",t} and arities n.
(N3) M f: 3x(N, (A)(x) t\B(x)) -+ Vx(N,(,\yA(y) t\ B(y))(x)
+-+
N, (A)(x) t\B(x))
Then we can define a mapping from a subset ofL,., on to L,.,k such that thefollOwing holds true:
M f: t/J ilf (M, R) f: q,# The proof is given below.
Theorem I2: Let L,.rm be a language with normality operators th4t moreover provides quantifiCJJtion over rekl�ions, and L,dnlr again be the corresponding klnguage with default implication. Let M be an L,.,-model which satisfies {N12) and (Nz2), and in which for variables A, B of type (e", t} the follOwing (N32) hold true (for all arities n):
(N3� M f: VAVB 3x (N, {A)(x) t\ {B (x)) -+ Vx(N,(,\yA (y) t\ B(y))(x)
+-+
N, (A)(x) t\ B(x))
Then M can be turned into a model Mfor L,.,Jr which remains identical with resped to the classical part ofL and where, for each definable set A
N (A) = {m i Vk e A (R,(m) :5 R,(k))}.
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This. however, is ensured by the definition of N, and the obvious obsexvation that all sets involved are definable. D The obsexvation that ranking should give rise to a notion of normality is not surprising. We can, however, prove the stronger proposition that (N 1}-(N 3), or their corresponding axiom schemes, are sufficient to restrict L...,.,.-models to those which have an equivalent L,.,�o-counterpart. It is this stronger proposition that we used in section 8 when discussing
272 The Meaning of Generic Sentences We will stan with the proof of the weaker Theorem 1 2• The argument corresponds closely to a similar construction in (Spohn 1988). Having accomplished this, we will proceed to prove the more intricate Theorem I 1• Proof of 1 2: We define a partition on the Cartesian products IY; of the domain De of individuals: m0 : = N,. (IY;) and m� :=
m0
m1 := N,. (IY;\m� ) and m; := m� U m1
m.>. := N,. (IY;\U;< >.m7 and ml := U;< >.m7 U m>. for limit ordinals. Let K be the smallest ordinal number such that m,. = 0. We can now define a mapping R,. from IY; into K, with the intention that (R,.),. e will be the ranking for L,.,.,.. For each m in IY; let (R,.)(m) := 1-' if and only if m E mw R� is thus a mapping of IY; into an ordered set, that is, an appropriate ranking function. We interpret the language L,d,.,. on the basis of this ranking function. It remains to be shown that for each set A � IY; . N,.(A) = {m I 'v'k E A(R,.(m) :::; R,.(k))}. Take an arbitrary subset A of nr;. We can now reason as in step (i) to (iv): w
(i)
There is a smallest ordinal number 1-' such that A n m�' =/: 0. According to construction, { m I 'v'k E A{R.(m) :::; R,.(k))} = A n mw (ii) According to construction, there is a set P � IY; such that m�' = N,. (P). We can even give P more precisely: P .0:\( Ui < �' m;). (iii) As m�' is the first set in the partition to have nonempty intersection with A, we know that A � P, that is, A n P = A. (iv) As A n N,.(P) =/: 0, it follows that =
M f= 'v'x(N,. (.Xy.P(y) t\ A (J) )(x)
+-+
N,. (P)(x) t\ A (x))
Therefore N,.(A} =(iii) N,.(A n P} =(iv) N,.(P) nA =(ii) m�' n A =(i) {m I 'v'k e A(R,. (m) :::; R,. (k))} as desired. 0
Note that the proof of 1 2 even validates the stronger claim that each model with functor based normality will be equivalent to a model ranked on the basis of an ordinal n. This class of ranked models cannot be characterised with a recursive set of first order axioms. We therefore know that the second order quantification which we allowed ourselves in the axiom schemes above were not only a convenient shortcut, but restricted the model class in a nontrivial way. It is mainly for this reason that the more general statement in ( 1 ) becomes interesting: We can show that not only a very limited class of operator-based models for normality are equivalent to (a subc;lass of) ranked models but that the equivalence holds in general Clearly, we will have to revise the original proo( because the construction made substantial use of the fact that the crucial property (N 3) held for all subsets of n: . In the following lemmas, we are always using models M of a language L•• without second order quantification. I will use the abbreviation A n B for .xx.A (x) t\ B (x). I will also ""
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m,. .. 1 := N,.(IY;\mZ) and mZ+ 1 := mZ U m,. .. 1 for successor ordinals
Regine Eckardt 2.73 occasionally omit the arity index n in the function N,.. All models are assumed to validate axiom schemes (NI} 1-(N3)1• The overall strategy of the proof is this: we show that N induces a linear ordering on the definable power set over U:. This is done in Lemma t --9; the linear ordering is given in Pefinition 2.. Using this ordering, we then construct an L,.n�o-model (M,R).
Lemma (i}
1:
Let M be as described. Itfollows thatfor all A, B � .U:
we
have that
N(A U B} n B = 0 and N(A U B} = N(A} or
(ii) N(A U B} n A = 0 and N(A U B} = N(B} or (iii) both intersections are nonempty and N(A « B) = N(A) U N(B�
N(A} = N(A U B} n A. Finally, we know that N(A U B} � A U B and therefore N(A U B} = [N{A U B} nA] U [N(A U B} n B]. Thus, the three constellations listed above are the only possible ones. 0
Lemma z: If N(B} � A and N(A U B) = N(A} U N(B} then N(B} � N(A� Proof We assumed that N(A U B} = N(A} U N(B) ( * ). Therefore, N(A U B} n A � 0. due to Lemma I. Thus, N(A U B) n A = N(A) (again from Lemma I}. Yet, as N(B} � A, we know moreover, due to ( * ) and N(B} � A , that N(A U B} n A = N(A} U N(B). Thus, N(A} = N(A} U N(B} and therefore N(B) � N(A). 0
Lemma 3: If N(A U B) � A, then N(A U B) = N(A). Proof In case that N(A U B} n B = 0, the claim follows from Lemma I. Else N(A U B} n B � 0. and therefore N(B) � N(A U B} � A. The claim then follows from Lenima 2.. 0
Definition
I:
We define N(A} < N(B} to be an abbreviation for
N(A U B} � A and N(A V B} n N(B} = 0 Remark: It follows easily that N(A} < N(B} if and only if N(A U B) n B = 0.
Lemma
4!
The relation < is transitive.
Proof Assume that for nonempty sets A, B, C,
(I}
(z)
N(A UB} � A and N(A UB} n N(B} = 0 and N(B U C) � B and N(B U C) n N(C} = 0.
We have to show that N(A U C} � A and N(A U C} n N(C} = 0. We do this by contraposition: assume that N(A U C} n C � 0. We will now compute N(A U B U C}. According to Lemma I, we know that N(A U B U C) = N(A U B} or N(A U B U C) = N(C} or N(A U B U C) = N(A U B) U N(C). Due to our assumption, we can conclude that in any . case
(*)
N(B} i N(A U B U C}
We will now distinguish two
cases:
N(A) n C =F 0 or N(A)n C = 0.
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Proof IfN(A U B} n B � 0 , then (N3) will yield N((A U B} n B} = N(B} = N(A U B} n B. In the same way, if N(A U B} n A � 0. it follows from (N3) that N((A U B} n A} =
274 The Meaning of Generic Sentences In the case that N(A) n C ::/= 0. we can immediately infer that N(A U B) n {B U C) :1: 0. well According to Lemma I, we know for arbitrary definable sets X, Y that N{X} n Y :1: 0 implies N(Y} � N(X U Y). Take now X to be A U B and Y to be B U C. We thus get that N{B U C} � N({A UB} U {B U C}). Due to assumption (.z), we can conclude that N(B) � N{A U B U C). This contradicts ( * ). If it was such that N{A) n C= 0. we would get that N(A U C} = N(C} (by Lemma I}. In that case, however, we can show that neither N{A} nor N(B} nor N(C} are in N(A UB U C), thus coming to a contradiction: we know that N{AU(B U C}} = N(A} or N{A U (B U C)) = N(B U C) or N(A U (B U C)) = N(A) U N(B U C). Because N(B U C} nN{C} = 0. we know that N(C} � N(A U {B U C)). Analogously, we can argue that N(B) � N( {A UB) U C} and that N(C} � N((A U C} UB). This means that N(A U B U C} = 0. which is only possible if A. B and C are = 0. We have shown that N(A U C) n C= 0. Thus, N(A) < N(C) which completes the proo£ 0 as
s:
If N{A) < N(B) and N(C) � N(B), we infer that N(A) < N(C}.
Lemma 6: If N(B} � N(A} and N(B) < N(C� we can infer that N(A} < N(C). In order to prove Lemma 5 and 6, we first have to cover a number of intermediate
observations.
Lemma 7: IfN(B) � N(A U B U C) and N(B) � N(A),
then N(A) � N(A U B U C).
Proof: N(B} � N(A UB U C} implies that N(A} n N(A U B U C} :F 0. Hence, A n N(A U B U C) :1: 0. We infer that N(A U B U C) nA = N(A) and therefore {by Lemma I} N(A} � N{A U B U C}. Lemma 8: IfN(B) < N(C) then N{C} n N{A UB U C} = 0.
Proof: We know that N(B U C) = N(B) and N(B U C) n N(C) = 0. Assume that N(C} n N(A U B U C} ::/= 0. Because we generally know that N(C} � C � (B U C}, we can conclude that (B U C) nN(A UB U C) :F 0. If this is the case, we know by (N3) that we can conclude that N(B U C) = N((A UB U C} n {B U C) ) = N(A U B U C) n (B U C). We assumed that N(C) n N(A UB U C) :F 0. We therefore know that N(C) n N(A U B U C) � (B U C} n N{A UB U C} where N(B U C) = (B U C} n N(A U B U C} (because N(C} � (B U C}). Therefore N(C)n N(B U C) :i= 0 which contradicts our assumptions. Therefore N(C) n N(A U B U C) = 0. Proof of Lemma
s:
(i) N{A U B U C} = N(A U B) or N (C) or N(A U B} U N(C). (Lemma r) (ii) Due to Lemma 8, we know that N(B) n N(A U B U C) = 0, because N(A) < N(B). (iii} Therefore, N(C) n N(A UB U C) = 0, as N(C} � N(B} (due to assumption). (iv) However, = N( (A U C) U B) = N(A U C) or N(B) or N(A U C) UN (B). (v) From (ii), we can infer that N( (A U C) U B) = N(A U C).
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Lemma
Regine Eckardt 275 (vi)
Assume now that N (A U C) n C ::f: 0. It follows that N({A U B) U C) n C::f:0 and thus N((A UB) U C) n C= N(C). This means that N{A U B U C) n N(C) ::f: 0, in contradiction with (iii). Thus, N(A U C) n C = 0, and thus N{A) < N(C� as required.
Proof of Lemma 6: N(A U (B U C)) = N(A) or = N(BU C) or = N(A) U N(B U C� N(B) = N(B U C) and N(B) � N(A). Therefore, N(B) � N(A U B U C� Due to assumption and Lemma 7, N(A) � N(A UB U C� As N(B) = N(B U C) and N(B) � N(A� we know that N{A U B U C) = N(A). N({A U C) UB) = N(A U C) or =N(B) or = N(A U C) U N(B). N(A) � A � A U C and N(A) � N{A U B U C), thus, N{A UB U C) n (A U C) ::f:0, and therefore N{A U B U C) n (A U C) = N(A U C� With Lemma I, we know that N(A U C) � N(A U B U C). Thus, N(A) = N(A U C) = N(A U B U C) (using (iv) ). (vii) If N(A U C) n C was ::f: 0; we'd get that N{A UB U C) n C ::f:0, that is C n N((A U B) U C) ::f: 0 and thus N(C) � N(A UB U C). (viii) However, with Lemma 8: N(C) n N(A U B U C) = 0. in contradiction to (vii). Thus, N(A U C) n C) = 0. that is N(A) < N(C), as required. Definition
2.:
For all A..B � M let A ::; B !ff N(A U B) � A .
We say that A "' B !ffA ::; B and B ::; A . Lemma 9: The relation ::; is transitive and reflexive.
Proof We assumed that N(A) � A for all A. This shows reflexivity. �ume that A ::; B and B ::; C. We have to show that A ::; C. Let us spell out what this means. If A ::; B, this means that N(A U B) � A and either N(B) n N(A U B) = 0 (that is, N(A) < N(B)) or else N(B) nN(A UB) ::f: 0, which means that N(B) � N(A). The same holds true for B ::; C. Thus, we have four subcases to consider: N{A) < N(B) and N(B) < N(C). This implies N(A) < N(C) which means that N(A U B) = N(A) and C n N(A U C) = 0. Especially, N(A U C) � A aiid therefore A ::; C Case 2: N(A) < N(B) and N(C) � N(B). We conclude that N(A) < N(C) by Lemma S· As in case I, we infer from N(A) < N(C) that A ::; C. Case 3: N(B) � N(A) and N(B) < N(C). We conclude with Lemma 6 that N(A) < N(C) and thus A ::; C. Case 4= N(B) � N(A) and N(C) � N(B). This implies N(C) � N(A) and so A ::; C. Case
I:
Lemma 9 shows that even in the weaker first-order variant, axiom (N3) is strong enough to considerably limit the shape of the normality function. It ensures that the normality function implicidy introduces an ordering at least on all L-definable sets in the model. This suffices to show that the normality function corresponds to a suitable ranking on the definable subsets of the mod�l-which are all sets we can talk about, anyway. We can now proceed to prove Theorem I 1, which I repeat here for the sake of convenience: Theorem I I: Let L be a logic language and L�rm its extension to a language with normality functors. Let L,,t/, its extension to a language with default implication based on ranking. We will use
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(i) (ii) (iii) (iv) (v) (vi)
276 The Meaning of Generic Sentences q/ to denote the inverse mapping to • which we defined above. Evidently, # does not map the entire of L,_, into L,.,.,.lt but only the range of ": This shows that L'"""' is richer than ranked languages because explicit reference to normality can be made. We can now prove the following: for each L_-rnodel M which satisfies N(l) to N(J), we can define an Lra•lt-rnodel (M, R) such that
M I= ¢ i.lf(M, R) I= ¢# Proof. Let M be an L...,,-model which satisfies {N1) to {N3� Let me use Pt�(D:) to denote
the definable part of the power set over D:. We adopt definition Pt�(D:): For all A, B Pt�(D:) let
2.
to get a preorder on
A $ B iff N{A UB) � A A - B iff A $ B and B $ A
[A] := {X IX E Pt� (D:) & X "' A}
6 := { { [A] lA e P.i {D:)}. $ }
where the relation $ is lifted to the set of equivalence classes in the evident way. The resulting relation is a linear ordering relation (reflexive, transitive, antisymmetric, and any two elements are mutually comparable) on 6. We are now in the position to be able to map all definable sets into the ordered set 6 in a way which respects the normality relations: If A is a definable subset of D:. let R,. (A) : = [A] . However, this is not enough: n order to come to a ranking, we have to map all elements in D: into an ordered set. As the singleton sets in D: need not be definable, this mapping is not yet immediate. Let {n; $} denote the Dedekind completion of the linear ordering 9 = ( { [AJ jA E Pt1 (D:)}; $}. Thus for each m E D:. the supremum sup{[A] IA e Pt� (D:)m e A} exists in e. Define R,(m) : = sup{ [A] IA E Pd (U.) 1\ m E A}. The rationale behind this defiilition is quite simple: if m is in A, it can't have lower rank than A as a whole, because the normality ofA is measured by the normality of its most normal elements. Clearly, R = (R,), e .., is a ranking function. Thus, (M, R) is a model of Lra,/e . We will show the equivalence of the two models by induction on the complexity of formulae. As the models are equal in their atomic part, the interesting step is the inductive step from formulae ¢11, ¢ and '¢11, '¢ to formulae of the shape Vx (N(¢)(x) - '1/J(x)) and ¢# -, 1/J# respectively. Let A be a definable subset of U,. We know that for any B definable in {U,) the following holds true: If [A] < [BJ, then N{A) � A n -.B. Therefore N(A) � A n n B t14i��abk.
[A ) < (B)
..., B
Assume that a E A and a E N{A). It follows that a is in no set B such that [A] < [BJ. Thus, R,.(a ) = [A], according to definition. We therefore know that N(A) � { a I Vb E A: R(b) ;::: R(a)J For this reason (M, R) I= ¢# -, ,p# implies M I= Vx (N(¢)(x) ..-+ 1/J(x)) Generally, the set A n n B tJ4i��abk. (A) < (B) -, B might contain more elements than N{A).
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Now we take the set of equivalence classes of sets: For all A EPt� (D:) define
Regine Eckardt 277 However, we can show that this does not make any difference with respect to what can be expressed in 1..-a...k We will show this by contraposition. Assume that (*j held true:
(*j
M f= "'" (N(t/>)(") - 1/J�
but
(M,R) f=
..., (4>#
-.. 1/J#)
This would mean that there is an a in 4> such that a ¢ N(tf>), a ,¢ '1/J , and R(a) = [t/>], which means that a E t/> n n B t�p....ble, [4>) < [B) -,B. But N(t/>) � 1/J and therefore N(t/>) = N((t/> n ..., 1/;) U (t/> n 1/;)) = N(tf> n 1/J). That is, N(t/> n 1/;) < N(t/> n ..., 1/;), As a e (t/> n -, 1/;) and R(a) =sup{[X] I X dfjinable, a e X}, we can conclude that R(a) is not minimal in tf>, in contradiction to the assumption that R(a) = [4>1. Therefore, there can not be an element a of minimal rank in 4> which falsifies 4># -,. 1/J¥ in (M,R) while 'V%(N(f/>)(") - 1/J(" ) ) holds true in M. This shows the converse direction of the main claim:
Which finishes the proo£
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M f= 'Vx(N(t/>)(") - 1/J(x)) implies that (M,R) f= 4># -,. 1/J#
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