JOURNAL OF SEMANTICS
Volume 16 1999
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JOURNAL OF SEMANTICS Volume 16 Number
I
CONTENTS HENRiihTB
SwART AND AluE MoLBNDIJX Negation and the Temporal StrUCtUre of Narrative D iscourse DB
1
ARIEL COHEN
How are Alternatives Computed?
fRANCESCA
PAOU Comparative Logic Natural Language
Please
as
an Approach to Comparison in
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43
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·
Negation and the Temporal Structure of Narrative Discourse · HENRIETTE DE SWART Utrecht University ARIE MOLENDIJK University of Groningen Downloaded from jos.oxfordjournals.org by guest on January 1, 2011
Abstract This paper develops an analysis of the temj,oral role of negative sentences in. narrative discourse in English and French. The analysis focwes on differences in the aspectual systems of English and French, and their consequences for the interpretation of
negation and quantification. A recursive rule for the introduction of discourse referents
characterizes both quantificational and negated sequences as complex states. The notion of
coercion explains why states (including complex states) can behave as events at the level of narrative discourse. The analysis is implemented in the framework of Discourse Representation Theory (DR'!) developed by Kamp & Reyle (1993�
1 NEGATION IN SENTENCE AND DISCOURSE
In tempc)ral semantics, affirmative sentences are taken to introduce new eventualities into the discourse representation. 1 Eventualities are located in time with respect to the speech time and other events that are part of the narrative structure. We extend this model of interpretation to negative sentences, and claim that they refer to negative states of affairs. This position requires some motivation, for the predominant view seems to be that negative sentences do not describe eventualities (Kamp & Reyle 1993; Asher 1993). Given that we implement our analysis in the framework of Discourse Representation Theory "{DRT} developed by Kamp & Reyle (1993), it is worthwhile running through their argumentation. Kamp & Reyle develop a theory of tense and aspect which assumes the general grammatical structure of tensed clauses in ( x): (r) [Tense [Aspect* [eventuality description]]J The predicate-argument structure of the sentence provides the description of some eventuality. Aspectual operators are modifiers whose input and output is a set of eventualities. The tense operator locates the eventuality in 1 Bach
(1986) introduces the term •eventailiry to gencnlizc over states. processes. and
events.
·
Negation and the Temporal Structure of Narrative Discourse
2
time. In English. aspectual operators are optional, and they allow recursion, which is why we use the Kleene star to indicate zero, one, or more operatiotl$. According to Kamp & Reyle (1993: S46-Ss), negation is not an aspectual operator, and we do not need to introduce a discourse referent for negative sentences as a whole in order to obtain the correct interpretation for sentences containing frame adverbials such as on .Sunday, compare:
(2)
a. Mary wrote a letter on Sunday. b. Mary didn't write a letter on Sunday.
n e t t' x y t-
I
Figure
1
I
a
Mary wrote letter n t t' X
on
Sunday.
t-
....
letter(y) e�t e: x write y
I
I
Figure 2. Mary didn't write a letter
on
Sunday.
We assume sufficient familiarity with DRT that the reader an interpret these DRSs with the informal explanation provided. The construction rules and the verification procedure are spelled out in the Appendix. 2
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In the Discourse Representation Theory (DRT) of Kamp & Reyle, these sentences are interpreted in Figures I and 2. These Discotirse Representa tion Structures (DRSs) contain discourse referents, and conditions which assign properties to the discourse referents::z
Henriette
de Swart and Arie Molendijk
3
·.
Mary looked at Bill. He smiled. b. Mary looked at Bill He was smiling. (4) a. Mary looked at Bill He didn't smile. b. Mary looked at Bill He wasn't smiling.
(3)
a.
·
(3a) suggests that Bill smiled after Mary looked at him, perhaps as a reaction to her doing so. In DRT, this is derived by a rule which says that events usually happen in succession. Accordingly, a condition e -< e' is introduced into the DRS. (3b) describes Bill as already smiling when Mary looked at him. Kamp & Reyle treat the Progressive as an aspectual operator which applies to the predicate smile to derive the corresponding state of the event of smiling as being in progress. They assume that states temporally include the event which functions as their antecedent in the discourse, which introduces a condition e � s. The examples in (4b) show that the contrast between the Simple Past and the Progressive is preserved under negation. This is reflected in the DRSs 3 and 4. which Kamp & Reyle propose for (�) and (4b):
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Tempqral nouns like Surulay introduce� interval t' into the DRS, and the condition that t' is a Sunday (typically the most recent Sudnay in a past tense context). The preposition on identifies t' with the location time t of the eventuality e. The past tense introduces the condition that t precedes the speech time now: t -< n. Events are included in their location time, which is captured by the condition: e C t. This leads to an inclusive or existential interpretation of the frame adverbial in Figure I. The negation operator in Figure :z. is generated lower than te�e, but high enough to take scope ov�r the event variable. .& a result, we obtain the interpretation that no event of Mary writing a letter is included in last Sunday. This corresponds with a durative or universal interpretation of _on Surulay in (:z.b): the negative sentence is true for the entire Sunday. Given that stative sentences trigger durative readings of locating time adverbials, the universal reading of (:z.b) has been taken as an argument in favor. of the claim that negative sentences denote states (Verkuyl 1993: 163). However; in Kamp & Reyle's approach, it is not necessary to interpret negative sentence$ as states to derive the correct meaning of (:z.b). Negation introduces a subordinate box, but the temporal structure of the negative sentence is the same as that of its affirmative counterpart. Kamp & Reyle further argue that the interpretation of negative sentences as reporting states would yield the wrong results in discourse contexts. It would blur the distinction between (4a) and (b), which are to be compared to their positive counterparts in (3):
4
Negation and the Temporal St:rUCtDre of Narrative Discourse e
n
t
X
y
t'
u
t -< n ekt Mary(x) Bill(y)
e:
I x look y I at
t' -< n u=y
...,
e'k t' e -< e' e': u smile
I
Figure
3
I
Mary looked :lt Bill He didn't smile.
n
e
t
X
y
t' u
t -< n ekt Mary(x) Bill(y)
e:
I x look at y I t' -< n u=y s
..,
I
sot' eks
s: u PROG(smile)
I
Figure 4 Mary looked at Bill He W2SD't smiling. In Figure 3, we negate the existence of an event e1 following e, so there is no event of Bill smiling for some contextually determined period of time t'
after Mary's looking at him (Kamp & Reyle
1993:
552). Figure 4 claims that
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e'
Henriette de Swan and Arie Molendijk
s
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there is no state of Bill smiling which includes the event of Mary looking at him. If we assume that discourse relations are established between discourse referents in the main box, and that negation has the aspectual effect of transforming any sentence into a stative one, the introduction of a higher level discourse referent for negatiye sentences would not account for the contrast in (4). Both sentences would qualify as states, and we would be unable to explain why the not.;.sm.iling in (4-a) is interpreted as a reaction to Mary looking at Bill. whereas it overlaps with this event in (4b). In addition. we would lose the parallel between (3 ) and (4). Kamp & Reyle take these observations as evidence against the interpretation of negation a5 an aspectual operator. In their view, negation has no effect -on temporal structure. We will refer to this claim as Kamp & Reyle's null hypothesis, and refer to the sentences in (3 ) and (4) which support this claim as the 'smile' examples. Although the null hypothesis is attractive at first sight, we will argue that the introduction of a stative discourse referent- corresponding to the negative sentence as a whole is ultimately preferable because it accounts for a wider range of data. Arguments in favour of this idea have already bee_n given by de Swart ( 1995, 1996). They are based on data involving the combination of negative sentences with aspectual adverbials introduced by for and until, the ability of negative clauses to provide a reference time, and . the availability of a wide-scope interpretation of frequency adverbials like often, always over negation. For lack of space we cannot repeat these arguments, but note that our claim represents a weaker ontological commitment than it might seem. Kamp & Reyle ( r993 ). Amsili & Le Draoulec ( 1995 ), and others do not deny that we_ need an external referent for negative sentences. Their idea is that the external referent of negative sentences ·is a time, rather than an eventuality (compare Figures 2-4 above). The only claim that we will make with respect to this point is that the negative condition holding for t can be labelled as a state which co-extends with t (section 3 below). The recursive rule that we will propose allows us .to switch from times to eventualities whenever needed. As a result, we have an e\rent-based semantics of negative sentences which incorporates all the insights of an analysis in which the external referent is a time. Note that our treatment of negation differs from the one proposed by Higginbotham ( I983 ), who claims that the only negative events which �e allowed in perception reports are those which somehow describe some positive, active behaviour. But as argued by van der Does ( r992: IJJ), this constraint is the result of more general aspectual restrictions on perception reports. Negative sentences are more limited in their use than affirmative sentences (c£ Hom 1989 for discussion). We take these restrictions to be pragmatic, rather � semantic in nature.
6 Negation and the
Temporal Structure of Narrative Discourse
In section 2., we will raise some problems for K.amp and Reyle's analysis of English, and ·go on to discuss negative sentences in French. French is interesting because its aspectUal system interacts with negation in ways which support the existence of an eventuality referent oqtside the scope of negation. Section 3 treats both negation.and adverbs of quantification like always as eventuality description modifiers, which map sets of eventualities on to sets of eventualities. A recursive rule guarantees that they introduce their oWn stative discourse referent. Section 4 shows that aspectUal operators like the Perfect/Progressive enter into scope ambiguities with logicil operators like negation and quantification because they are of the same semantic type. No scope ambiguities are observed in FrenCh: the Passe Simple and the Imparfait always take wide scope. This motivates their treatment as aspectually sensitive tense operators. Section s works out the consequences of these aspectual differences for negative sentences at the discourse level In both English and French, a context-sensitive process of coercion is available. Rhetorical relations between sentences at the discourse level can trigger reinterpretation of the negative state as an event. However, rhetorical relations are constrained by aspect. Our account of the smile examples in the two languages is then based on our analysis of the differences between the aspectUal systems of English and French. ·
NEGATION IN ENGLISH AND FRENCH 2. I
Discourse structure of negative sentences
Kamp & Reyle (1993) base their analysis on a relatively small set of examples. If we take into account a wider range of data, we find that the temporal structure of negative sentences cannot always be derived from their affirmative counterparts. Compare (sa) and (b):
(s)
Paul switched on the l�ht. The room was very bright. · b. Paul switched on the light. The room was not very bright.
a.
pointed out by Hinrichs (1986) and Lascarides & Asher (1993), the room being bright is a result of the event of switching on the light. This interpretation triggers a temporal structure in which the state s immediately follows the event e. We will note this with Kamp & Reyle's 'abut' relation: e ::>C s. Admittedly, Kamp & Reyle's (1993) rules for the interpretation of narrative discourse do not generate this structure, but we can assume that it is possible to develop a more refined version of DRT which yields the desired interpretation. However, this is not sufficient to interpret (sb). (sb) As
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2.
Henriette de Swart and Arie Molendijk
7
·(6)
The plane exploded. But John didn't panic. On the contrary, he was very calm. b. John invited all his friends. They didn't show up. John decided to go out into the street and bring in all the homeless from the neighbourhood. c. John invited all his friends. They didn't show up, although they had promised to come. a.
In (6a) the state of being calm overlaps with the not panicking. rather than with the event of the plane exploding. (6b) relates a sequence of events, one of which is a negative one. It is the friends not. showing up which triggers John's decision to go out in the street and bring in all the homeless. This makes it impossible to directly attach the last sentence to the first one. Finally, the pluperfect in (6c) locates the promise before the not showing up, and world knowledge suggests that the promise follows the invitation.
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allows dUferent temporal stn�ctures, depending on the ·rhetorical relation established between the two sentences. The room not being very bright can be the result of Paul switching on the light, if the light is not strong enough. This reading can be derived by negating the abut relation between s and e: -, (e ::>C s) . This is the only option within Kamp & Reyle's system of negation. However, there is another, perhaps more likely interpretation of (sb), namely one in which the room not being very bright is the reason for Paul to switch on the light. In that case, we have a relation of explanation between the two sentences in Lascarides' & Asher's ·terms, and the temporal structure is one of immediate precedence: the· room not being bright immediately precedes the event of switching on the light, and ends there. We could represent this as ...,s ::>C e. Given that a relation of explanation can be established in (sb), but not in (sa); it is impossible to derive the temporal structure of the negative sentence from its affirmative counterpart. Further problems arise when we consider the way the discourse continues after the negative sentence has. updated the context. Kamp & Reyle's null hypothesis predicts that the temporal structure of negative sentences does not play a role in the further discourse. This prediction follows from the claim that the eventuality variable is embedded under negation, and the negative sentence itself does not denote a state or an event (compare Figures 3 and 4). Accordingly, we expect subsequent sentences to be attached to the same reference point as the negative sentence. Thus Kamp & Reyle's treatment of negative events as subordinated to the main story-line implies that the negative sentences should be dispensable as far as temporal structure is concerned. However, the discourses in (6) become incoherent if we leave out the negative sentence:
8 Negation and the Temporal Struct111'e of Narrative Discourse
Under the null hypothesis, it is unclear how we can account for the way negative sentences update the context in narrative sequences like those in (6) because negative sentences are not supposed to shift the reference time. 2.2
Negation and aspectuality in French
Marie regarda Paul. ll lui sourit .Marie looked-PS at Paul. He smiled-PS at her b. Marie regarda Paul. ll lui souriait Marie looked-PS at Paul. He smiled-IMP at her (8) :l. Marie regarda Paul. ll ne lui sourit pas Marie looked-PS at Paul. He NEG smiled-PS NEG at her b. Marie regarda Paul. n ne lui souriait pas Marie looked-PS at Paul. He NEG smiled-IMP NEG at her
(7)
a.
The Passe Simple (PS) in (7a) suggests that Paul smiled after Marie looked at him, possibly as a reaction to her looking at him. The Imparfait (IMP) in the second sentence of(7b) describes a state ofPaul smiling which overlaps with the event of Marie looking at him. This is in line with the well-established view that Passe Simple sentences refer to events in the DRS, and lmparfait sentences denote states (Kamp 1981; Kamp & Rohrer 19S3; Molendijk 1990, de Swart I99Sa). States include the current reference time provided by a preceding event sentence. In a sequence of event sentences, an event typically moves the reference time, and gives a further development of the story-line. There is no difference between affirmative and negative sentences in this respect. The sequence of Passe Simple sentences in (Sa) describes succession of events, whereas (Sb) States that Paul's not-smiling temporally includes .Marie's looking at him. In order to represent these examples in DRT, we have to decide where the information contributed by the Passe Simple and the Imparfait comes in. We assume with Vet (1994) that French also follows the general schema in (1). If we propose a treatment of the Passe Simple/Imparfait as aspectual operators which function in ways similar to the Progressive, we could develop DRSs for (Sa) and .(D) which look similar to the ones given in Figures 3 and 4t respectively. As a result, we would predict ·that the temporal relations which hold between the two sentences in (7) are not affected when the second sentence is replaced by its negative counterpart, as
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More problems for the null hypothesis arise if we take into account the discourse structtire of negative sentences in a language like French. Let us first point out that the French discourses in (7} and (8) roughly behave as their English counterparts in (3) and (4):
Henriette de Swart and Arie Molendijk
9
(9)
Paul alluma Ia lampe. ll faisait tres clair dans Ia piece {maintenant). Paul switched-PS on the light. It_ was-IMP very bright in the room (now). b. Paul alluma Ia lampe. ll ne faisait pas tres clair dans la piece. Paul switched-PS on the light. It NEG was-IMP NEG very bright in the room. a.
Molendijk (1990: So) points out that the Imparfait sentence in sequences like (9a) describes a state which coincides with the result state of the event of turning on the light, described by the Passe Simple sentence. Thus we have the same temporal structure e :::>C s as in (sa). Just as (sb), the discourse involving the negative sentence in (9b) is ambiguous. It can either take the room not being very bright to describe the result state of Paul switching on the light (under the assumption that the lamp is not strong enough), or describe it as the state which explains why Paul switched on the light. Thus, in French as well as in English, not all examples allow derivation of the temporal structure of the negative sentence from its affirmative counterpart. Furthermore, we already pointed out that the null hypothesis faces serious problems when we consider how the temporal structure of a discourse is updated after the negative sentence has been processed. The examples can be repeated for French:
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in (8). An analysis of negative sentences along these lines would be compatible with Kamp & Reyle's null hypothesis. However, it is equally possible to provide an alternative analysis of these examples in which the negative Passe Simple sentence in (8a) would describe an event, and the negative Imparfait sentence in (8b) would describe a state. Under this analysis, we would predict the negative event to follow the event of looking, and the negative state to overlap with the event of looking. The two approaches would yield different, but equivalent representations of the discourse. The French version of the smile examples thus weakens Kamp & Reyle's evidence in favour of the null hypothesis. This raises the question whether French provides evidence which helps us really to decide between the null hypothesis and the hypothesis according to which negation is an aspectUal operator. Crucially, the null hypothesis implies that aspectual information always takes narrow scope with respect to negation. This assumption is necessary to preserve the claim that the temporal structure of a discourse involving a · negative sentence is derived from the one involving its affirmative counterpart. We already observed that this assumption is problematic for English (c£ s). Similar problems arise with respect to the French discourses in (9):
10 Negation and the Temporal Smu:ture of Narrative Discourse (Io)
Jean ne paniqua pas. Au contraire, il avait l'air tout calme. There was-PS an enormous explosion. But Jean NEG panicked-PS NEG. On the contrary, he seemed-IMP quite calm. b. Paul s'enerva. Ses invites ne venaient pas. D'ailleurs, ils ne vinrent pas. Paul fut d� Paul got-PS worked up. His guests NEG came-IMP. In fact, they NEG came-PS. John be-PS disappointed. . c. Jean invita tous ses amis. lls ne vinrent pas. Pou.rtant, ils _avaient promis d'etre li. John invite-PS all his friends. They NEG come-PS NEG. However, . they had promised to be there. a.
n y �Ut une enorme explosion. Mais
·
( u)
Paul s'enerva. Ses invites ne venaient pas. 11ll fut d� Paul got-PS worked up. His guests NEG came-IMP NEG. He was-PS disappointed.
If negative sentences do not introduce a top-l�el discourse referent, they
do not move the reference time forward. Therefore, the null hypothesis predicts that the reference time in both ( n) and(1ob) remains at the initial event reported in the Passe Simple. Paul's disappointment is not naturally interpreted as anaphorically dependent on his getting worked up, so this approach could explain the infelicitous nature of (u ). However, (tob) is perfectly coherent, and conveys that Paul became disappointed when it was clear that his guests were definitely not coming. The null hypothesis is unable to explain the contrast between (tob) and ( u). The discourses in (12) and (13) illustrate that we can go one step further and construct discourses which are incoherent if we use an affirmative sentence in the Imparfait, but which regain their coherence with a negative sentence. This is hard to explain under the view that the temporal structure of a negative sentence is derived from its affirmative counterpart.
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We observe that the French counterparts of(2) we a Passe Simple when the negative sentence is used to move on the story-line. In faet, the we of the Imparfait would be infelicitous in these contexts because it would describe the negative situation as stative rather than event-like. This means that the . distinction between the Imparfait and the Passe Simple is preserved in the context of negative sentences. In order to see more precisely how the difference in meaning between the Passe Simple and the Imparfait carries over to negative sentenCes, compare the incoherent (u ) with the well formed sequence (zo):
HenriEtte de Swart and Aric Molcndijk
(u)
II
Depuis quelque temps. Jean courait apres Pauline. 17ll l'attnpait. For. some time, Jean run-IMP after Pauline. He catch-IMP her b. Depuis quelque temps, Jean courait apres Pauline. ll ne.l'attrapait pas For some time, Jean run-IMP after Pauline. He NEG catch-IMP NEG her l'annee, Pauline etait une etudiante assidue. (13) a. Depuis le debut de . ?? Elle manquait un cours. Since the beginning of the year, Pauline be-IMP a diligent student. She miss-IMP a class. b. Depuis le debut de l'annee, Pauline etait une etudiante assidue. Elle ne manquait pas un cours. Since the beginning of the year, Pauline be-IMP a diligent student. She NEG miss-IMP NEG a class. a.
(14)
This morning, Jean was running after Pauline. 11He was catching her. b. This morning, Jean was running after Pauline. He caught her. c. This morniilg. Jean was running after Pauline. 11He was not catching her. d. This morning, Jean was running after Pauline. He didn't catch her. a.
(r�) describes a situation of global temporal overlap, just like (ua) above. For (r4b) we assume that the event sentence takes the location time of the progressive sentence as its reference time. The inclusion of the event in its
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In the oontexts of (12)-(13), there is no event sentence in the preceding discourse, so we assume that the reference point for the. second sentence is the location time of the stative first sentence (Kamp & Reyle 1993: 545). This location time t is given by identification with the time of the adverbial phrase introduced by depuis 'since' (compare on in example 2 above). Moreover, states include their reference time (Kamp & Reyle 1993: 544), so we introduce a condition t � s ', where s' is the state described by the second Imparfait sentence. In (ua) and (r3a) the sequence of state . sentences thus triggers an interpretation in terms of global overlap. This reading is pragmatically unnatural because it requires that Pauline is being caught all the time that Jean is running after her (ua), or that Pauline is missing a class all the time that she is being a serious student (13a). The Imparfait attrapait in (ua), for instance, has the same feeling about it as the Progressive in (r�). rather than the simple past in (r4b):
u
Negation and the Temporal Structure of Narrative Discourse
reference tinie implies that the event of catching occurs this morning. We add a negation and claim that there is no event of Jean catching Pauline during the entire time of Jean's running after her. In English, this would be expressed by (14-d), rather than (14-c). The aspectual difference. between the two sentences can be accounted for by the null hypothesis, as is reflected in the DRSs constructed for (14-c) and (14-d) in figures s and 6. The observation that the Imparfait sentence in (ua) has the same feeling about it as the Progressive sentence in (14.a) is compatible with the ·idea that both types of sentences are stative. According to the null hypothesis, the treatment of Imparfait sentences as referring to states automatically leads ·to the introduction of state variables under negation. In other words, if we treated the 1mparfait as an aspectual operator similar to the Progressive, we would give the discourse in (ub) the temporal structUre depicted in Figure s. and we would expect it to have the same degree of relative inacceptability as (14-c). but· this is not what we find. (ub) is perfectly coherent, and its meaning is more like the one of (14d), depicted in Figure 6. Under the null hypothesis it is not possible to derive this interpretation, for this analysis gives aspectual operators necessarily narrow scope with respect to negation. can
X
y
t"
t--< n s t this morning(t') t t' Jean(x) Pauline(y) s:( x PROG(run) after y I t"-< n s' s' t" s' o t s": j·x PROG(catch) y I 0
=
..,
FJgUR
S
0
This momi:ng. Jean was running after Pauline. He was not
catching her.
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n s t t'
Hcuriette
n
s
t t'
X
de Swart and Arie Molendijk
I3
y t"
t-
t this moming(t') t t' Jean(x) Pauline(y) s: I x PROG(run) after y I =
..,
e eCt e � t" e: I x catch y I
Figure 6 This morning. Jean was running after Pauline. He did not catch her.
'What these observations suggest is that the interaction between the Progressive and negation is different from the interaction between the Imparfait and negation. It may be true that the Progressive takes narrow scope with respect to negation (compare section 4-1 below for motivation), but this is not the case of the Imparfait. In fact, we will argue that the Passe Simple and the Imparfait always take wide scope with respect to negation (compare section 4-3 below). As a result, we cannot derive the temporal structUre of a negative sentence in the Passe Simple or the Imparfait from their affirmative counterpart. Moreover, the examples discussed in this section show that negative sentences in the Imparfait refer to states, whereas negative sentences in the Passe· Simple describe events. The difference in aspectual nature combined with the claim that the Passe Simple and the Imparfait take wide scope with respect to negation shows that negative sentences must refer to eventualities just like affirmative sentences. This raises the question how to formulate the construction rule for negative sentences in such a way that we obtain this discourse referent. In section J, we define a recursive rule for the introduction of eventuality variables, in ways similar to what has been proposed for quantification over events (Partee 1984; de Swart 1991).
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t" -< n
14 Negation and the Tempor.al Structure.ofNarrative Discourse
3 NEGATION AND QUANTIFICATION IN EVENT SEMANTICS 3. I The
domain of eventualities
4
3.2
Complex event descriptions
Logical operators introduce operations over eventualities, and build complex eventuality descriptions out of other-atomic or complex eventuality descriptions. Adverbs of quantification are the temporal counterparts of determiner expressions, and establish a relation between two sets of eventualities. As pointed out by Partee (1984) and De Swart (1991); tense takes wide scope over the adverb of quantification in sentences like (15): (15) Jane always came late. 3 In this paper we treat pronouns, proper names, and (in)de6nite NPs, but not quantified NPs. • See Kri£b (1989) and Verltuyl (1993) for compositional analyses. De Swut (1998a) intcgntes
their rcsula in a theory processes. and evena.
of aspect shift, while adopting a more fine-grained ontology of states,
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We interpret tenseless clauses as denoting sets of eventualities, that is members of the domain of eventualities £. A verb which has all its argU.ment places filled by either constants or variables describes an atomic eventuality description.3 Such eventualities are conceived of as minimal: the eventuality does not contain anything in addition to what is .supported by the predicate-argument structure. The domain £ of eventua lities is partially ordered by a part-whole relation �t and a precedence relation -<. There is furthermore a join operation ut so that £ has the structure of a complete join semilattice (Bach 1986). Following Kamp & Reyle (1993), we assume that the set£ of eventualities contains a subset E of events and a subset S of states. Every eventuality is either a state or an event, that is £ = E U S. The crucial difference between the two types of eventualities is that states have homogeneous reference and satisfy Krifka's (1989) postulates of divisive and cumulative reference, whereas events have non-homogeneous reference and satisfy Krifka's postulates of quantized reference (see Appendix, part C). We take it for granted that there is a compositional procedure, which yields either a state or an event variable for every atomic eventuality description. As usual in DRT, verification is defined in terms of embedding conditions. A DRS K is true in a model M iff there is an embedding function which associates members of the universe of discourse of M with the discourse referents of K such that each of the conditions inK is verified in M (see Appendix, part D);
Henriette de Swut and Arie Molendijk
rs
(16) Recursive rule for the introduction of discourse referents:
For 'Y a condition holding at time t, s: 11] and Max(s) and s. =, t define the maximal state consisting of the condition 'Y holding at time t.
'Y can either be an atomic condition(a property ascribed to an individual) or
a complex condition (that is, a negative or quantified condition). The application of this rule implies that every negative or quantified statement denotes a state description, namely the set ofstates which consist of the negative or quantified situation holding at some timet. Fernando (1994) suggests that there is a natural correlation between conditions and states. In DRT, conditions require verification of embeddings in the modeL and do not involve the notion of change. DRSs, like events, involve change, and are interpreted as a relation between input and output assignments. A condition can be understood as a special kind of DRS, whose output assignment does not differ from its input assignment. This makes it possible to reinterpret conditions as even�ties, as long as we label them as. states. For the condition on maximality, see below. An example like (15) can now be interpreted as in Figure 7: 5 There are further contcttual restrictions on the domain of quantification, e.g. in (rs) we only take into account situations in which Jane is alive. We assume that such restrictions are handled by the �� . 6 The phnse 's holds at t' suggests that s and tare coextensive rather than that they just overlap. Accordingly, we strengthen the DRT condition and take states to be coextensive with their location time, written =1• s =1 t is defined as the conjunction of s � t and t � s, whett � denotes temporal inclusion. This an be viewed as a temporal version of the 'strongest meaning hypothesis' (Dalrymple tt aL 1998�
.
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Tense restricts the domain of quantification in such a way that we o�y take into consideration past situations of Jane coming.5 Partee assumes that there is a location time t already present in the top box, so that the event in the antecedent box falls within the current reference time. She also proposes to label the quantificational statement as ·a whole as a state. In order to obtain the state corresponding to the q�tificational statement as a whole, we introduce a recursive rule for the introduction of discourse referents. As Kamp & Reyle(1993: 673) poiri.t out, it is straightforward to construct states out of triples {t,x, P}, where t is a period of time, x an individual and P a property such that t is an interval at which x has P. The state consists of the individual having the property P at time t. What properties of individuals have in common with quantified and negative statements is that they all introduce (static) conditions into the DRS. We use a generalized version of Kamp & Reyle's proposal to build states out of pairs ( t, 'Y) of a period of time t and a condition 'Y holding at that time:6
16 Negation and the Temporal St:rUCtUr'C of Narrative Discourse
_
n s t X t-
Jane(x)
s=rt Max(s)
e'
e s:
�
e Ce' e': I x come late I
Figure 7 Jane always came late.
The DRS is true iff every (contextually relevant) event of Jane coming the domain of quantification of the adverbial quantifier extends to her coming late. Adverbs of quantification take two arguments, but following Kamp & Reyle we will treat negation as a one-place operator.7 Consider example(17):
within
(17) Jane didn't come. According to( 17) Jane has the property of not coming at some time t in the past. We cannot view the complex negative state s that this sentence introduces as a minimal situation. It is too weak a statement to say that we can find at least one minimal situation holding at t at which Jane coming is not the case. There can be many of such eventualities without them guaranteeing the truth of(17). A similar problem arises in the interpretation of monotone decreasing indefinite NPs like at most three N (c£ Kri£ka 1989). In order to make a sentence like (18) true, it is not enough to find some group of students which has at most three members; we have to rule out the existence of a larger group: (18) At most
three
students failed the
exam.
The non-persistent character of negative sentences like(:i7) and(18) implies that we have to switch to maximal eventualities. Following Kri£ka, we interpret maximal eventualities e (generalizing over states and events) as the sum of all eventualities at some point in time: We only discuss i.nsunccs of negation which bear on the event as a whole. The analysis can be to c:asc:s like (i� in which ni:gation· is sensitive to the presupposition-focus fnmc of the sentence (Kmzer t989� 7
cxtcndcd
(Q Jane didn't come at three o'clock
(Q can mean thatJane came, but not at three o'clock (maybe at four� See de Swut ( 1998b) foe discussion.
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e�s e:l xcomel
Henriette de Swart and Arie Molendijk 17
(19) 'v'e[MAX(e)
�
3t [e
=
supc(>.e'(e' � t)})]
supc is the supremum in the join semilattice of eventualities E, which gives w the sum of all the eventualities temporally included in t. Kri£ka describes negative eventualities as the 'fusion' ·of all eventualities at a given time t which are not of the type described by the sentence.. The dependency on t accounts
for the intuition that negative sentences typically presuppose a contextually determined period of time for which the negative state of affairs holds. Krifka's proposals lead to the definition of event predicate negation in(zo): (zo)
>.P >.s 3t[MAX (s) As
=1 t
A -,3e [P(e) Ae � s]]
n s t X t--< n
Jane(x)
s=tl Max(s)
e s:
..,
e�s e:l xcomel
Figure 8 Jane didn't come. 4 T HE ASPECTUAL SYSTEM OF ENGLIS H AND FRENCH
The interpretation of negation and quantification as eventuality description modifiers whose input and output is a set of eventualities implies that adverbs of quantification and negation are of the same semantic type as aspectUal operators like the Progressive or the Perfect. This predicts that
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Negation is thw a modifier which operates on an eventuality description P and yields the maxin;lal states holding at t such that no eventuality e of type p· is included ins at t. This is very close to Kamp & Reyle's(1993) definition of negation in temporal semantics. The main difference is that they have jwt a location time outside the scope of the negation operator, whereas we define the negative condition holding at t as a state, building on the recursive rule in (x6) above. Accordingly, (17) generates the DRS in Figure 8. A DRS which contains a negated sub-box is true if and only if there is no embedding function which verifies the subordinate box (see Appendix, part D).
18 Negation and the Temporal Struetu.re of Narrative DiscOurse
their interaction can lead to scope ambiguities. In this section, we discuss such scope ambiguities in English, and show that similar ambiguities do not arise with the French Passe Simple and Imparfait. We use this as the basis for our claim that the French past tenses should not be treated as aspectual operators, but as tense operators which are sensitive to the aspectual character of their input description.
Scope ambiguities in English The relations · the temporal connective when establishes between the subordinate and the main clause depend on the aspectual properties of the clauses involved. When one of the clauses is stative, the relation is overlap. When two events are connected, it is possible .to get succession of events. The Progressive induces statitivity; compare: (21) a. When Mary came home, Max left. b. When Mary came home, Max was leaving. In (21a), the leaving can be interpreted as taking place after Mary came home. In (2zb), the situation of Max leaving is interpreted as already going on when Mary came home. Nothing changes in the temporal structure when we add an adverb of quantification like always: (22) a. When Mary came home, Max always left. b. When Mary came home, Max was always leaving. The same effects of succession of events (22a) or overlap (z.z.b) can be observed here as in (2 I ), so the Progressive takes narrow scope with respect to the quantifier. The Progressive cannot · take wide scope, because the quantificational relation as a whole describes a complex state, and the Progressive requires an event (or a process) as its input description. The interpretation of (z.z.b) is spelled ou� in Figure 9: 4-1
y
t�n Max(x) Mary(y) s=r t Max(s) c
s:
e:
I
c
y
�s·
come home I
=>
s' �s· ' s : PROG(x leave)
I
c
I
Figure 9 Max w.u always leaving when Mary came home.
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n s t X
Henriette de Swart and Arie Molendijk
19
Examples in which an aspectual operator takes narrow scope with respect to a logical operator do not provide evidence in favor of a higher-level discourse referent. For that we need aspectual operators which allow scope ambiguities. The English Perfect is such an operator. In (i.3a), the Perfect takes narrow scope with respect to negation: (23)
Mary hasn't met the president b. [Pres [....,[Perf [Mary_meet the president)]]]
a.
(24)
Mary hasn't seen her grandchild for over a year (now). b. MarY hasn't written a single poem for twelve years (now).
a.
The reading of (2�) with an explicit or implicit now means that Mary is in the state of not-seeing her grandchild, this state of affairs has been going on for more than a year now, and it hasn't ended yet. (24b) is similar. We propose (25) as the grammatical structure of (24a): (25) [Pres [Perf [...., [Mary see her grand-child]])) The peculiarity of this use of the Perfect is that it is limited to stative predicates, as Kamp & Reyle (1993: 567) observe. Interestingly, the negative sentences in (24) behave like the stative predicate in (26a) rather than the non-stative ones in (26b) and (26c): (26)
a. Mary has lived in Amsterdam for three years (now). b. #Mary has seen her grandchild for over a year (now). c. #Mary has written a poem for twelve years (now).
This is not just a motivation for the wide scope interpretation _of the Perfect over negation; it is also an additional argument in favour <;>f the view that negative sentences are stative. 4-2 In
Absence of scope ambiguities in French
contexts that do not involve quantification or negation, the French data involving an alternation of the 1mparfait and the Passe Simple are similar to the English data involving the Progressive/Simple Past distinction. Compare the contrast in (27) to the one in (21) above:
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(23a) denies that we are now in the state which results from the event of Mary meeting the president. This is naturally rendered by embedding the Perfect under the negation operator. Examples in which the Perfect takes wide scope with respect to negation are given in (24}.
zo
Negation and the Temporal Structure of Narrative Discourse
(27)
Quand Marie rentra, Max partit When Marie came-PS home, Max left-PS b. Quand Marie rentra, Max partait When Marie came-PS home, Max left-IMP a.
As observed in de Swart (1991: 221, 254), the role of the tenses changes when we add a quantifier like toujours 'always'. We can no longer use the combination of one Im.parfait and one Passe Simple, but we have to have either two Im.parfaits or two Passes Simples:
(28)
Max Max
both (28a) and (b), it remains vague whether the relation between the subordinate and the main clause event is one of overlap or immediate succession. although in b?th cases there seems to be a preference for a succession reading, due to the event-like character of the atomic predicates. The tense difference affects the presentation of the quantificational relation. The lmparfait in (28a) gives th� relation an open. unbou�ded character and describes it as an ongoing state of affairs. The Passe Simple in (28b), on the other hand, presents the quantificational relation as a bounded situation which is dosed off in time. This suggests that the Passe Simple and the Im.parfait take wide scope over the adverb of quantification. and the requirement on paralle l tenses is an agreement phenomenon. These observations motivate a treatment of the Passe Simple and the Im.parfait as aspectually sensitive tens_e operators rather than aspecrual operators, because· tense operators always take widest scope. In
4-3
Aspect and coercion
The intuition behind the claim that the Passe Simple and the Im.parfait are to be interpreted as aspectually sensitive tense operators can be illustrated by drawing a parallel with the nominal domain. We take the contrast between the two French past tenses to be similar to the one between determiners like much and many, little and few in examples like (29): (29)
a. There are many/few apples in the salad. b. There is much/little· apple in the salad.
the unmarked case, determiners like many, few combine with count nouns such as chair, student, whereas determiners like much, little operate on mass nouns such as water, gold. However, a contextually governed process of In
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partait touJours quand Marie rentrait, always left-IMP when Marie came-IMP home b. (Pendant deux mois) Max partit toujours quand Marie rentra (For two months) Max always left-PS when Marie came-PS home
a.
Henri'ette de Swart and Aric Molendijk
21
a. My program ran for a few minutes. b. My program ran in less than four minutes (this morning). c. I was rather tall for a number of years. d. Suddenly, I knew the answer. (3 I) a. John played the sonata for about eight hours. b. For months, the train arrived late. c. I read a book for a while, and then went for a walk
(Jo)
The sentences in (3 I) show that an event predicate can be coerced into a state by giving the sentence an iterative (3 Ia), a habitual (3 Ib) or a process reading (3 I c). Typically, a contextUal reinterpretation process is triggered whenever there is a conflict between the aspectual character ofthe eventuality
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reinterpretation may come in to satisfy the requirement on the determiner, and this will give rise to special meaning effects. Apple is a typical count noun (29a), but there is a systematic-although implicit-mapping from objects on to the 'stuff' they are made of which licenses its use as a mass noun in (29b) (Link 1983; Bach 1986). We follow Bach (1986) in interpreting aspect shift as analogous to shifts between mass and count readings. Along these lines, the Passe Simple and the Im.parfait can be viewed as similar to the determiners much and many, little and few, in the sense that they impose selection restrictions on the eventuality description they operate on: the Passe Simple only operates on events, and the Im.parfait is restricted to states. Otherwise, they have the same interpretation. That is, both the Im.parfait and the Passe Simple are past tenses and introduce a condition t � n into the DRS. The aspectual selection restrictions are treated as well-formedness conditions on the DRS: the Passe Simple presupposes a condition e: 111 and the Im.parfait is only felicitous ifthere is a condition s: 111 in the DRS. If the felicity condition is not · satisfied, a context-dependent process of reinterpretation comes in to coerce the eventuality description into a description of the right aspectual type. Moens (I988) introduces a general set of aspectual transitions, some of which are controlled by explicit aspectual operators, while others are free as long as their meaning effects are supported by the context (c£ also Pustejovski 1995). For instance, (Joa) describes a durative situation because it combines with a .for-adverbial. The same predicate-argument structure describes a non-durative event in (Job), as the combination with the in adverbial shows. (3oa) measures the time during which the program ran, (Job) describes how long it took for the program to start running or to do a complete run of the program. If we add a beginning point and an endpoint, the state becomes temporally bounded and therefore quantized (Krifka 1989) (Joe). If we focus on the starting point of the state, we obtain an inchoative reading (Jod):
2.2.
Negation and the Tcmponl Structure of Narrative Discourse
(p.) Soudain, Jeanne sut la reponse. Suddenly, Jeanne knew-PS the answer. e
D
t
X
t -< D
Jeanne(x) e �t
. e.·
c., .
s:
I
x
know 5the answer
I
FJgUre IO Jeanne sut (PS) Ia reponse.
I
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description and the aspectual constraints of some other expression in the context. In the examples in (3o) and (3 1) above, the triggers are durative time adverbials introduced gby for and in, and adverbs like suddenly. If coercion forces an eventuality description to switch from one aspectual category to another, it is an operation of the same semantic type as an aspectual operator like the Perfect or the Progressive. Coercion operators are thtis interpreted as eventuality description modifiers which map sets of events on to sets of states or vice versa. Given that the ontological nature of the eventuality is specified in the DRS by means of variables s and e, we need to make aspectual transitions explicit in the DRS, even if the mapping has no morp�ological or syntactic reflection. To this end. we introduce coercion operators into the DRS. We write Ces for the coercion of an event into a state, and � for coercion which maps a state onto an event. Aspectually sensitive tense operators such as the Passe Simple and the Imparfait in French count as triggers for coercion. When a typical event description is combined with the Passe Simple, the input conditions on the tense operator are satisfied, and the DRS-construction proceeds as usual. When the Passe Simple operates on a state description, the construction rules insert an operator C.rr to resolve the aspectual conflict. Similarly, the combination of the Imparfait with a stative predicate is straightforward, but for the Imparfait to operate on an event description, the event needs to be coerced into a state. This is handled by the introduction of a coercion operator Ceso As in (3o) and (3 1) above, it is the linguistic and extra linguistic context which tells us how to interpret the coercion operator. The embedding function requires the . meaning effects of at least one of the possible · mappings to be supported by the conteXt. For instance, (32) combines a stative description with the Passe Simple, and the introduction of a coercion operator C.rr leads to its interpretation in Figure 10:
Henriette de Swart and Arie Molendijk
l3
The state of knowing the answer is coerced into an event by giving the sentence an inchoative reading. The result is ail event which, in the usual way, is included in its location time. In the discourse, it will function as an event, and establish temporal relations accordingly. French has no Progressive verb form, so the transition of an event to the state of the event being in progress is a free transition as long as · the meaning is supported by the context (3 3a): (33)
Jeanne ecrivait Un.e lettre Jeanne wrote-IMP a letter b. Jeanne ecrivit une lettre Jeanne wrote-PS a letter a.
n
s
t
X
t -< n s =t t Jeanne(x)
s:
Figure
Cu
II
e y letter(y} e: x write y
I
I
Jeanne ecrivait {IMP) une lettrc:.
The examples worked out in Figures IO and I I illustrate that in the DRS representation, we just need to make sure we have a variable of the right aspectual type to satisfy the aspectual constraints of the tense operator. Accordingly, we define construction rules which check the input conditions
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The usual rules for the calculation of spectual class characterize the predicate-argument structure of (3 3) as an event description. This satisfies the input conditions of the Passe Simple, so in (3 3b) the writing of the letter is presented as a (completed) event. The aspectual constraints on the Im.parfait trigger a coercion of the event into the state of an event in progress in (3 3a). The conflict between the eventuality description and the tense operator is resolved by the introduction of the coercion operator Ca. which leads to the representation in Figu�e I I:
.34 Negation and the Temporal Structure of Narrative Discourse
for the operator, and insert a coercion operator Cst or Ca whenever the aspectual constraints are not satisfied. We assume that the value of the coercion operation is uniquely determined in the context, even though this might be an oversimplification. The analysis is thus set ·up in such· a ·way that the well-formedness of the DRS is saved by . the introduction of a coercion operator, but the verification of the DRS is dependent on the felicity of specific aspectual transitions in the context (compare de Swart 1998a for more details of this analysis of the Passe Simple . and the Iniparfait).
Quantificational contexts in French
We can now return to the quantificational contexts, and explain the meaning effects which arise in (28 ), repeated here as (34) on the basis of the semantics we adopt for the French past tenSes: (34)
Max partait toujours quand Marie rentrait Max always left-IMP when Marie came-IMP home b. (Pendant deux mois) Max partit toujours quand Marie rentra (For two months) Max always left-PS when Marie came-PS home a.
The Passe Simple and the Imparfait are tense operators which operate on the highest discourse referent. Thus, they cannot take narrow scope with respect to the adverb of quantification. The recursive rule for the introduction of discourse referents introduces a state variable for the quantificational relation as a whole. This satisfies the input conditions on the Imparfait (c£ Figure u ). In order for the quantificational relation to be compatible with the Passe Simple in (34b), a coercion operator must be inserted. The addition of boundaries is sufficient to give the eventuality quantized reference (Kri£ka 1989), and to treat it as an event in the DRS in Figure I 3· Given that the Passe Simple and the Imparfait are interpreted at the level of the quantificational relation as a whole, the temporal relation between the eventualities described by the main and subordinate clause remains underspecified. This is indicated by the symbol �. which indicates succession or overlap. The type distinction between aspectual operators and tense operators predicts that .the Progressive applies at a 'lower' level than the Passe Simple/lmparfait. The scoping relations explain why the contrast between Simple Past and Progressive is preserved under quantifica tion in English, whereas the related contrast in F:rench disappears in such . contexts.
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44
Henriette de Swut and Aric Molcndijk 2.5 n
s
t
X
y
t-
Marie(y) s t Max(s) =t
s: Figure
12. Max
e' =}
e -< e' e': I x leave I
partait {IMP) toujours quand Marie rentrait {IMP).
n
e t
X
Y.
t -< n Max(x) Marie(y) e�t s t' Max(s) s = , ·t' e:
Cse
e' s:
Figure I 3
e' � s e': y come home
I
Max
e" =>
j
e' � e" e": x leave
I
I
partit (PS) toujours quand Marie rentn (PS).
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e eCs e: 1 y come home 1
.z6 Negation and the Temporal StructUre s
of Narrative Discoune
NEGA TION IN NARRATIVE DISC OURSE 5.1
Negation and aspectuality in French
(3 s)
Marie regarda Paul. ll lUi sourit Marie looked-PS at Paul. He smiled-PS at her b. Marie regarda Paul. ll lui souriait Marie looked-PS at Paul. He smiled-IMP at her (3 6) a. Marie regarda Paul. ll ne lui sourit pas · Marie looked-PS at Paul. He NEG smiled-PS NEG at her b. Marie regarda Paul. ll ne lui souriait pas Marie looked-PS at Paul. He NEG smiled-IMP NEG at her. a.
Assuming that the Passe Simple and the Imparfait are tense operators, which take wide scope over negation, we predict that the negative event in (36a) follows Marie's looking at Paul, whereas the negative state in {36b) includes Marie's looking at Paul. The fact that the Passe Simple triggers the reinterpretation of the negative state as an event is reflected in the grammatical structures for {36a) and (b) in (37). The DRSs these sentences give rise to are derived in a straightforward way from the results in previous sections. They are given in Figures 14 and 1 5: {37)
a. PS: (Past [ Cst [..., (Paul smile at Marie])]) b. IMP: [Past [--. [Paul smile at Marie])]
Just as in the quantificational cases, we observe that the contribution of the Passe Simple and the 1mparfait comes in at the level of the highest discourse referent. Other French examples are analysed along the same lines. Consider (1ob), repeated here as (38a) with i� grammatical structure in (3 8b): (3 8)
Jean courait apres Pauline. ll ne l'attrapait pas· Jean run-IMP after Pauline. He NEG catch-IMP NEG her b. [Past [..., [Jean catch Pauline]]] a.
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The Passe Simple and the Imparfait apply to event and state variables, respectively. According to the recursive rule for the introduction of discourse referents, negative sentences refer to states. Building on these two insights, we predict that the combination of negation with the Imparfait is straightforward, whereas combination with the Passe Simple requires coercion of the negative state. The addition of boundaries is sufficient to give the eventuality quantized reference, and to treat it as an event in the DRS. This analysis accounts for ·the French version of the 'smile' examples in (7) and (8), repeated here as (3 5) and (36):
Hcnri.Ctte de SW2rt and Arie Molendijk 27 t
e
n
X
y
t'
s
t -< n e�t Mary(x) Paul(y)
e:
I x look at y I
s:
Figure
14
e' e' c s e': y smile
...,
I
I
Marie regarda {PS) Paul 0 ne lui souriait {IMP) pas.
n
e
t
X
y
e'
t'
t -< n e � t Mary(x) Paul(y)
e:
I x look at y I t' -< n e ' � t' e -< e' s t" Max(s) s = t t"
e':
Cse
s:
...,
e" e" � s e": y smile
Figure IS Marie regarda {PS) Paul U ne lui sourit {PS) pas.
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t' -< n s = t t' e� s Max(s)
�8 Negation and the Temporal Structure of Narrative Discourse
The negative sentence fulfils the aspectual requirements of the Imparfait, so Figure 16 correctly describes the state of Jean not catching Pauline as including the state ofJean mnning after her: n
s
t
X
y
t -<
n
s
'
t'
s =r t
Jean(x) Pauline(y)
I x run after y I t' -<
n
s' = t
t'
· Max(s') t c s
'
e s':
..,
I
e c s
'
e: x catch y
I
Ftgare I6 Jean courait {IMP) apres Pauline. U ne l'attrapait (IMP) pas.
The DRS captures the intuition that there is no event of catching during the time of the run. Thus, this DRS resembles the one in Figure 6, rather than the one in Figure 5, as desired.8 We have shown in this section that the recursive rule for the introduction of discourse referents combined with the interpretation of the Passe Simple and the Imparfait as aspectUally sensitive tenses provides the right interpretation for all the French sentences discussed. Given that all the components of this analysis have been independently motivated, this is an interesting result. The examples discussed so far have a relatively simple discourse structure. They illustrate the default situation in which two event sentences describe a succession of events (36a), an event sentence followed by a stative sentence describes a strUCtUre of temporal inclusion of the event in the state (36b), and two stative sentences describe global simultaneity of two states (3 Sa). However, as .we already observed in section 2 above, these ' The English examples come out slightly diH'erent in oar theory than the DRSs constructed after the null hypothesis. In pan:icuhr, the negated boxes will be labelled as states. However, this docs not affect the comparison.
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s:
HcmiCttc de Swart and Arie Molendijk (
.7.9
default relations can be over-ruled. Accordingly, Lascarides & Asher (1993) and others have argued that we need to take into account the rhetorical structure of the discourse. In section 5.2, we sketch an approach to French along these·lines. In section 5·3· the analysis is extended to include English. .
5.2
Discourse structure and negation in French
(39)
Discourse interpretation rule for the Passe Simple: a describes an eventuality e1 and a is taken as the temporal antecedent of a sentence {3 in the Passe Simple, which describes an event e,., then *e,. -< e1• b. Discourse interpretation rule for the Imparfait: If a describes an eventuality e1 and a is taken as the temporal antecedent of a sentence {3 in the Imparfait, which describes a state s,., then *S,_ V e1, where V means temporal discontinuity. a.
If
·
The sequences in (4o) illustrate the predictions made by the discourse rule (39a): (4o)
Marie regarda Paul. ll lui sourit. Marie looked-PS at Paul. He smiled-PS at her b. Marie entra a Ia demiere minute. Paul fut tout content de Ia voir. Marie came-PS in at the last minute. Paul was-PS quite happy to see her. . c. Paul donna une grande tete. ll invita tous ses vieux amis. Paul threw-PS a big party. He invited-PS all his old &iends d. Paul tomba. Jules le poussa. Paul fell-PS. Jules pushed-PS him. a.
In (4oa) and (4ob� we have an instance of the Occasion relation. The Occasion relation is a 'strong' form of narration in which there must be a plan or a natural sequence of events in which an event of the type described by the first sentence causes, provokes, inspires or leads to an event of the type described by the second sentence (Moens 1988; Glasbey 1995; Asher 1996). If an Occasion relation holds between event-denoting sentences a
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Lascarides & Asher (1993) develop a framework in which a mixture of linguistic and non-linguistic information involving causal relations and other world knowledge determines the temporal structure of narrative discourse. Their work on English has been extended to French by Asher & Bras (1993) and Landeweerd (1998). With these researchers, we assume that aspectual information constrains the set of adlnissible rhetorical relations. In particular, the Passe Simple does not allow us to go back in time, whereas the linparfait blocks temporal discontinuity:
30
Negation and the Temporal Structure of Narrative Discourse
and {3, we infer Narration (41a) by default {>). Axiom (41b} reflects that Narration leads to succession of e-Vents: (41) Occasion a. {r, a, /3) A Event(ecr} A Event(ep} A Occasion(a, /3) > Narration(a, {3) where ( T, a, {3) means that {3 is attached to a to update the discourse structure T. b. 0 (Narration(a, .B) ea -< ep) _.
·
(42) Elaboration a. ( r, a, /3) A ep C ecr > Elaboration{ a, /3) b. 0 {Elaboration( a, /3) ep � ecr} _.
·
The action of inviting the guests in (4ob) is taken to be part of what it means to throw a party, and is temporally included in this bigger event. The English counterpart of (4od) in (43) is often cited as an illustration of the relation of Explanation, defined in (44): (43) John fell. Max pushed him. (44) Explanation a. ( T, a, /3) A Cause(ep, ecr} > Explanation( a, ,B) b. Causes precede effects: 0 (Cause(e, , e3) -,e, -< e3) . c. 0 (Explanation{a, ,B} -,ecr -< ep) _.
_.
If a sentence {3 is attached
to a, and the event described by {3 is taken to be the caq.se of the event described by a, then we establish the rhetorical relation ofExplanation (w). Given that causes �ot precede their effects (44b), Explanation typically shifts back in time: Max pushed John, and caused him to fall (43). In French, we cannot relate a Passe Simple sentence to a preceding one by means of the relation of Explanation, because the conditions in (44b) and (c) would require us to go back in time, a move which is incompatible with the discourse rule for the Passe Simple, formulated in (39a) above. The only interpretation we can provide for (4od) is one in which the sentences are related by Narration or Occasion, and the discourse describes a succession of events. We can give a similar story for the interpretation of a Passe Simple •
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Because Occasion only holds between events, the use of the Passe Simple is required. In (4oa) the Occasion relation leads to Paul smiling as a reaction to Marie looking at him. In the context of (41b), this leads to coercion of the state predicate, with the effect of an inchoative reading. (4oc) exemplifies an Elaboration relat;ion. Elaboration(a, {3) holds if the event in {3 is part of the bigger event described by a {Moens & Steedman 1988} (42a). The part-of relation leads to temporal inclusion (42b):
Henriette
de Swart and Arie Molendijk
3I
sentence followed by an Imparfait sentence. Examples of relations we can establish are Background (45a� Explanation (45b) and Rest,Ut (45b): Julie entra a Ia demiere minute. Elle portait un suberbe manteau rouge. Julie entered-PS at the last minute. She wore..:.IMP a superb red coat. b. Julie rentra. Le soleil lui bnilait les epaules. Julie came-PS in. The sun burned-IMP her shoulders. c. Julie alluma Ia lampe. n faisait tres clair dans Ia piece (maintenant). Julie switched-PS · · on the lamp. It was-IMP very light in the room (now). d. #ll se mit a pleuvoir. Jean etait tout mouille. It started-PS raining. Jean was-IMP all wet. The Background relation in (45a) is the default interpretation in which the state provides background information about some event. As a result, the state temporally surrounds the event. The Imparfait is not in conflict with the temporal structUre of Explanation, as long as the constraint on contiguity is respected (45b). The Result relation we need to interpret (4sc) is defined in (46): (45)
a.
·
(46) Result a. {-r , a, /3) /\ Cause{e0, ep ) /\ State {ep) > Result{a, P) · b. 0 Result{a, /3) e0 � ep) The Result ·relation causes the state of the light giving the room a sad air to follow the event of switching on the light. The constraint on contiguity (4oh) causes the state to start immediately after the event. Note �t one is not all wet right after it starts raining. The violation of the constraint on contiguity thus leads to incoherence of the discourse in (45d) (from Landeweerd 1998, 186). Negation is interpreted lower than the Passe Simple and the Imparfait, so negative sentences obey the same discourse rules as affirmative ones, and allow the same rhetorical relations. Some examples we gave in section 2 can now be fully interpreted. -+
(47)
Paul alluma Ia lumpe. ll ne faisait pas tres clair dans Ia piece. Paul tumed-PS on the lamp. It NEG was-IMP NEG very bright in the room. b. ll y eut une enorme explosion. Mais Jean ne paniqua pas. Au contraire, il avait l'air tout calme. There was-PS an enomious explosion. But Jean NEG panicked-PS NEG. On the contrary, he seem-IMP quite calm. c. Paul s'enerva. Ses invites ne venaient pas. D'ailleurs, ilS ne vinrent pas. n fut de� a.
·
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·
32. Negation and the Temporal Structure of Narrative Discourse
John got-PS worked up. His guests NEG came-IMP NEG. In fact, they NEG came-PS NEG. He was-PS disappointed. d. Paul s'enerva. Ses invites ne venaient pas. 1?Jl fut de� Paul invited-PS all his friends. They NEG came-IMP NEG. He was-PS disappointed.
5·3
Discourse structure and negation in English
The English 'smile' examples are important, becawe they provide an argument in favour of Kamp & Reyle's null hypothesis. The relevant examples are repeated as (48) and (49):
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·
In (47a), we can either derive a relation of Explanation (the room not being very bright ·cawes Paul to tum on the light) or a relation of Result (Paul turned on the light, but the room is still not very bright). Both relations are compatible with · the discourse rule for the Imparfait, so it depends on the context which interpretation is most appropriate. In (47b) the last Imparfait sentence establishes a Background relation with the preceding Passe Simple sentence. The observation that Jean was quite calm is not the Background for the explosion, but for the event of not panicking. The analysis also provides a straightforward explanation of the contrast between (4'7c) and (47d� The negative state of the guests not coming reported by the lmparfait sentence in (47d) provides the Back ground for and thw overlaps with the event of Paul getting worked up reported in the Passe Simple. The unbounded nature of the negative state of affairs suggests that Paul is still waiting. In that context, Paul's disappointment is not naturally related to his getting worked up or him still waiting, so the discourse is not coherent. In (47c), the negative Imparfait sentence also establishes a Background relation, but it is followed by another negative sentence, this time in the Passe Simple. We can establish an Occasion relation between the last two Passe Simple sentences: it is the event of his friends definitely not showing up which leads to his disappointment. The quantized nature of the negative event closes off the period of waiting. The natural interpretation is that Paul is disappointed as a reaction to the guests definitely not-arriving. and follows it in time. The analysis sketched in this section combines a theory of rhetorical structure with a set of temporal constraints on the French past tenses to account for the temporal structure of narrative discourses i:nvolving affirmative and negative sentences alike. In the next section, it will become clear that the English 'smile' examples can be interpreted in a similar way.
Henriette de Swart and Aric Molcndijk 3 3 a. Mary looked at Bill He smiled. b. Mary looked at Bill He was smiling. (49) a. Mary looked at Bill He didn't smile. b. Mary looked at Bill He wasn't smiling. If the application of a logical operator always creates a (complex) state, we have to explain why the negation of the event in (49a) can be interpreted as posterior to the event described by the first sentence, whereas the negation of the state in (49b) cannot We have assumed that discourse structure is constrained by aspect. Accordingly, we will argue that simple past and past progressive sentences do not enter into the same kind of rhetorical relations. In the discourse in (48a), the event of Mary looking at Bill 'occasions' the event of Bill smiling, because the action-response structure forms a natural course of events. States can also b� responses (c£ Hinrichs 1986 for examples):
(48)
a. Mary entered the room. John was happy to see her. b. Susan told Bill the story. He was impressed.
The state is here interpreted as a reaction to the event. We can treat the examples in (so) along the same lines as (48a) if we assume that the Occasion relation can trigger coercion of the state into an event. This is not so strange if we realize that similar discourses in French would require the use. of the Passe Simple, rather than the Imparfait (c£ 4ob). In both languages, the coercion operator triggers an inchoative reading for the stative sentence, which focuses on the onset of the state. The main difference is that contextual reinterpre�tion can be deduced from the use of the tense in French, but not in English. However, in both languages, the contextual reinterpretation process leads to the inference that Bill being impressed is his reaction to Susan telling the story, and is thus located later. in time than that event. This approach makes it possible to treat (49a) as involving coercion as well Although (49a) involves a complex, negative state, and not an atomic state, it is essentially of the same type as the examples in (so). In order for the Occasion relation to apply, the state is coerced into a bounded, quantized negative event, which makes the not-smiling valid for a dosed period of time t, the beginning point of which lies after the looking event The representation of (49a) under this analysis is strictly identical to the DRS given for the French Passe Simple sentence {36a) in Figure 1 s above. Thus English and French develop identical meanings. but on the basis of a different syntax-semantics and semantics-pragmatics interface. If we compare Figure Is with Kamp & Reyle's representation of (49a) in
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{so)
34 Negation and the Temponl Structure of Narrative Discourse
(s I ) Negation of states
The discourse relations that a sentence describing the negation of a state enters into constitute a (not necessarily proper) subset of the admissible rhetorical relations for its affirmative counterpart.
As Dowty (1986) points out, the Progressive tends to express overlap with the events described in the previous discourse. He suggests .that this is due to the semantics of the Progressive, which abstracts away from the start and finish of the situation. The feeling of being 'in the middle of things' does not easily allow in inchoative or a bounded event reading.9 If a Progressive sentence is not easily reinterpreted as an event, it will not normally describe a response to another event. Thus, unlike (48a), (48b) forces temporal overlap. If it is true that stative sentences preserve aspectual structure under negation, we expect the same relation of temporal overlap to hold for the negative counterpart of the progressive sentences in (49b). This interpretation of (49b) is spelled out in Figure 17: exception Dowty notes coru:cms the kind of. cwnple in (i� An anonymous reviewer similar example with a negative sentena: (ii): (Q The president began the interview in a coldly oflicial iiWUlCf, much as Mary had expected. But the nat thing she knew, the president was offering her the ambassador post. (ii) ( . . . � The nat thing she knew, she wasn't sitting in her chair. Dowty argues that the perception of the event already in progress is the nat salient event in the · discourse. This leads to a quasi-inchoative reading of the Progtessive. Similar remarks can be made for Freuch in relation to certain narrative uses of the Impar&it (the so-called 'Impar&it pitton:sque'). the ase of which is scverdy constrained (compare Molendijlt 1990 fOr discussion). 9 One suggests a
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. Figure 3, we observe that the two proposals lead to different but equivalent DRSs. This confirms our claim that an event-based semantics of negative sentences incorporates all the insights of an analysis which treats the external referent as a time. This still leaves the past progressive version . of the smile example to be accounted for. The introduction of rhetorical relations triggering coercion might suggest that posteriority should be a possible interpreta tion in (49b) as well, but of course it is not. We base our explanatio1;1 of this fact on Asher's (1993) observation that states are closed under negation, but events are not. We take the fact that states are closed under negation to mean that negation applied to states has no aspectual effects. We assume that the negation ofa state is 'transparent' in the sense that the negative sentence inherits its aspectual nature from its affirmative counterpart. Given that aspect constrains the set of admissible rhetorical relations, this means that negative sentences inherit constraints on discourse structure from their affirmative counterparts. This is formulated in the following inheritance principle:
Henriette de Swart and Arie Molcndijk 3 S n
e
t
X
y
s
t'
u
t -< n
I
e {; t Mary(x) B i ll(y)
e: x look at y
I
t ' -< n s = t t'
u=y
s' s:
Figure
...,
17
s' c s s': PROG(u sinile)
I
I
Mary looked at Bill. He wasn't smiling.
Figure 17 iS not identical to the representation of the French example (36b) in Figure 14- This reflects the fact that we treat the Imparfait and the Progressive in essentially different ways. However, the DRSs in Figures 14 and I 7 both rule out an occurrence Qf smiling over a period of time including the event of looking. Although they differ in grammatical structure, the sentences can therefore be used as translational equivalents. The reason that (49b) expresses temporal overlap, whereas (49a) does not (necessarily) is that (49b), as the negation of a progressive sentence, does not allow any other temporal relation than its affirmative counterpart in (48b). Kamp & Reyle's claim that the restrictions on the interpretation of the negative sentence in (49b) are related to its affirmative counterpart in (48b) is thus preserved under our analysis. However, unlike Kamp & Reyle, we allow more room for variation. Aspect constrains the set of admissible rhetorical relations, but it does not usually fully determine the discourse structure. In a particular context, this allows a negative sentence to make a different choice from among the set of admissible discourse relations than its affirmative counterpart. This was illustrated above in (s), repeated her� as (s2a and b):
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" Max(s) e {; s
36 Negation and the Temporal StrUCtUre of Narrative Discourse
(52}
Paul switched on the light. The room was very brigh� b. Paul switched on the light. The room was not very bright. c. Paul switched off the light. The room was very bright. a.
6 CONCLUSION In
this paper, we have shown that there are il,tteresting differences
between the aspectUal systems of English and French. We argued that the Passe Simple and the Imparfait are not interpreted as aspectual operators, but as tense operators which are sensitive to the aspectual nature of the eventuality description they operate on. As a result, the French past tenses always take wide scope with respect to logical operators like negation and quantification. We have shown that the treatment of negative sentences as states does not exclude the possibility of them being presented as events, as long as we recognize the crucial role coercion plays in both English and French. Our proposal can be viewed as a generalized version of the analysis of negative sentences developed by Kamp & Reyle (1993). Acknowledgements ·
The research of the first author was partially supported by a fellowship &om the Royal Netherl.andJ Academy of Arts and Sciences (KNAW), which is hereby gr:atcfu.l.ly acknowledged. Many thanks to Tim Fernando, Hans Kamp, Rita Landeweerd, Sja.ak de Mey, Hotze Rnllmann, Co Vet; two anonymous reviewers, and the audiences at the Tilburg workshop on computational semantics and the linguistics colloquia at the University ofTUbingeu, the University of Washington at Seattle, and Stanford University for helpful discussion and comments. We are equally grateful to Brigitte Kampers-Manhe for her help with the French data.
·
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In section 2 above, we showed that the temporal structure of (52b} cannot be derived &om the actual discourse structure of (52a). If (52b) involves the relation of Result, this triggers an interpretation in terms of succession. If we take the room not being very bright to be the reason for Paul to switch on the light, we establish a relation of Explanation, which locates the negative state before the event of the first · sentence. Knowledge of the world makes a relation of Explanation unlikely in the context of (52a), but (sx) shows that it is quite plausible in other contexts. This means that both Result and Explanation are among the possible discourse relations the affirmative stative sentence can establish. The discourses in (52) confirm our view that the inheritance principles needs to be defined in terms of admissible, rather than actual rhetorical relations.
Henriette de Swart and Arie Molendijk 37 HEN1UETrE DE SWART . UiL-OTS/RDrflllna I..tmguages Utrtcht Univmity
Final
Received: I o. u..97 version received: 2.t.o8.g8
Kromme Niaiwtgracht Z9
35 u HD
Utrtcht 71re Neth61anJ.s
MOLENDIJK Dep4rtmmt ofRomance LangutJges Univmity of Croningen PO Box 716 9700 AS Croningen 71re Neth61anJ.s ARIE
In the Appendix we introduce the construction rules, the DRS-conditions, and a verification procedure necessary to construct and interpret the DRSs given in the teXt, in so far as they are different &om the DRSs constructed by .Kamp & Reyle (1993). A.
DR.S-conditious for coercion operaton (i) If e: 11] is an event description then
is a state description. (ii} If s: UJ is a state description, then
is an event description. Construction rules for complex sentences Introduction of discourse referents for sentences modified by an aspectual operator
B.
•
ippropite
Asp(S� type (e or s). Introduce in Ux: a new discourse referent of the b. Introduce in Conx a condition: e: I Asp K, l (or s: Asp K, } Introduce in Ux, : a new discourse referent of the appropriate type (e' or s '). c. d. Introduce in Conx, a condition: t': [1] (or s': [1]) For state sentences modified by a coercion operator c.{S}: Introduce in U_i: a new event discourse referent t a. b. Introduce in Conx: e: I c. K, I c. Introduce ·in Ux, : a new discourse referent s. . d. Introduce in Conx, : s: [1] a.
·
•
·
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APPENDIX
3 8 Negation and the Temporal Structure of Narrative Disco� •
•
For event sentences modified by a coercion operatQr C.,(S): Introduce in Ux: a new stative discourse referent s b. Introduce in CoDK: s: C.. K, c. Introduce in UK, : a new discourse referent t. d. Introduce in CoDK, : t: r-rl For negative sentences of theform uot(S): a. Introduce in Ux: a new state discourse referent s and a new time t b. Introduce in Conx: Max(s), s =1 t and s: c. Introduce in Ux, : a new discourse referent of the appropriate type (t or s1 d. Introduce in Conx, : t � s (or s' � s) e. Introduce in CoDK, : e: [1] (or s': r-rl) For (universally) quantified sentences ofthe form Always(S, S'): a. Introduce in Ux: a new state discourse referent s and a new time ·t b. Introduce in CoDK: Max(s) and s =1 t � s: � .. c. Introduce in Ux, : a new discourse referent of the appropriate type (t or s1 d. Introduce in Conx, : t � s (or s' � s) e. Introduce in Conx, : t: [1] (or s': [1]) £ Introduce in Ux.: a new discourse referent of the appropriate type (e' or s'1 g. Introduce in CODK. : t !;;;; t1 (or s !;;;; s") h. Introduce in Conx. : e': (or s": Tense operators (i) Past tenSe (English} a. If S is the first sentence of the discourse, introduce in Ux: a new discourse referent n which is identified with the speech time. If S is not the first sentence of the discourse, continue with b. b. Introduce in Ux: a new time discourse referent t where t is the location time of the sentence c. Introduce in Conx: t -< n d. If "( is a state desaiption, introduce in Conx: s =1 t e. If "( is an event descriptio.n. introduce in Conx: t � t (ii) Passe Simple (French} a. IfS is the first sentence of the discourse, introduce in Ux: a new discourse referent n which is identified with the speech time. If S is not the first sentence of the discourse, continue with b. b. Introduce in Ux: a new time discourse referent t c. Introduce in Conx: t -< n d. If "Y is an event description, introduce in Conx: t � t e. If "Y is not an event description, but a state description introduce in �nx: a.
I
I
, ..., K, I
l K, K l
•
EJ
Ej)
and continue with d. (iii} Imparfait (French) a. If S is the first sentence of the discourse, introduce in UK: a new discourse referent n which is identified with the speech time. If S is not the first sentence of the discourse, continue with b.
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•
Henriette de Swart and Arie Molenclijk 39
·
b. Inaoduce in U.rc: a new time discourse referent t Inaoduce in Con.rc: t � n d. If "Y is a state description, inaoduce in Con.rc: s =1 t e. If "Y is not a state description, inaoduce in Con.rc: c.
and continue with d.
•
•
{ii) VP[DIVe (P) ..,.. \fx\fy[P(x) 1\y !;e x - P(y)]) {iii) VP[QUAe {P} ..,.. \fx\fy[P(x) 1\ P(y) - -.y !;e x)) A part-whole relation !;, a precedence relation �. an identity relation =, a temporal inclusion relation �. a temporal co-extension relation =1 (defined as the conjunction of � and 2), and an 'overlap relation o are defined over events, states and times in the usual way, so that the eventuality domain satisfies the seven axioms on event structure defined by Kamp 8c Reyle (1993: 667). A function PredM which maps predicates onto their denotation such that: (i) For each one-place predicate TA; over times, PredM{TA;) is a subset of T {ii) For each atomic event description P;{t;, a, , . . . , a8 ) , PredM{P;) is a set of tuples { e;, a, , . . . , aa) where e; E EM and a, . . . a, E UM. {iii) For each atomic state desaiption P;{s; , a, , . . . , a8 ) , PredM (P;) is a set of tuples (s;, a,, . . , a,) where 1; E SM and a, . . aa E UM. (iv) {s;, a,, . . , a,) belongs to PredM{PROG{P;)) for P; an event description iff 1; E SM and {s;, a, . . . a . ) belongs to {PROG{PredM {P;))). (v) The coercion operator C., is multiply ambiguous and has senSes C.,, . . . C.,. for the n free aspectual transitions defined as possible mappings from events to states in the language under consideration. {a;, a, , . . . , aa) belongs to PredM ( C.,, {P;)) for .P; an event description iff 1; E SM and ( s;, a, . . . , a,) belongs to ITER(PredM {P;)); ( s;, a, , . . . , a,) belongs to PredM{c... {P;)) for P; an event description iff 1; E SM and (s;, a, , . . , a,) belongs to PROC{PredM {P;)}; {s;, a,, . . . , a,) belongs to PredM (C.,, (P;)} for P; an event description iff 1; E S,v and ( s;, a, , . . , a,) belongs to HAB{PredM{P;)}, etc. (vi) The coercion operator C,. is multiply ambiguous and has senses � C... for the n free aspectual transitions defined as possible mappings from states to events in the language under consideration. { e;, a, , . . . , a,) belongs to C.., {�{P;)} for P; a state description iff e; E EM and { e;, a, . . . , a,) belongs to ADD-TO{PredM{P;)): {e;, a,, . . . , a,) belongs to �. {PredM{P;)} for P; a state description iff e; E EM · and ( e,, a,, . . . , a,) belongs to INCHO{PredM (P;)}; { e;, a, . . . , a,) belongs to C,., {PredM (P;)} for P; a .
.
.
.
.
•
. . .
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C. The model The model M is a structure which consists among others oE • A set T of times •. A set EM qfeventualities. {EM, Ue) is a completejoin semi-lattice, with Ue as thejoin operation on eventualities. EM = EM U SM where EM is the set ofevents and SM is the set of states. States have homogeneous reference, that is they satisfy K.rifka's (1989) postulates on cumulative (i) and divisive (ii) reference. Events have non-homogeneous reference and satisfy .Krifka's (1989) postulate on quantized reference (iii): (i) VP[CUMe {P) ..,.. \fx\fy[P(x) 1\ P(y) - P{x U y)))
40 Negation and the Temporal Structure of Narrative Discourse
D. Verification Let K be a DRS. let "Y be a DRS-condition. and letf be a possibly partial embedding from the set R of discourse referents into the model M. then: (i) f verifies the DRS K in M iff verifies each condition belonging to ConK in M (ii) If 'Y is ofthe form �= IT( a, . . . a., ) thenf verifies 'Y ifff maps � on to an element e ofEM and a, . . . a, on to elements a1 a, of UK such that ( e, a1 a,) belongs to PredM(IT) (iii) If 'Y is of the form s: IT(a 1 a, ) thenf verifies 'Y ifff maps s onto an element s ofSM and a, . . . a, on to elements a, . . a, ofUK such that ( s, a1 a,) belongs to PredM(n) K , then f verifies 'Y ifff maps s on to an element s of SM (iv) If 'Y is of the form s: such that MAX(•) and there is no embeddin.g g from R into M which extends/, such that Dom{g) = Do (f) U U and g verifies K,. (v) If 'Y is of the form s: K, => K� thenf verifies 'Y ifff maps s on to an element s ofSM such that MAX(s� and for every extension g of J such that Dom{g} = Dom(J) U UK, and g verifies K, in M. there is an extension h of g such that Dom(lr) = Dom{g) U UK. and h verifies K� in M.
l
• • •
j -. I
i
j'
• • •
• • •
.
• • •
I
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state description iff � E EM and ( e, , a, , . . . , a,} belongs to BOUND(PredM (P;)), etc. • Semantic efFect of aspectual transitions (informal) (i) PROG is a function from EM to SM which maps event descriptions onto state descriptions in such a way that the state is temporally included in the event and · describes the action as being in progress. as a development which would eventually lead to a culmination point (although that need not be reached). Com� Parsons (1991� Asher (1992), and I.andman (1992) for more discussion. (ii) PROC is a function from EM to SM which maps event descriptions on � state descriptions in such a .way that the state describes the homogeneow process underlying the event predicate without any reference to an inherent culmination point. (iii) ITER is a function from EM to SM which maps event descriptions on to state descriptions in such a way that the state describes an unbounded number of events of the type described by the event predicate; seeJackendoff(t990: 29) for more discussion. (i:v) HAB is a function from EM to SM which map5 event descriptions on to state descriptions. HAB functions like an implicit adverb of quantification similar to always, and is interpreted in terms of universal quantification with exceptions. (v) ADD-TO is a function from SM to EM which maps state descriptions onto event descriptions which consist of the preparatory phase leading up to plw the change in state from non-s to s as a culminated process. (vi) INCHO is a function from SM to EM which maps state descriptions on to event descriptions in such a way that the event describes the start of the state (that is, the change from noe-s to s� see Dowty (1979) and Hinrichs (1985) for more discussion. (vii) BOUND is a function from SM to EM which maps state descriptions onto event descriptions in such a way that the event consists of a bounded (quantized) portion of the state.
Henriette de Swart and Arie Molcndijk 41
REFERENCES & Le Draoulec, A. (1995� 'Contribution to the event negation problem', in Pro&«tlingr of the Worlulwp
Amsili, P.
Hinrichs, E. (r986� "Temporal anaphora in
,
•
·
,
Amsterdam.
(1979), Word Meaning and Montague Grammar, Reidel, Dordrecht. Dowty, D. (1986), "The effects of asp.:�
Dowty, D.
on the temporal structure of d�owse: semantics or pragmatics',
class
Linguistics and Philosophy, 9, 38-61. (1994� 'What is a drsr, in Proaedings of the Intmlalional Workshop on CompulliliDnal Semantics, Tilburg, 61-70. Glasbey, S. (1995� 'When, discourse rela Fernando, T.
tions and the thematic structure of events', in Proatdings of the Worlulwp
on Tmse, Spaa and Movtmmt, VoL
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Toulouse, 91-104Higginbo� J. (1983), "The logic of
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roo-.2.7.
Hinrichs,
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'A compositional for Aktionsarten and NP reference in English', Ph.D. thesis, Ohio State University, Columbus.
semantics
"Temporal interpretation, discourse . relations and commonsense entailment',
Linguistics and Philosophy, 16, 437-93·
Link, G. (1983),
"The logical analysis of plurals and mass terms: a lattice theoretic approach', in R Bauerle, C. Schwarze, & A. von Stechow (eds),
MMning, Use and Intnprttalion of LDnguagt, de Gruyter, Berlin, 30.2.-.2.3. Moens, M (1988� "Tense, aspect and tem poral reference', PhD. thesis, University of Edinburgh. Mol�jk, A. (1990� Lt
Passi Simple tt 17mparfoit: une approack rtichtnbachimne,
Rodopi, Amsterdam.
Parsons, T. {1991� Events in the Stmanlia of English, MIT Press, Cambridge, MA. Partee, B. (1984� 'Nominal and temporal anaphora', Linguistia and Philosophy. 7, .2.43-86. Pwtcjovsky, J. (1995� '1M Gtnnativt
Lexiam,
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discourses of English', Linguistics and Philosophy, 9, 63-8.2... on Tmse, Spaa and Momnent, VoL 5, Jackcndoff; R (1990� Snnantit Struaures, Toulouse, 17-29MIT Press, Cambridge, MA. Asher, N. (1992� 'A de&ult, truth condi Kamp, H (198 1), 'Evenements, representa tions discursives et refCrence temporelle', tional semantics for the progressive', lAnguages, 64. 39-64Linguistics and PhilOsophy, IS, 463-508. Asher, N. (1993), &jft'tfl« to Abstract Objects Kamp, H & Reyle, U. (1993), From Discouru to LDgi&, Kluwer, Dordrecht. in Discouru, Kluwer, Dordrecht. Asher N. (1996� 'Mathematical treatments Kamp, H & Rohrer. C. (1983), "Tense in of discoune contexts', in Proaedingr of texts', in R Bauerle, C. Schwarze, & A. von Stech� (eds� Meaning, Use and the Amstmlam Colloquium, VoL Jo, ILLC, Interpretation of lAnguage, de Gruyter, University of Amsterdam, Amsterdam, .2.1-40. Berlin, .2. 5o-69 Bach, E. (1986� "The algebra of events', Kri£b, M (1989� 'Nominal reference, tem poral constitution, and quantification in Linguistics and Philosophy, 9, s-16. event semantics', in R Bartsch, J. van Dalrymple, M., Kamzawa, M., Kim. Y., Benthem, & P. van Emde Boas (eds), Mchombo, S., & Peters, S. (1998), 'Reci Semantics anJ Conttxtual Expressions, procal expressions and the concept of reciprocity', Linguistics and Philosophy, 2.1, Foris, Dordrecht, 7S-I I S. r ssr:uo. I.andman, F. (199.2.� "The Progressive', Natural lAnguage Semantics, I, 1-3.2.. van der Does, J. (199.2.), 'Applied quantifier logics', Ph.D. thesis, University of Lascarides, A. & Asher N. . (1993�
�
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the temporal interpretation of narrative disorders', Ph.D. Thesis, University of Umci, Report no. 34de Swart, H (1991� 'Adverbs of quantifica tion: a generalized quantifier approach', Ph.D. thesis, University of Groningen, published by Garlarul, New York, 1993· de Swart, H (1994� 'Time adverbials in sentence
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H (1998a� 'Aspect shift and coercion', Natural La� arul Linguistic Theory, 16, 347-Ss. de Swan, H (1998b), 'Position and mean ing: time adverbials in context', in P. Bosch & R R van der Sandt (eels� Foais
arul Natural La� Processing,
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tuality: The lntera&tion bttlvttn Tmaporal arul Attmporal Structurt, Cambridge University Press, Cambridge. Vet, C (1992.� 'Petite grammaire de l'Aktionsart et de l'aspect', Cahim dt Grammairt, 19, 1-17.
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Natural Langua� Processing, IBM Institute for Logic and Linguistics. Heidelberg. 4 1 S-4 de Swart, H (1995), 'Negation, aspect and polarity', in Proaeding,r of the Worlulrop on Tense, Spaa arul Movmrmt, VoL 5, Toulouse, 3-16. de Swart, H (1996), 'MeaniDg and use of
not . . . until', journal of Stm4ntits, 13,
How Are Alternatives Computed? ARIEL COHEN Ben-Gurion Univmity
Abstract
I
THE INTERPRETATION OF FOCUS I. I
Focus and alternatives
In his dissertation and subsequent work. Rooth (I985; 199i.) proposes that focus indicates a set of alternatives under discussion. Rooth suggests that every phrase 4> has, in "addition to the Montagovian ordinary semantic value, [t/>r. also afocus semantic value, [t/>f. The focus semantic value is the set of all terms derived from 4> by substituting objects of appropriate types for the focused constituents of t/J. More precisely, [4>f is· defined by Rooth as follows: '
Definition I (Focus semantic value) I. If t/> is a non-focused simple phrase, [t/>jl = .{ [4>]" } . For example, [MaryJ' = {m}.
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It is widely assumed that focusing a phrase indicates that alternatives to the phrase are considered. The question is, how are alternatives to a given phrase determined? There arc a number of proposed answers to this question (Rooth 19Bs, 199z; von Stcchow 1989;]acobs 1983, among others� These accounts, however, typically deal only with logically simple phrases; when more complex phrases are considered, they turn out to be· inadequate. Current theories f.W to provide a principled relation between the altciuatives inducccl by a complex phrase and those induced by its component parts; moreover, they predict incorrect truth conditions in some c;ases. The heart of the problem with· these accounts lies in the assumption that the same combinatory rules used in dctcrmining the meanings of expressions also apply in determining the alternatives induced by them. Instead, I argue that alternatives are induced by presupposition, and that focus induces alternatives only to the extent that it gives rise to presuppositions. The problem of determining the alternatives is thereby reduced to the problem of determining pre supposition in context: the rules for computing alternatives arc the same rules that govern the derivation of presupposition. These rules arc different from the combinatory rules used to compute the ordinary meaning. and thus avoid the problems which plague previous approaches.
44
How Aze Alternatives Computed?
Rooth (1985) proposes that for any expression ¢, the set· of alternatives to
is provided by the focus semantic value of
Definition 2 (Alternatives, Rootb. 1985)
ALT(
]f.
There are cases when the set of alternatives affects truth conditions. For example, Rooth points out that (n) and (2b) have different truth conditions: {2)
I only introduced [Bill)p to Sue. b. I only introduced Bill to [Sue)F. a.
If the speaker mtroduced Bill and Tom to Sue, and performed no other introductions, (2a) is false and (2b) is true. The reason is, according to Rooth, that the speaker who utters (za) considers alternative individuals who could be introduced to Sue, and. asserts that only one of them, Bill, was in fact introduced to her. In contrast, the speaker who utters (2b) considers alternative individuals to whom Bill could have been introduced, and asserts that Bill was, in fact, introduced only to one of them, Sue. Another function of the set of alternatives, according to Rooth, is to restrict the domain of adverbs of quantification. For example, (3a) and (3b) have different truth conditions: ·
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2. If¢> is afocused s imple phrase, [()pf is the set ofobjects matching [¢>]0 in type. For example, [[TohnJFY = B, where B is the set ofindividuals. 3· If¢> is a complex phrase, with components ¢>,, . . . , ¢>, ·the definition ofthefocus semantie value is applied recursively to each compone nt. Then each member of [¢>f is produced by picking o ne element of each of [¢>1f, . . . , [¢,]', and combining them using the ordinary semantic rule for
Arid Cohen 4S
(3)
a. Mary always takes Uohn)F to the movies. b. [Mary)F always takes John to the movies.
If Mary goes
(4)
V ALT(take-to-movies(m, (j)F, s)) b. V {take-to-movies(m, x, s) I x e E} c. alwaysr,[V {take-to-movies(m, X, s) lx e E}] [take-to-movies(m, j, s)] d. alwaysr,[3x: take-to-movies(m, x, s)] [take-to-movies(m, j, s)] a.
Thus we get the desired interpretation, namely that whenever Mary takes someone to the movies, it is always John. The restrictor of(3b), on the other hand is (sa), equivalent to (sb). So the logical form of (3b) is (sc), which boils down to (sd):
(s)
a. V ALT(take-to-movies([m)F, j, s)) b. V {take-to-movies(x, j, s) I x e E} c. alwaysr, [V {take-to-movies(x, j, s) I x e E}] [take-to-movies(m, j, s)] d. alwaysr,[3x: take-to-movies(x, j, s)] [take-to-movies(m, j� s)]
That is how the desired interpretation of (3 b) is derived, namely that whenever someone takes John to the movi� it is invariably Mary. 1
See also de Swart (1991� who formulates this account within a dynamic logic approach.
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to the movies often, with all sorts of people, but John sits at home and never goes out unless Mary takes him, (3a) would be false, but (3b) would be true. Rooth explains that the domain of quantification is restricted by the disjunction of the alternatives. Since (3a) and (3b) have different focus structures, they introduce different sets of alternatives; hence their respective domainS of quantification are different, and consequently they have different auth conditions. Partee (1991) has formulated this claim in the framework of tripartite logical forms proposed by Kamp (1981) and Heim (1982). 1 She proposes that the disjunction of the alternatives is mapped on to the restrictor. For example, the restrictor of (3a) would be (4-a), i.e. situations in which Mary takes some alternative to John to the movies. By definition, this is equivalent to (4b). So the logical form of (3a) would be (4c), which is logically equivalent to (4d�
46 How Aze Alternatives Computed?
I .2
Restriction
by context
It is well known, and, in fact, has already been pointed out by Rooth (1985) himsel£ that definition 2, as it stands, is itiadequate. Consider (6): (6) John always [agrees]F with Mary. .
According to definition
I,
(7) [John [agrees)p with Mary] ! =
{R (j , m, s)},
(8) Q: Did SirJohn already introduce each gentleman to his partner at table? A: No, Sir John only introduced (Bill]p to (Mary]p.
Suppose we restrict the focus semantic value by the context, so as to obtain something like: (9) a. [[Bill]F] f = {xlx is the male partner at the table of a lady} b. [[Mary]F]f · = {y IY is the female partner at table of a gentleman}
2 Compare de Swan (1991� who reinterprets Rooth's definition of the focus seu:antic value: in cxacdy this Wolf.
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where s is a situation variable. In words, the focus semantic value is equal to . the set of all ·possible relations between John, Mary, and a situation. I£ as definition 2 maintains, the set of alternatives is equal to the focus semantic value, (6) would be true just in case wheneverJohn satisfies some relation to Mary, he always agrees with her. But since John always satisfies some relation to Mary, e.g. occupying a location in space different from Mary's, the truth of (6) would require that John agree with Mary every single second of their lives. Clearly, this is incorrect; we should only require that John agree with Mary in all suitably restricted situations,· say when they have a discussion. It follows, then, that the set of alternatives must somehow be restricted by the context. One possible way to do this would be to change the definition of the focus semantic value, so that, if 4> is a focused simple phrase, [[r/>]p]f will not contain all objects OJ.atclpng [rt>Y in type, but a smaller set, restricted by the context.2 Thus, [agree]f will not be the set ofall relations between two individuals and a situation, but only the contextl.ially relevant ones, perhaps {agree , argue, ignore} . In this way, we can maintain definition 2, and keep the compositional definition of the computation of alternatives. However, this approach will not work: if the set of alternatives is identified by the focus semantic value, it is impossible to compute it compositionally. This can be demonstrated by an argument which von Stechow (1989) ascribes to Ede Zimmermann. He considers the following · mini-dialogue:
Ariel
Cohen 47
Definition 3 (Structured meaning) Let P be an expression. Then (Ax1 • • • AX11.Q(x1 , • • • , X11) ,.a1, • • • , a11 ) is a structured meaning for any a1 , • • • , a11 s.t. AX1 • • • AX11.Q(a1 , • • • , a11}
= P.
The idea is that foc:uS determines the structUred meaning: a1 , • • • , a11 are the focused phrases of P. For example, the structUred meanings of (I oa) and (ua) are (xob) and (ub), respectively (where the situation variable- is left free): (xo) (u)
Mary takes Uohn}F to the movies. b. (>.x.take-to-movies(m, x, s}, j ) a.
·
[Mary) takes Johnp to the movies. b. (>.x.take-to-movies {x, j , s) , m)
a.
Von Stechow's theory (1989) can easily accommodate the contextual restriction of alternatives: Definition 4 (Alternatives, von Stechow 1989)3 . The set of alternatives obtained .ftom a structured meaning (>.x1 • • • AX11• Q(x1 , • • • , x,), a1 , • • • , a11 ) is the set
{ Q (y. , . . . ,yll ) IR(y. , . � . ,yll )}, where the relation R is determined contextually. ' This definition docs not aCtually appear in his papet, but it follows &om it
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The sentence ought_ to be true iff there is exactly one pair (x,y) s.t. X E [ [Bill] p] f, r E [ [Mary]p] f, and Sir John introduced X to Y· Now, suppose Mary is Bill's partner at table, and Erica is Jeffrey's partner at table. Suppose further that Sir John introduced Bill to Mary and Jeffrey to Mary, and performed no other introductions. Intuitively, (8) is true, since Sir John only made one introduction between a gentleman and his partner at table; Jeffrey and Mary are not partners at table. However, Jeffrey is the male partner of a lady, and Mary is the female partner of a gentleman, so that Jeffrey_ is in [ [Bill]p] f and Mary is in [ [Mary]F] f, and·the sentence is predicted to be false. Therefore, the set of alternatives to the VP cannot be a function of the sets of alternatives to its parts, and compositionality fails. Von Stechow's (1989) solution · to this problem is in a different framework from Rooth's-the theory of structured meanings (see also Jacobs 1983). StructUred meanings are defined as follows:
48 How Are Alternatives Computed?
FOr example, the sets of alternatives induced by (rob) and (nb) are (ua) and (ub), respectively:
{take-to-movies(m, x, s) IR(x)} b. {take-to-movies(x, j , s)IR(x)} If we take R(x) to indicate simply that x is an individual, these are exactly (12)
a.
the sets of alternatives derived by Roath's approach. Now, the structured meaning of (13a) is (13b), and the set of alternatives is (rJc): a.
Sir John introduced [Bill)F to (Mary)F.
(.Xx. .Xy. introduce(j, x,y) , b, m) ) {introduce(j, x ,y) I R(x,y) } If we take R (x, r) to indicate that X is the male partner at table of lady y, we b. c.
get the desired truth conditions. In a more recent article, Roath (1992) makes a similar proposal, but within his framework of alternative semantics. The definition of the focus semantic value remains the same as in Roath (r985), but ALT(
Definition s (Alternatives, Rooth 1992) Thus, for example, [ [agree]F] f is still the set of all relations between two individuals and a situation; but the set of alternatives will contain only a restricted subset of these relations. In the context of(6), this might be the set
{ agree , argue , ignore} .
To take another relevant example, while (14) [introduce [Bill)F to [Mary]F] f = { introduce(x,y) lx,y E E} , the set of alternatives will be restricted by the context to (r s) ALT(introduce (Bill)F to (Mary)F) = {introduce(x,y) lx is a gendeman & y is a lady & X and f are partners at table}. We thus get the same set of alternatives as that proposed by von Stechow (1989)·
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(13)
Ariel Cohen 49 2 PROBLEMS
The same goes for the structured meaning approach, since structured meanings are generated wing the ordinary semantic rules (and then A abstracting over the focused phrases). Indeed, von Stechow (I989: 20) explicidy states that the alternatives to a disjunction are in the form of disjunctions.• Now consider {I7): •
More prec:isdy, fm von Srcchow altcmativc:s are sc:a ofworlcU; fm example, ALT([Ede]F walks)
is the set of worlds where some altcmati.vc: to Edc walks. He sutcs that ALT{IP V ¢) = ALT{IP} U ALl'(¢}. In words, ALT(IP V T/1) is the union of the set of worlds where some alternative to tP holds with the set of worlds where some altm12tive to 1/J holds, i.e. the set of worlds where some disjunction of the form IP' V ¢' holds, where IP' is an altm12tive to tP and ¢' is an alternative to T/J.
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The combination of definitions I and s seems quite plawible; it provides a compositional definition of the focw semantic value, and allows context to restrict the resulting alternatives. Similarly, von Stechow (I989) . proposes a compositional definition of structured meanings, while leaving the restriction R to be determined by the context. There are, none the less, two difficulties with these accounts: a theoretical one and an empirical one. The theoretical problem is �t neither Rooth nor von Stechow provides a way to characterize the set of alternatives given a particular context, and the role of context _remains completely unspecified. Consequendy, the restriction of alternatives by the context is not defined compositionally; the context is not taken into account when the alternatives to a complex phrase are computed as a function of its parts, but only after the compositional computation (i.e.' the focw semantic value or the structured meaning) has been completed. This leads to the result that, while [] ! or the structured meaning of are defined compositionally, there is no corresponding compositional definition of ALT(
). Suppose 4> is a complex phrase, with components 1 . and 2• Then ALT() is not a function ofALT(1) and ALT(2), but rather a function of [.] l and [2] 1, or 'the corresponding structured meanings (and the context). So the set of alternatives to a phrase is not a function of the alternatives to its parts. For example, it is not clear how the alternatives to -, ¢, 4> /\ 1/J, and 4> V 1/J are related to the alternatives to 4> and 1/J. Moreover, the idea that the set of alternatives to 4> is a subset of [4>]f sometimes predicts incorrect truth conditions. According to definition I, it is crucially the ordinary semantic combinatory rules which are used in the computation of the focus semantic value. Thus, for example,
so How Are Alternatives Computed?
(17) Usually, people obey the law, or they are sentenced to [death)F.
than or: (18)
a. Usually, people obey the law, or else they are sentenced to [death)F. b. Usually, people are sentenced to [death)F unless they obey the law.
For this reason, in addition to and and or, I will use a variety of particles throughout the paper to express conjunctions and disjunctions. Let us now return to (17) and its paraphrases in (18). The focus semantic value of the VP is:
( I 9) {obey the law or be sentenced to x}. Presumably, context restricts x to range over puniShments, so that the set of alternatives is: · (2o) {obey the law or be sentenced to x l x is some punishment.} Consequently, Rooth's theory (and, similarly, von Stechow's) would predict that (17) expresses quantification over people who obey the law or are sentenced to some punishment or other, and states that the majority of those obey the law or are sentenced to death. Assuming that the majority of people do obey the law, (17) is predicted to be true, but it is, in fact, clearly false. Sentence (17), I claim, expresses quantification over people who do not obey the law and are sentenced to some punishment, and states that this punishment is usually death. Since this is not the case, (17) is false. The desired set of alternatives, is, in faa, in the form of conjunctions, not disjunctions: (21) {disobey the law and be sentenced to x l x is some punishment}.
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Before I discuss (17), a methodological point is in .order. Crucially for (17) and other examples throughout the paper, only one element of a coordinated structure is focused. It may not be easy to produce this stress pattern felicitously; a brief pause seems to be required between the disjuncts. The reason is that conjunctions and disjunctions are represented as a parallel structure (Goodall 1987), and it is hard to focus one element in a parallel structure without also foCusing its counterpart (Erteschik-Shir 1997). Goodall claims that only and, or, and but may (though not always) express 'real' coordination, represented as a parallel structure. He distin guishes between these cases and and other ways to conjoin clauses, such as while, after, whereas, etc., which do not result in a parallel structure, even though their meanings comprise a conjunction or a disjunction.· Conse quently, it is easier to focus . only one element of the disjunction if the disjunction is expressed using the noncoordinating or else or unless, rather
Ariel Cohen S r
Note that the set of alternatives is not a subset of the focus semantic value of the VP, in contrast with Rooth's and von Stechow's proposals. In this paper I will address both problems with Rooth's and von Stechow's theories: a consideration of the theoretical problem will suggest a solution to the empirical one. 3
ALTERNATIVES AND PRESUPPOSITION
(22)
a. John trinkt nur [Pils]p. b. John only drinks �ager]p.
Blok & Eberle state: All our German and English infoiDWlts confirmed that, even if we accept lagu as translation of Pils, [(zza)] and its possible translation [(zzb)] differ with respect to the alternatives of the focused constituent. Assuming the empty context, the alternatives of which the nativt sptalur is aware in [(zza)], are those kinds of beer that have a name in German, and correspondingly for [(zzb)]. Normally, neither is Kolsch an alternative of lagu, nor is ale an alternative of Pils (p. I I I, emphasis added).5
According to Blok & Eberle, then, alternatives are computed at the level of words, not concepts; if a concept does not have a name in a particular language, it cannot serve as a member of a set of alternatives. They base this claim on thejudgements of their informants: although each word of(22b) is a direct translation of the corresponding word in (22a), the whole sentence is not a , completely faithful translation, in that it induces different alternatives in the hearer. While this difference between the two sentences is unquestionably an important one, Blok & Eberle have not shown that the two sentences differ in their truth conditions. It may very well be true that, upon hearing a sentence in some language, one would initially imagine only those alternatives which have a name in thatlanguage. It does not follow, however, that these are, indeed, the alternatives with respect to which the sentence is evaluated when its truth value is assessed. Suppose that John drinks only Kolsch and lager, and no other kind of beer. I think it is clear 5 KDlsch and Pi1s are kinds of German beer. Interestingly, Ia� in current Getman does not denote a kind of beer (its meaning is actaally swrt, swrthowse� although � etymology of the English word lager is, in bet, German, 'Lager-Bier' originally denoting a beer brewed for the stordwuse to mature before use (c£ The Nt111 Slllrttr Oxford English DiaiDnary, Oxford University Press, 1 993�
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When considering the definition of the set of alternatives, an important question to consider is whether the set of alternatives is dependent on the particular language used. Bl.ok & Eberle (forthcoming) suggest that it is. They consider the following German sentence and its .ED.glish translation:
S'-
How Aie ·Alternatives Computed?
(23)
a. Does John agree or argue with Mary? b. John always [agrees]F with Mary.
In most cases, however, the alternatives have to be inferred somehow. Consider, for example, the following sentence:
(24) Roses are usually [red)p. What is the set of alternatives with respect to which (24) is evaluated? Intuitively, it is the set of colours. A speaker who utters (24) is perceived to be considering all colours which roses might have, and asserting that the majority of roses are red. But why should colours be the set of alternatives? One possible answer is that this is simply because red is a colour-being red entails being coloured. Perhaps, then, alternatives are based on entailment. This solution will not do in general, however. SaJrlet is a colour, but it is also a shade of red. Being scarlet, then, entails being red, but it also entails being a colour. Would the alternatives to scarlet be all colours or just shades of red? To answer this question, suppose Mary owns three hats: one blue, one green, and one scarlet. Every cJ.ay she wears one of these hats. Therefore, there are times when she wears a hat that it not scarlet; but whenever she wears a red hat, it is invariably scarlet, because she does not own any other red hat. Given this scenario, (25) is false: · (25) Mary always wears a [scarlet]p hat. If the alternatives to scarlet were shades of red only, (25) would have to be true; since it is false, we can conclude that all colours, and not just shades of red, are alternatives to scarlet. The question is, then, why is that? Perhaps this claim needs to be quali6cd a little. If some form of linguistic determinism r:ums out be true, then a language may affect the ontology of the language user, and hence affect the set of alternatives. But eva1 if such influcnce does exist, it can only be indirect, by affecting the world view to
6
of the language user.
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that, in this case, both (22a) and (22b) are false. I£: as Blok & Eberle claim, sentences in a given language were evaluated only with respect to sets of alternatives which correspond to words in that language, (22b) would be judged true. I propose, therefore, that alternatives are language independent; they depend on the meaning of a sentence, on the context and world knowledge, but not on the language used.6 The question, then, is how the alternatives are determined in a given case. Sometimes the solution is easy-the alternatives are explicidy stated. For example, if (6), repeated below as (23b), is uttered in response to (23a), the alternative$ are overdy given, namely the set { agree, argue } .
Ariel Cohen s 3
In a different context, 5earle (1959) discusses this problem, and suggests considering presupposition rather than entailment. Being scarlet and being red both presuppose being coloured; hence the alternatives to both are the set of colours. However, although being scarlet entails being red, it does not presuppose it; hence shades of red arc not alternatives to scarlet. Note that I am guilty of some abuse of language here, since, strictly speaking. semantic relations such as entailment and presupposition only hold between propositions, and not between other semantic objeCts, such as the property of being red or being coloured. This abuse of language, however, is harmless, since semantic relations can easily be extended to hold between arbitrary expressions, as I show in the appendix. The proposal that alternatives arc based on presupposition is, however, still inadequate. Being red presupposes being coloured, but it also presupposes being a physical object. Yet other properties of physical objeCts, e.g. specific sizes, weights or shapes, are not alternatives to red. If they were, it would be impossible for {2.6) to be true: ·
Sentence {26) says that the hats which Mary wears have none of the properties which constitute an alternative to red, except for red itsel£ If size, weight, shape, etc. were considered to be such alternatives, the truth of (26) would imply that Mary's hats have no sizes, weights, or shapes. Note that being coloured entails being a physical object, but not vice versa. In fact, there does not seem to be any other presupposition of being red which entails being coloured. In this sense, being coloured is a minimal presupposition of being red: ·
Definition 6 (Minimal presupposition)
.
1/J is a minimal presupposition of rp iff 'i/J is a· presupposition of tP andfor every 1/J' presupposed by rp1 if .,P' => 1/J then .,P' � 1/J. Note that a minimal presupposition need not be unique; it may be the case that an expression has two or more minimal presuppositions, none of which entails the other. For example, being a bachelor presupposes both being male and being an adult (c£ McCawley 1968). This is why the following sentences are odd: (27) a. Mary
{ �'t }_ a bacbdor.
b. The baby
{ ISn. t } a bachelor. is ,
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(26) Mary only wears [red}F hats.
54 How Arc Alternatives Computed?
However, neither presupposition entails. the other, nor do they entail any other presupposition of being a bachelor. Hence, they are both minimal presuppositions of bachelor. I propose, then. that alternatives are determined by minimal presupposi tions. More precisely, alternatives are expressions which share a minimal presupposition:
Definition 7 (Alternatives)
Let ¢> and '1/J be expressions, ¢> minimally presupposes 1/J. Then
ALT(¢>)
=44
1
1
{ ¢> I ¢> minimally presupposes 1/J}.
(28) Bears usually ride unicycles. In the context ofa discussion ofvarious animals and their forms oflocomotion. however, (28) would be false, since few bears use unicycles to move about. Consequently, it is clear that we should use a pragmatic (i.e. context dependent) definition of presupposition, rather than· a semantic one. Numerous such definitions have been proposed, and they generally make use of concepts such as folicity (or appropriateness) and ·mutual knowledge (or common ground� The following definition, from Levinson (1983: 205, original emphases), is .typical:
Definition 8 {Pragmatic presupposition) An utterance A pragmatically presuppo�es a proposition B iJJA is appropriate only if B is mutually known by participants. Such a definition is, needless to say, vague; unless the notions of appropriateness and common ground are given a precise formulation. it does· not really enable us to prediCt, given an expression and a context, the pragmatic presuppositions of the expression in that context. Providing such a precise formulation of presupposition is a deep problem, one which this paper does not attempt to solve. What I do claim is that, as far as computing alternatives is concerned. nothing further is needed. Assuming that there is some mechanism which derives the presuppositions of a sentence in context, we do not need any additional device in order to derive the set of alternatives induced by the sentence in that context.
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Note that the choice of a set of alternatives to ¢> depends on the choice of the minimal presupposition 1/J, which. in turn. is provided by the context. For example, in the context ofa discussion of acts performed in the Great Russian Circus, the alternatives to the VP in (28) would all presuppose performing some circus act (e.g. juggle, walk the tightrope, ride a unicycle, etc.). Given this context, (28) would be true, s�ce, presumably, the majority of bears which perform in the Great Russian Circus ride unicycles.
Ariel Cohen S S
(29) Q: Which shade of red
does Mary like best? A: Well, she always wears [scarlet]p hats.
Since sets of alternatives share a presupposition, it is reasonable to expect that a presupposition of the term inducing the alternatives would be shared by the alternatives.8 For example, the verb manage is an implicative verb (Karttunen I97I); saying that x managed p normally presupposes that x attempted or needed to accomplish p. For example, (30) presupposes that Ed tried to become friends with the prison guards, or that becoming friends with them was something that Ed needed to accomplish. {3o) Ed
{ =.::g } di
e
to become friends with the prison
guards. ·
';['he alternatives induced by the property manage p, then, are plausibly possible outcomes of the attempt to accomplish p--success or failure. Now consider the following sentence: (3 I ) People usually manage to survive a week without food. Sentence (3 I) is true iff people who ·have to survive a week with no food usually make it. Note that many people do not survive a week without food, for the simple reason that they do not have to; they are never put in this predicament. In other words, these people do not satisfy the presupposition; hence they do not satisfy any of the alternatives and, therefore, are not included in the quantification domain of the adverb.
dcsiSn a sort hierarchy, where alternatives are daughters of '&amcd' nodes, and claim that context helps to determine which nodes are framed. Blok ac Eberle do not, however, provide any account of what it is about the context which brings about this d£cct. 1 Schubert ac Pelletier ( 1987) also claim that 'it is often presupPDsitions oftk mb phrase which suggest the (set ofalternatives)' (p. 44Z, original emphasis). They do not, however, explain why this should be so. 7 Sec also Blok ac Eberle (forthcoming). They
immediate
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The common ground, then, determines the presuppositions, hence the m;n;mal presuppositions, in a given context.7 For example, in the context of the Great Russian Circus, predicating of an individual that it rides, or does not ride, a unicycle, presupposes that this individual performs some act in the circus. Hence only bears performing in the circus will be considered in this context, and (28) would be judged true. This contrasts with Searle's (I9S9) use of semantic presupposition, a definition of presupposition which is context independent. Indeed, iii the right context, shades of red, and not colours, may be considered alternatives to scarlet. For example, in the following mini-dialogue, the answer is true even if Mary also wears blue and green hats, so long as she does not wear hats of any other shade of �ed:
s6 How Arc Alternatives Computed?
Additional examples are not hard to find; the following are from Schubert & Pelletier (1987: 440): (32)
a. Cats usually land. on their feet. . b. A student always admires a fair professor. c. Men usually notice pretty women.
4
FOCUS
An idea common to many theories of focus9 is that the focused element is 'new,' 'informative', or 'unexpected', whereas the unfocused part of the sentence is 'old', 'given', 'known', or 'in the common ground'. This is reminiscent of pragmatic definitions of p"resupposition, where the presupposed information is considered to be part of the common ground. Since the alternatives share a presupposition, it follows that what they differ on is the material which is not presupposed, i.e. the focus. It is for this reason that focus induces alternatives. The primary · role of focus, then, is not to induce alternatives; it only induceS alternatives to the extent that it . rise to pragmatic presuppositions. gives As we have seen, the domain of a quantifier may be restricted by a set of alternatives provided by the presupposition induced by focus. In such cases, focus is, in Rooth's (1985) terms, associllted with the quantifier. However, the alternatives which restrict the domain of the quantifier may be provided in some other way, in particular, by lexical presuppositions of the verb. In such cases, focus is not associated with_ the quantifier, and may be associated with another operator, or remain free. 10
9 Sec Valldavi (1992) md Enacbilt-Shir (1997) for ovcniewl.
10
Compare Schubert ac Pelletier (1987� who consider bow determining the restriction of the domain of the quanri£icr interaCtS with presuppositions based on presuppositional verbs {as opposed to presuppositions based on mess pancms� Our impression is that verb presuppositiOn tcrub to 'win out' if there is a conflict (p. 44 3� ·
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Landing on one's feet presupposes that one drops to the ground; therefore, (32a) is about cats which drop to the ground, and is true just in case they usually land on their feet. Admiring a person presupposes that one knows that person; hence (32b) is evaluated with respect to students who know a fair professor, and states that such students always admire the professor. As · for (32c), noticing a person presupposes that one is near that person; hence (32c) would be true just in case. when a man is near a pretty woman, he usually notices her.
Ariel Cohen 57
For example, consider (3 1) when uttered with a different intonation:
(33) People usually manage to survive a week without [food]F. If the role of focus were to provide a set of alternatives which restricts the
(34) What do people manage to survive a week without? Alternatively, focus may have a contrastive role, perhaps by being associated with the contrast operator (Partee 1991). This is the plausible interpretation of the focus in a context such as the following:
(3 5)
A; B:
People manage to survive a week without water. No! People manage to survive a week without [food]F.
B is correcting A's claim by stating that it is food, rather than water, that people manage to survive a week without.
5
THE CALCULUS OF ALTERNATIVES
The proposal that alternatives are determined by presupposition has concrete implications for the way alternatives to complex phrases are determined as a function of their component phrases. It turns out that this way of computing alternatives avoids the problems with Rooth's and von Stechow's proposals. That is to say, it is possible to provide a principled method to compute alternatives which is empirically adequate. This is in contrast with Blok's (1994) view that alternatives cannot be derived compositionally. He considers the set ofalternatives to the VP in (36):
(36) John only [eats :pples]F. Blok writes: } and the (set of The (set of alternatives) of ut will be something like" {drink, chew alternatives) of apple will be (banana. lime }. But then, it may be dear that the (set of .
.
.
.
.
.
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domain of uiually, we would expect (33) to mean that people who manage to survive a week without something, usually survive a week without food. However, (33) has no such reading; rather, just like {31), it means that people who have to survive a week without food usually make it. In this case, the set of alternatives introduced by the focus does not restrict the domain of the quantifier, but is associated with a different operator, depending on the context. It might be associated with the illocutionary operator assert, Gacobs 1988, as described by von Stechow 1991), iil.dicating that alternatives to food (perhaps various basic necessities) are under discussion. Thus, (33) may be an appropriate answer to (34):
sS How Are Alternatives
Computed?
alternatives) we arc looking for does not contain drink 4 banana. Tb.e relevmt word may be work or ltiss 4 woman, who knows. But it certainly has nothing to do with the subconstituelits of the focusecl expression. !u such, this problem seems unsolvable to me from a logical or linguistic point of view (p. 8� It is not clear � me what makes Blok reach this pessimistic conclusion.
Drinking a banana may just be too bizarre a property to be normally
s.I
Negation
Since alternatives are determined by presupposition, and since, in a given context, presuppositions are unaffected by negation, I propose that the set of alternatives induced by a negated expression is equal to the the set induced by its non-negated counterpart:
Rule 1 {Negation) Thus, for example, both (37a) and (37b) are evaluated with respect to the same set of alternatives: (37)
a. Usually, John [agrees]F with Mary. . b. Usually, John doesn't [agree]F with Mary.1 1 .
Since agreeing presupposes the occurrence of a discussion, both and ALT(., (agree]p) constitute the set of possible reac tions to a discussion, e.g. {agree, argue, ignore}. Sentences (37a) and (37b), then, are about situations in which John and Mary have a discussion: (37a) is true iff in most of these situations, John agrees with Mary; (37b)-if he does not.
ALT((agree]p)
1 1 Readers
who 6nd (37b) odd may wmt to consider the second conjunct of (i) instead: (i) John usually l.i.stcns to Mary, but he doesn't (agrec]F with her. 'The poinls made here will apply equally well
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considered; but chew a lime is certainly an acceptable alternative. I believe Blok is, indeed. correct in pointing out that, in some contexts, the relevant alternatives to eat apples may include work or kiss a woman; eat apPles may, given a suitable context, presuppose a property such as do something in the morning or engage in ·one ofjohn•s favorite activities, etc. Surely, however, there are contexts where decomposing eat apples into the alternatives of eat and apples is preferred; it certainly seems to be what we do in the null context, if the whole phrase . eat apples is focused. In this section I will concentrate on a test case for the computation of alternatives: the alternatives induced by logically complex phrases.
Arief Cohen S9
5.2
Conjunction
In the case of conjunctions, the predictions of the theory presented here coincide with those of Rooth and von Stechow. The alternatives to conjunctions are, straightforwardly, in the form of conjunctions:
Rule 2 (Conjugation)
ALT([c/J]p A 1/J)
=
{c/J ' A 1/J i c/J' E ALT{c/J)}
For example, (3 Sa) is about situations in which Bill drives too fast and receives some punishment, stating that in most such situations he gets a ticket; (3 8b), on the other hand, is about situations in which Bill breaks the law somehow and gets a ticket. stating that in most such situations he drives too fast: (38)
a.
Bill usually drives too fast and (gets a ticket)p (but yesterday he was
arrested). b. Bill usually [drives too fast]F and gets a ticket (but yesterday he crossed a solid white line). Since, as mentioned above, it is hard to focus only one conjunct. the effect is stronger when conjunction is expressed by (part of the meaning of)
after: (39)
a.
Bill usually [gets a ticket)p. after he drives too fast (but yesterday he
was arrested). . b. Bill usually gets a ticket after he [drives too fast]F (but yesterday he crossed a solid white line).
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It should be emphasized that we cannot. as Rooth and von Stechow propose, use . the ordinary combinatory rule for negation to derive ALT(..., [agree]p)· That is to say, the set of alternatives cannot be {-,agree, ..., argue, ..., ignore}. If this were the case, (37b) would ·be evaluated with respect to all situations, since John and Mary always satisfy at least one of those alternatives: when they argue, they satisfy ..., agree{j, m, s) , and when they do not argue, they satisfy ..., argue{j, m, s) . Hence, even if John and Mary are in complete agreement whenever they have a discussion, (37b) will be true, since most of the time (including times when John and Mary sleep, when they are at their respective workplaces, etc.) they do not have a discussion, and, trivially, do not agree on . an�.
60 How Me Alternatives Computed?
Disjunctions
5. 3
Assuming that natural language obeys de Morgan's laws, we can apply them to rules I and
2 to
derive the set of alternatives induced by disjunctions:
Rule 3 (Disjunction) ALT([]p V '1/J) = ALT( -,(-, []� A -,.,p)} = ALT(.., []p A -,.,p) = {¢' A ..,t/11 4>' E ALT(¢)} ·
(4o)
Usually, people obey the law or they are sentenced to (death]p. b. Usually, people obey the law, or else they are sentenced to (death]F. c. Usually, people are sentenced to [death)p unless they obey the law. a.
The set of alternatives to the
VP
is:
{41) ALT(obey the law or be sentenced to (death]p) = {disobey the law and be sentenced to x l_x is some punishment} Consequendy, the sentences in (4o} are true iff the majority of people who are punished for disobeying the law, are sentenced to death; since this is happily not the case, (4o) are false, as desired. To take another example, consider the f!lmiliar adage that all good things are either illegal, immoral or fattening. Suppose that So% of all good things were fattening, 1 9% immoral, and I% illegal. Given this scenario; the sentences in (42), where illegal is focused, are false: (42)
Usually, good things are fattening, or they are [illegal]F. b. Usually, good things are fattening, or else they are [illegal]F. c. Usually, good things are [illegal]F, unless they are fattening. a.
The alternatives to the
VP
are:
= {¢ A -,fattening!¢ E {fattening, immoral, illegal}}
(43) ALT([illegal]F or fatte�)
Hence (42) would be true jwt in case the majority of good things which are not fattening are illegal Since only one-twentieth of those are illegal, the
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In words, the alternatives to a disjunction take the form of conJunctions. This claim contrasts with Rooth and von Stechow, whose rules predict alternatives to a disjunction to be in the form of disjunctions. This may seem counterintuitive at first, but is, in fact, precisely what is needed to account for the truth conditions of sentences like {17) and (z8), repeated below:
Ariel Cohen 61
sentence is false. U: on the other hand. we followed Rooth and von Stechow in deriving disjunctions as alternatives to . disjunctions, (42) would be satisfied iff the majority of good things were illegal or fattening. Since, .in the scenatio desCribed, this is true of 8r% of good things, we would, . incorrectly, predict (42) to be true. S -4
Alternatives and presupposition projection
(44)
a. Fred will kiss Cecilia again. b. Fred has managed to kiss Cecilia and Fred will kiss Cecilia again. c. It is possible that Fred has managed to kiss Cecilia and that he will kiss her again.
Now consider the following example:
(45) People usually attempt to survive a week without food and manage to do so. 12 Let me emphasize that I am using Karmmcn's gencral.izitions as descriptive statements only, making no comment on their explanatory adequacy.
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In this paper I propose that alternatives are induced by presuppositions. If so, the manne r in which alternatives are derived should be the same as that in which presuppositions are projected. It is well known that presupposi tions of parts of a complex phrase are often inherited by the phrase as a whole, but not always. Detennining the constraints guiding such inheri tance is known as the projection problem. I will not attempt a solution to this problem here. My aim is to use this phenomenon to show that whatever principles constrain the projection of presupposition also constrain the computation of alternatives. That is to say, I claim that the problem of computing alternatives is the same problem as that of detennining presuppositions. In the previous section I have proposed three rules for the computation of alternatives; it turns out there are exceptions to these rules, and these exceptions are exactly those cases where presuppositions of the parts are not inherited by the whole. Consider corYunctions. The presupposition of a conjunction is usually the conjunction of the presuppositions of the conjuncts. However, if the first conjunct entails a presupposition of the second conjunct, this presupposition is not inherited by the conjunction, as first observed by Karttunen (1973)!2 He notes that although (w) presupposes that fred has kissed Cecilia before, (44b) does not presuppose this statement (though it does entail it). This can be seen by the fact that (44c) may be true even if Fred has never kissed Cecilia in the actual worlci
62. How Are Altcmativcs Coinputed?
The first conjunct of the verb phrase entails the presupposition of its second conjunct, namely an attempt to survive a week without food. Hence, this presupposition is not inherited by the verb phrase as awhole and, consequendy, (45) is not only about people who attempt to survive· a week without food, but about people in general. Therefore, unlike (3 1), sentence (45) is false. This is also the case with alternatives induced by focus. We have seen above that (6), repeated below, is evaluated only with respect to situations in which John and Mary are having a discussion. ItS truth requires that John agree with Mary in all these situations, but not that he agree with her 24 hours a day. .
But note that the same is not true of (47): (47) John always has a discussion with Mary and (agrees]p with her.
This sentence can only get the bizarre interpretation that, in all situations, John has a discussion with Mary and that, furthermore, he agrees with her in all these situations. " Note that the first conjunct entails the presupposition of the second conjunct, namely that John and Mary are having a discussion; this presupposition, therefore, is not projected, and does not induce a set of alternatives to restrict the domain of quantification. Note that Rooth and von Stechow, as well the exceptionless rule 2 above, would erroneously predict that (47),just like (46), is only about situations where John and Mary are having a discussion. A similar point can be made regarding disjunctions. Again, the relevant observation is first made by Karttunen (1973). He notes that a presupposi tion of 1/J will be inherited by the disjunction QJ V 1/J unless it is entailed by ..., r/J. For example, while (48a) presupposes thatJack has children, (48b) does · not, since the presupposition of the second. disjunct is entailed by the negation of the first disjunct (48) a. All of Jack's children are bald. b. Either Jack has no children or all ofJack's children are bald. Now consider (49): (49)
a. Usually, politicians are crooked, or they (hide}F their honesty. b. Usually, politicians are crooked, or else they (hide]F their honesty. c. Usually, politicians [l;Ude)p their honesty unless they are crooked.
Rule 3 would predict that (49) quantifies over politicians who are not crooked; (49) ought to be true just in case the majority of honest politicians hide their honesty. However, this is incorrect; suppose the majority of
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(46) John always (agrees)F with Mary
Ariel Cohen 63
Received: 2.2..0'].98
ARIEL COHEN
Final version received: o8.01.99
Department of Foreign Literatures and Linguistia Ben-Gurion University Beer Sheva 84 105 Israel e-mail: [email protected]
APPENDIX A: SEMANTIC RELATIONS BETWEEN ARBITRARY EXPRESSIONS Defmitions 6 and 7 above make use of semantic relations; specifically, the definitions refer to entailment, equivalence, and presupposition. Such relations are only defined between truth carriers, ie. propositions; however, focused expressions often denote other semantic objects, such as properties, relations, individuals, functions, etc. In order to provide an account of alternatives in terms of presupposition, semantic relations need to be formally defined for expressions which are not propositions. Let us first deal with expressions whose type is propositional, ie. a function from zero or more types to the type of propositions. 14
Definition 9. (Semantic relations) Let 4> and 'r/J be expressions ofthe � (71 , , Tn , t), where t is the type ofpropositions. Let � be a semantic relation, e.g. entailment or presupposition. T1ms 4> � 'r/J holds ifffor every assignment of values to the variables x:• , . . . ,x�·, • • •
4>(x:• ,
. • .
,x�·) � 'r/J(x:
•
1 3 See Beaver
solutions
to
it.
.�
• •
,x�·).
(rm) for a thorough overview of
•• J remain agnostic
the
projection problem ·
here regarding what the rype ofpropmitions aanal1y iJ.
and the
proposed
.
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politicians are crooked, but the few honest ones flaunt their honesty rather than hide it In this scenario, rule 3 would predict (49) to be false, but it is clearly true. Now note that the presupposition of the second disjunct of (49) is entailed by the negation of the first: hiding one's honesty presupposes being honest, and being honest is entailed by the negation of being· crooked. The fact that (49) is not only about honest politicians, but about all politicians, shows that the presupposition of the second disjunct is not inherited by the disjunction, in accordance with Karttunen's observation. Hence, this phenomenon provides additional evidence that alternatives are determined by presupposition. We may not know the exact nature of the rules which govern the derivation of presuppositions, but we do know that this behaviour is not arbitrary, but is governed by some rules.13 What I have attempted to show in this paper is that those rules, rather than the regular semantic combinatory rules, are the rules that determine the computation of alternatives. Thus, any solution to the problem of presupposition would immediately provide a principled, empitically adequate account of the computation of alternatives.
64 How Aie Alternatives Computed? '
For example, we y.till say that the property scarlet entails the property reel, jwt in case. for every assignment of values to the individual variable x, scarlet{x) entails red{x). To give another example, the relation manage presupposes the relation attempt iff for every assignment of values to the individual variable x and the proposition variable y, manage(x,y) presupposes attempt{x,y). What about an expression whose type is not propositional? The type of such expressions is a function from zero or more types to the type of individuals. In order to define a semantic relation for such an expression, we first need to transform it into an expression of a propositional type. This will be accomplished by Quining, 15 which I define as follows:
De6nition 10 .(Quining) Let ¢> bt of the typt (T. , . . . , T,, e), whert e is the typt of ituliviJuals. Quining ¢> rtst�lts in: �
=
,P(x?"• , . . . , x'!"• }.
After ¢> is Quined, we can use definition 9 to define semantic relations betWeen ¢> and another expression. for example, the individual Socrates will be Quined into the property of being equal to Socrates: >.x.x = Socrates. Thus, being Socrates presupposes being human just in case for . every assignment of values to the individual variable x, x = Socrates presupposes human{x). As another example, consider the function father-o£ When it is Quined, a two-place relation results: >.x.>.y.x = father-of(r). Therefore, being a father entails being a parent just in case, for every assignment of values to the individual . variables x and y, x = father-of(y) entails parent(x,y). 15 Named :Uter Quine's (t96o) 'rcpaning' of names as properties, so that, for example, Socrates is repancd into the property of being Socrates.
REFERENCES Beaver, D. (1997), 'Presupposition', in J. Erteschik-Shir, N. (1997), The Dynamics of van Benthem & A. ter Meulen (eds), Focus Structurt, Cambridge University Hatulbook ofLogit and LAnguagt, Elsevier, Press, Cambridge. . Goodall, G. (1987), Paralltl Structures in Amsterdam, 939-1008. Blok, P. L (1994), 'On the contribution of Syntax, Cambridge U�versity Press, contextual information to the semantics . Cambridge. and pragmatics of foeus', in H. J. Heixn, L (1982), "I'he semantics of definite Biirchert & W. Nutt (eds� Modeling and indefinite NPs', dissertation, Uni Episttmic Propositions, German AI Society, versity of Massachusetts at Amherst. 5-17. Jacobs, J. (1983), Fokus utul Skaltn, Block, Peter L & Eberle, Kurt (forth Niemeyer, Tiibingen. coming), What is the alternative? The Jacobs, J. (1988� 'Fokus-hintergrund computation of focus alternatives from gliederung und grammatik', in H. lexical and sortal information', in Focus: Altmann (ed.), Intonations-Joruhungtn, Linguistic, Cognitive, atul ComPutational Niemeyer, Tiibingen, Sg- I 34Ptr�ctives, P. Bo.$Ch & R van der Kamp, H. (1981), 'A theory of truth Sandt (eds), Cambridge University and semantic representation', in J. Press, Cambridge, 105-19. Gronendijk, T. Janssen & M Stokhof
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>.xe.>.x?"• . . . >.x?:• .
Ariel Cohen .6s 'Problems in the representation of the logical form of generics, plurals, and mass nouns', in E. LePore .(ed.), New Di�ctions in &mantia, Academic Press, London. 385-45 1. Searle, ]. {19s 9), 'On determinables and the notion ofresemblance', in Proceedings of the Aristotelian Society, Supplmrmtary Volume 33, Harrison & Sons, London, I4I-S8. von Stechow, A. (1989), 'Focusing and back grounding operators', Technical Report 6 Fachgruppe Sprachwissenschaft, Uni versitit Konstanz, Germany. von Steehow, A. ( 1991), 'Curient issues in the theory of focus', in A. von Stechow & D. Wunderlich (eds), &mantik/ Semantics: An International Handbook of Contemporary Research, de Gruyter, Berlin. 804-25. de Swart, H ( 1991 ), Adverbs of QIWltifi cation: A Generalized . QIWltifier Approach, dissertation, Groningen Uni versity. Also published by Garland, New York, 1993· Vallduvi, E. ( 1992), 71le Informational Component, Garland, New York. .
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(eds), Mathematical Centre, Amsterdam, 1-41Ka.rttun.en. L {1971), 'Implicative verbs', LAngua�. 47, 34o-s8. Karttun.en. L { 1973� 'Presuppositions of compound sentences', Linguistu Inquiry, 4. 169-93 · Levinson. S. C. { 1983� Pragmatics, Cam bridge University Press, Cambridge. McCawley, J. D. ( 1968� 'Concerning the base component of a transformational grammar', Foundations of LAnguage, 4. 243-69. Partee, B. H { 1991 ), 'Topic, focus and qiWltification', in S. Moore & A. Z. Wyner (eds), Proceedings of the First Conference on Semantics and Linguistic 71leory, Cornell University, 1 59-87. Quine, W. V. 0. ( 196o), Word and Object, MIT Press, Cambridge, MA. Rooth, M. E. (1985), 'Association with focus', dissertation. University of Massachusetts at Amherst. Rooth, M. E. · { 1992), 'A theory of focus interpretation', Natural LAngua� &mantia, I, 7S-I I6. Schubert, L K. & Pelletier, F. J. { 1987),
]ovt1141 ofSmwmda 16: 67-96
Comparative Logic Natural Language
as an
Approach to Comparison in
FRANCESCO PAOLI
University ofMilan
Abstract
I INTROD UCTI O N Montague grammar (Montague I 974. especially I 8 8-270) is generally regarded as a considerable refinement of the traditional logical analysis of natural languages, as usually outlined in the first chapters of any textbook of elementary logic. This is of course true. The orthodox analysis works fme for simple and mosdy ad hoc examples, but breaks down as soon as slighdy more complicated constructions are examined. Think of adjectives like big, wide, good; etc. According to Reichenbach (1947), an advocate of the traditional view, complex noun phrases consisting of an adjective in prenominal position and a coinmon noun can be given a conjunctive reading. ( I) and (2) below, for example, are equivalent: (I) Horses are four-legged animals; (2) Horses are four-legged and horses are animals. But this is no longer the caSe with the forementioned adjectives, sometimes called relative (in contrast with four-legged and the like, termed absolute) since they involve a 'referen.ce class'. (3) and (4) are by no means equivalent: Dumbo is a small elephant; (4) Dumbo is small and Dumbo is an elephant.
(3)
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Montague grammarians claim that the logical analysis of language sketched in standard textbooks of6rst order logic is too coarse to be appropriate to such natural languages as English. In the followiJ:Ig we shall defend the opposite view-at least as far as a very narrow fragment ofEnglish is concerned, traditional logical analysis is perfectly adequate provided we are ready to pay the price of abandoning classical logic in favour of logia containing a sufficiendy rich stock of logical constants. Within the framework of Casari's comparative logic, we shall outline a model of a restricted fragment of English. We shall address such issues as the construction of complex noun phrases, the distinction between gradable and vague adjectives, the structure of comparative sentences and of antonym adjectival pairs.
68 Comparative Logic as an Approach to Comparison in Natural Langtiage
1989, 1990· 1 997)·
It must be emphasized that the primary focus of the present pr�pbsal is on suggesting a new field of application for comparative logic, rather than on providing new insights into the semanrical structure of adjectives and comparison in natural language. Hence, we shall be rather selective in the choice of the examined natural language constructions (see, however, section 6 for a broader discussion). The formal build-up of the corresponding model will only be outlined. In section 2, we shall briefly·review the main rival theories of adjectives. In section 3 we shall do the same for theories of comparison. In section 4 we shall discuss the philosophical basis of comparative logic, focusing on its application to the reconstruction of comparative sentences and complex
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Montague concludes that adjectives cannot be treated as ordinary predicates. For traditional logic, adjectives, intransitive verbs and common nouns are nothing but one-place predicates. According to Montague, however, they fall under three distinct syntactic categories. In particular, adjectives are syntactical operators mapping noun phrases to other noun phrases; semantically, they correspond to functions from properties to properties (properties, in tum, are sets of individual concepts, ie. sets of functions from possible worlds to individuals). Is this multiplication of categories praeter necessitatem? It would be nice if we could prove that it is. In what follows, we shall endorse a 'generalized conjunctive reading' of such sentences as (3) above. For this purpose, we shall introduce a noncommutative conjunction A * B, to be read approxi mately as 'A, taking into account that B'. This connective will be shown to have a rather natural algebraic interpretation. Comparative sentences are considered as yet another shortcoming of the· received view. Given the sentence a is more P than b, traditional logic regards 'is more P than' as a standard two-place predicate, ie. it leaves it unanalysed. Several theories, which we shall review b�low, have been proposed to amend this apparent flaw, each one of them considerably enlarging the logical apparatus of first order theories or their ontological background. Here we shall suggest an 'implicative reading' of comparative sentences, involving nothing else than standard predicates (corresponding to positive adjectives) and . a nonstandard implication connective. Generally speaking, we shall try to account for the above-mentioned natural language constrUctions not by increasing the number of the ordinary categories of logical morphology and syntax, but by proceeding to a more fine-grained investigation of propositional connection. In order to save classical logical analysis, we shall give up classical logic and adopt a different frame-Work, based on Casari's comparative logic (Casari 1987,
noun phrases. In section s we shall sketch a formal model, and in section 6 we shall point out some limits of our investigations.
2. ADJE CTIVES : THE ATTRIBUTIVE AND THE PREDI CATIVE APPRO A C H
·
2.1
Attributive theories
Attributive theories (Parsons 1 972.; Montague 1 974: Cresswell 1 976; Hoepelman 1986) are sometimes also called NM-iheories ('NM' stands for 'noun modifier1. The best-known theory in this group is Montague's. As already remarked, Montague handles adjectives as operators trans forming noun phrases into other noun phrases. Although all kinds of adjectives admit of such a treatment, absolute adjectives have strong invariance properties and therefore resemble in their behaviour the ordinary predicates of logic. Montague accounts for the predicative use of adjectives introducing a dummy noun ('entity'). (s) and (6) are thus claimed to be equivalent:
(s) This is red;
.(6)
This
is a red entity.
Other forms of deletion are sometimes invoked. Keenan & Faltz (198 5); for example, translate (s ) as: There is a property Q such that This is a red Q is true. Hoepelman {1986) provides a somewhat different account. Adjectives are viewed as functions whose domain and ra.nge is the powerset of the universe of discourse. For example, fTAIL (MAN) gives the set of tall men,
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Most adjectives have in English both attributive (prenominal) and predicative occurrences, although a few only occur either as attributives (e.g. former) or as predicates (e.g. asleep). Does this syntactic dichotomy show up even on the semantic level? A number of authors who share this opinion (like Siegel 1 979) postulate indeed two different semantic classes, attributive adjectives and predicative adjectives. However, if this were correct, the same adjective, e.g. good, should belong to different classes according to circumstances. Hence, most writers agree that a more uniform treatment is needed. There are two main families of uniform semantic theories of adjectives: attributive theories suggest that even syntactically predicative occurrences of a given adjective can be treated as semantic attributives, whereas predicative theories lay the opposite claim.
70 Comparative Logic: as a:n Approach to Comparison in Natural Language
fTAI.L(TALL MAN} yields the set of very tall men. Common nouns are themselves functions with the same domain and range. There is a single basic predicate, T ('thing'); the remaining predicates result from the inductive application of common nouns to other predicates. . Here are some objections usually raised against attributive theories:
A) Unsatiifactory analysis ofcomparatives (Kamp 1975). We shall return later on such an issue.
(7) The cathedral tower is high compared to the surrounding buildings. In such a case, we should have two distinct and incompatible reference classes.
C) Problems in deriving the reference class for attributive occurrences (Beesley
1982). According to attributive theories, the reference class of an attributive should invariably be given by its argument, i.e. the noun it is attached to. However, suppose you hear the following exc�e at a cocktail party: (8) 'Who is Quang?' 'Quang is the short Vietnamese'. · In such a circumstance, probably, short means 'short for a man'.
D) Lack of unifOrmity (Beesley 1982). Treating adjectives as functions, Montague canno t even show that A red bam is red is a valid sentence. To make up for this, he is forced to introduce appropriate meaning postulates, splitting up the class of adjectives according to three distinct inference patterns. So, his theory is only seemingly uniform. E) Problems with composition {Siegel 1979). If adjectives are functions, it should be possible to compose them unambiguously. However, John is a tall old man has two different readings (John is tall for an old man'; John is both tall and old for a man'). 2.2
Predicative theories
According to predicative theories, also known as P-theories {Reichenbach 1947; Kamp 1975; Klein 1 980; Beesley 1982), adjectives can be looked upon
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B) Problems in deriving the reference class for predicative occurrences (Hamann · 1991). Deletion of a dummy noun does not properly explain the predicative use of adjectives. Probably, John is tall does not mean that John is tall as an entity, but that John is tall as a man. Some NM-theorists retort that the proper reference class is supplied by the grammatical subject of the sentence, but this does not work either. Take the following example:
Francesco Paoli
71
2.3
NM-theories, P-theories and adjective degrees
All of us learned in grammar books that descriptive adjectives have three degrees: positive, comparative and superlative. Leaving aside the superlative, one could wonder which of the remaining two degrees is logically prior to the other. It turns out that most partisans of the attributive approach agree on the logical priority of the comparative-or at least claim that both forms derive from a third, more basic, structure-whereas most predicative theorists embrace the opposite view. According to the proponents of the first alternative (Langford I 942.; Sapir 1949; Katz 1972.; Wierzbicka 1972.; Wallace 1972.; Bartsch & Vennemann 1972.; Cresswell 1976), dimension and value adjectives are intrinsically relational. The attribution of'tallness' to an individual presupposes the basic and primitive c�tive operation of comparison between her height and the one of another individual. Moreover, a is taller than b seems more precise than a is. tall, hence we cannot found the former on the latter. How, then, can we express a positive adjective in terms of the corresponding comparative? There are- several alternatives: a is tall can be rendered as either a is taller than most (Langford) or a is taller than one would expect (Wierzbicka) or else a is tall to degree d and d is to'wards the top of the scalefor tallness (Cresswell; see below for further explanations). ·
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as ordinary predicates. Kam.p suggests that adjectives, in particular relative adjeCtives, are to be understood as indexed by contextual parameters giving the appropriate reference class. If we assume that the only relevant contextUal factor is the noun to which the adjective is (possibly) attached, we get back the attributive theory as a special case. Indeed, Montague's theory can be subsumed under Kamp's. Prenominal occurer nces of adjectives are explained therein by postulating an operator transforming predicates into noun modifiers. Predicative theories, too, are not short of unsolved problems. The main one, as we have seen, is given by the unfeasibility of a conjunctive reading when a relative adjective occurs in prenominal position. Moreover, there are adjectives-such asformer-that occur only as attributes. Finally, an adjective modifies, if anything, nothing else than common nouns (Hoepelman 1 986). In what follows, we shall try to plead the cause �f the predicative approach by trying to show that complex noun phrases involving relative adjectives (at least when dimension adjectives such as tall or big are at issue) can actually be given a 'conjunctive' reading. though in a rather peculiar sense (remark that also Beesley 1982., although from a completely different standpoint, advocated the feasibility of a conjunctive reading).
72. Comparative Logic as an Approach to Comparison in Natural Language
Hereafter we list some common replies of writers who champion the second view:
·
A) Compositionality (Klein 1980; Keenan & Faltz 1985� In all known languages comparative constructions are given as syntactical functions of the corresponding positive adjective, i.e. they 'contain it as a discernible part' (Keenan & Faltz). Therefore, taking comparatives as basic would violate Frege's principle of compositionality. However, von Stechow (1984) claims that compositionality is not violated if we regard the positive degree of the adjective A a5 made up by the A-stem plus an invisible morpheme for the positive, which is seen as the unmarked form of the adjective.
C) Argument from learning (Kamp 1975; Wi�rzbicka 1996). Developmental psycholinguistics teaches us that the competent use of dimension and value adjectives antedates the mastering of comparative constructions. As Kamp puts it: When we learn a. language such as English we learn the meaning of individual adjectives and, moreover, the semantic function which (the] comparative-fonning operation per forms in general, so that we have no d®culty in understanding . . . the meaning of the comparative of an adjective of which we had thus far only encountered the positive.
D) Coun�examples (Dummett 1978; Siegel 1979; Keenan & Faltz 1985). If we define a is P as a is Per than most, we get some counterexamples. If a large natural number were a natural number larger than most natural numbers, there would be no such number. Moreover, we can consistendy suppose that John is the only driver around and a good one as well. According to the previous definition, however, he could not be such. Lasdy, someone who is c.razier than most people might be just a bit odd, not really crazy. In the following, we shall try to defend the plausibility of the vi_ew
according to which the positive degree is logically prior to the comparative. Indeed, .we shall entirely resolve comparative constructions in terms of the corresponding positive adjectives and the propositional connectives. 24
Gradability arul vagueness
Before moving on to the issue of comparison, let us briefly consider the distinction between gradability and vagueness. Relative adjectives, most typically dimension adjectives such as big or wide, are susceptible of being
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B) Semantic universals (Wierzbicka 1996). Equivalents of big, tall, etc. are available in all known languages, whereas some languages lack comparative constructions.
Francesco Paoli 73
graded, i.e. they come in · degrees. At a first sight. this might seem to mean that they are vague. Upon a closer look, hC>wever, profound differences emerge between the two notions. We shall no.t address, except for some casually scattered remarks, the thorny philosbphical issues concerning vagueness (the most exhaustive source of information on the topic is Williamson 1 994), confin;ng ourselves to some purely linguistical observations. It is generally agreed that the distinctive marks of gcldable adjectives are that: a) they can occur in predicative position after copular verbs; b) they can be preceded by degree modifiers (rather, very, etc.: Klein 1 98o); c) they are order inducing {Klein 1 991); d) they possess a comparative (Hoepelman 1986; Hamann 1 991). Vague adjectives, on the other hand, typically occur in antonym pairs sharing a basic quality (tall-short, big-small, etc.), which cannot be used to partition their domain of application for they admit of extension gaps (for instance, some individuals may be neither tall nor short). If, following Hamann (1991), we call '+pole' the adjective asserting that the common basic quality holds and '-pole' the adjective asserting that the common quality does not hold, we have: ·
{9) {+pole) => not (-pole); (1o) ( -pole) => not (+pole), but not the converse implications, holding instead for sharp adjectives. Bierwisch (1989) remarkS that gradability implies vagueness, but not conversely. Value adjectives, indeed, admit of extension gaps but only become gradable once referred to a suitable comparison class. Williamson (1994) argues that gradability does not imply vagueness either. Acute. and obtuse, taken as adjectives expressing properties of angles, have no extension gap (ifwe agree to include right angles in either class) and are separated by a definite cut-off point, but it is certainly correct to say that a J0° angle is more acute than a 6o0 one. Engel (1989) makes a similar point for acid and basic as expressing properties of chemical compounds. In our theory we shall not be concerned with value adjectives. Within our model, then, vagueness will imply gradability but the converse will not hold. Semantically speaking, gradable and vague adjectives will corresp.ond to functions having different ranges.
3
THEORIES OF COMPARISON
Before going on to examine the main rival theories o f comparison, let us circumscribe our topic. We shall not discuss, in the following, some
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·
74 Comparative Logic
as an
Approach
to
Comparison in Natural Language
important kinds of comparative expressions, including comparatives in opaque contexts (like Russell's well-known I thought your yacht was longer than it is), differential comparatives (john is J em taller than Bill ), compara tives with negative polarity items (Cindy is more beautiful �n anyone here) or nominal comparatives (john ate more apples than oranges), confining ourselves to ordinary adjectival comparatives. We shall, however, briefly return tc;> some of these issues in the final section, where we shall try to discuss what further developments would be needed to improve our theory.
Supervaluational theories·
Van fraases n's method of supervaluations, originally devised to tackle problems related to free logics, was simultaneously adopted in the mid I970S by Fine {I97S) and Kamp (I97S) in order to give an account of the· phenomena, respectively, of vagueness and comparison. The theoretical core of this approach is the rejection of bivalence. Vague predicates, resp. dimension and value adjectives, imply the existence of extension gaps in correspondence of borderline cases. When I evaluate a sentence like a is P, the outcome can be .either one of the two classical truth values (T or F), or a third value, N, to be assigned whenever the meaning of a falls in the extension gap of the meaning of P. A valuation of a given language, therefore, is a mapping of its propositions to the set {T, N, F}. Among all valuations, we can select those whose range is {T, F}-we agree to call them classicaL The completion of a valuation v is a classical valuation v• preserving the classical truth values · already assigned by v: i.e. if v(A) = T(F), then v* (A) = T(F). A proposition of such a language is supertrue iff, for every valuation v, v*(A) = T for every completion v• of v. Intuitively, a completion of a valuation can be seen as a context dependent parameter rendering perfectly precise such distinctions as, for example, the one between old and young. Think. for example, of a competition where a minimal (or· maximal) age limit is set for participation. The application to comparatives, due to Kamp, is nearly straightforward: a is at least as P as b (more P than b) is true in the valuation v iff the set of completions of . v where b is P is (properly) included in the set of completions of v where a is P. So, John is taller than Bill iff there is a context-dependent parameter aC:cording to which John is tall and Bill is not.' Lewis's (r970) delineation semantics exploits the same idea, although Note that such reading is correct only if we add further stipulation, namely that the set of all 1
a
a
sets of completions that satisfy our propositions (ie. that render them true) is linearly ordered by set theoretical inclusion.
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3.1
Francesco
Paoli 7S
dispensing With the apparatus of supervaluations; The role of contextual factors is further emphasized by Klein (1980). The supervaluational approach to comparison lays itself open to a number of criticisms. The first three concern the general treatment of gradable and vague adjectives, whereas the remaining ones specifically regard comparatives. Williamson 1994). Supervalua tionists raise to three the number of truth values because vague predicates determine borderline cases. Suppose that we call 'medium-sized' those objects that are neither definitely big nor definitely small Where should we include, then, objects that are neither definitely big nor definitely medium-sized? Supervaluationists could retort (as Kamp and Klein actually did) that the impossibility to spot a sharp cut-off point, in these cases, by no means implies that it does not exist. But this same line of thought can be used to validate a sharp distinction 'between big and small objects.
A) Second order vagueness (Putnam 1983;
position to rescue all classical tautologies but have to abandon generally _accepted inference rules of classical logic, e.g. contraposition.
C) Gradability and vagueness (Williamson I994). Consider the sentence: (I I) An angle C?f 6o0
is acute.
(I I) is absolutely precise, and hence should receive the value valuations. {rz) below is just as precise: (12) An
T
in all
° angle of 30 is acute.
° Intuitively, an angle of 30 is strictly more acute than an angle of 6o0• According to our previous definitions, however, it is not.
{Williamson I994). Let John be the tallest person in the world and Bill the second tallest. Then
D) Non-borderline comparatives (13)
John is taller than Bill
cannot be true, for the undeniable truth of Bill is tall is incompatible with the truth ofJohn is (strictly) taller than Bill on the supervaluational account. But this is highly counterintuitive. "' E) Mixed comparatives (Kamp 1975; von Stechow 1984). Kamp concedes that such sentences as:
(14)
Jones is more intelligent than he is kind
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B) Mutilation ofclassical logic (Williamson I994). Supervaluationists are in a
76 Comparative Logic as an Approach to Comparison in Natural Language
are 'considerably more difficult to treat . . . Their analysis requires more mathematical strUcture than has been built into the models here con sidered.' But mixed comparativeS often occur in everyday language and should be accounted for. F) Nested comparative$. The supervaluational approach does not allow iterated nestings of comparatives, but it would be desirable to express, e.g. propositions such as The age difference between John and Bill is greater than the age difference between Tom and Dick (orJohn is older than Bill more than Tom is older than Dick).
The semantics of degree
Among the best-known alternatives to Kamp's model of comparatives there is undoubtedly Cresswell's (1976} semantics ofdegree. The underlying idea is simple. As dimerision and value adjectives typically come in degrees, to compare the Pness of a and b is to compare the degree to which a is P with the degree to which b is P. . More precisely, a degree is a pair (u, :5),. where :5 is a p3rtial order relation and u belongs to its field. Basic sentences are of the form a is P to . degree (u, :5), whence the semantics of the positive can be extracted as shown in section 2. 3 above. Comparison is only allowed between degrees of the same scale, i.e. the partial order relation must be identical in both cases. So, if a is P to degree (u, :5) and b is P to degree (v, :5), then a is more P than b is true iff v < u. To accommodate in his theory multidimensional adjectives (clever and the like), which usually do not have a naturally associated metric, Cresswell must however define degrees of P-ness as equivalence classes modulo the relation induced by the comparative of equality associated with P. Some writers have highlighted the following flaws of Cresswell's semantics of degree: .
A} Circularity (Klein 1980). Cresswell's theory is circular because compara tives are defined in terms of degrees, and degrees are in tum defined as equivalence classes modulo the relation induced by the comparative of equality. Moreover, it is counterintuitive: to compare Bill's and Tom's tallness-an extremely simple task-we should involve in the comparison all the individuals belonging to their equivalence classes. B) Separate scales (Hamann 1991). According to Hamann:
·
Cresswell builds up a separate scale for each adjective-noun combination, so that the scales for tall man, tall boy etc. are not the same. One expectS. however, that for dimension adjectives the scale remains constant and only the norm varies with the comparison class.
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3 .2
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77
C) Non-borderline comparatives (Keenan & Faltz 1985). IfJohn and Bill are both definitely short, it might be sensible to maintain that they are tall to degree zero, hence exactly as tall as each other according to Cresswell Yet John might be taller than Bill ·
(Cresswell 1976; von Stechow, 1984). Mixed co� paratives are acceptable in Cresswell's theory only in so far as the adjectives involved share the same degree scale (e.g. tall and long). A sentence such as ( 14) above would count as meaningless. However, Hamann et al. ( 1980) have tried to amend the semantics of Cresswell in order to accommodate mixed comparatives.
D) Mixed comparatives
3·3
Many-valued and fuzzy approaches
The employment of many-valued and fuzzy logics permits an analysis of comparatives in terms of propositional connection (Goguen 1969; Lakoff
1975)·
.
If supervaluationists had raised to three the number of truth values, many-valued logicians are far more liberal and admit even infinite partially ordered s.ets of truth values, ranging from absolute truth to absolute falsity through a more or less densely populated spectrum of intermediate values (of course, not all many-valued logics lend themselves to such a reading. but most fuzzy l
A} Number of values (Lakoff 1975). One
·
might find it rather unconvincing to have an infinite set of truth values, because human beings can only perceive finitely many distinctions among degrees. Lakoff: however, calls this a 'surface phenomenon'.
B) Generalized truthfunctionality (Kamp 1975; Williamson 1994). All many
valued logics generalize classical truth functionality to the new contexts
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(Cresswell 1976). Cresswell asserts that 'on the semantics given for er than, it is not possible to have degrees of "taller than-ness" '. Hence nested comparisons are not feasible.
E) Nested comparatives
78 Comparative Logic as
an
Approach to Comparison in Natural Language
where more than two values are at issue. But this usually implies that principles such as the excluded middle and even the law of noncontra diction are invalidated. Moreover, many-valued logics are amenable to the objection of higher-order .vagueness.
C) Circularity (Klein 1980). The many-valued approach is open to the same charge of circularity already advanced for Cresswell's theory. In fact, the only plausible definition of degrees of 'membership' in a property is in terms of equivalence classes modulo the relation induced by the comparative of equality associated with that property. 1980; Williamson 1994). The sentence of the example (13) above caimot be true, since for it to be so the truth · value of Bill is tall should be strictly smaller than the truth value ofJohn is tall. Intuitively, however, the two sentences should have the same truth value, corresponding to absolute truth.
E) Nested comparatives. Suppose that John is taller than Bill and Bill is taller . than Tom. Hence, John is taller than Bill andJohn is taller than Tom have the same truth value, i.e. they are absolutely true. From an intuitive point of view, however, the latter should be truer than the former. F) Comparative trichotomy (Lakoff 1975; Morton 1 984). If we have a linearly
ordered set of truth values (e.g. if we resort to numerical truth values, as it is often done in many-valued lagics), any two sentences are mutually comparable. But our linguistical intuitions lead us to perceive syntactically well-formed sentences like (1s) John is more intelligent than this wall is long as semantically anomalous.
4 C O MPARATIVE L O G I C AS A THE ORY O F ADJE CTIVES AND COMPARI SON Comparative logic was originally introduced by Casari (see e.g. 1 987) in order to reconstruct the theory of comparison developed in Aristode's Topics. The examples discussed by Aristode already provided a wide ranging and rather intricated sample of comparative sentences, including many nested and mixed comparatives. Later (see· Casari 1 989 and 1 997), it turned out that by suitably relaxing some inessentially strong assumptions one could develop a logic having as models algebraic structUres of intrinsical interest, while getting at the same time an
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D) Non-borderline comparatives (Klein
Fr.ancesco Paoli 79
extremely flexible framework where many well-known logics could be encompassed. Nowadays, comparative logic is a sufficiendy well-established area of research, linked to the fields of substructural and paraconsistent logics (see also Minari 1988, unpublished; Paoli 1996, in press; unpublished); its proof theory and semantics are currendy being investigated by the present writer. 4-1
· Comparative implication
·
(SA) A implies B iff the ·truth degree of A is smaller than or equal to the truth degree of B. (RA) The sentence a is at most as P as b is Q means that the truth degree of a is P is smaller than or equal to the truth degree of b is Q.
The first sentence quoted in (RA) is extremely general The truth conditions of such propositions as a is at most as P as b is or a is at most as P as Q can be recovered therefrom as special cases. By suitably adapting (RA), i.e. by substituting in the obvious ways smaller or equal with smaller, greater or equal, etc., we are in a position to treat comparatives involving more, less, at least as , and so on. P as The identification of implication and comparison of truth degrees leads to deliberate ambiguiti� For example, (16) and (17) below have the same logical form according to our analysis: .
.
.
(16) John is at most as tall as Bill; (17) IfJohn is tall, then Bill is tall
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Relevance logicians claim that implication is an inherently relational, non truth functional notion. We agree with the first claim and disagree with the second. We think that it is truth functional because it involves a compariSon between the degrees of truth of the antecedent and of the consequent. We also suggest that it is inherendy relational because such a 'comparison cannot be avoided even in extreme cases, such as whenever the antecedent is absolutely false or the consequent is absolutely true-contrary to what happens in classical and Lukasiewicz's infinite-valued logics, where under such circumstances the value of the corresponding implication can be established completely disregarding one of the two propositions (notice that, in classical logic, such 'extreme cases' already exhaust as many as three quarters of all possible cases!). We share with many-valued logicians two assumptions: the simplification assumption (SA), equating the concepts of implication and comparison of truth values, and the reduction assumption (RA). reducing to such a fundamental operation every other (basic) kind of comparison:.
8o Comparative Logic: as an Approach 4.2 The
to
Comparison in N�tural Language
ontology of truth degrees
Our semantics of comparatives is based on a number of ontological hypotheses, hereafter listed (cp. Casari 1997).
(A) Every proposition is semantically interpreted by a degree of truth. Truth degrees should form at least a partially ordered set; in the following. however, we further assume that they form a conditionally complete lattice.2
bounded. These are the most striking features which distinguish compara tive logic from many-valued and fuzzy logics, whose sets of truth degrees usually contain just one positive degree (absolute truth) and just one negative degree (absolute falsity}. as well as a more or less numerous stock of intermediate degrees. Ironically, fuzzy logicians do not really take fuzziness at face value. They only admit degrees of approximate truth, not however degrees of 'defmite' truth; correspondingly, they only admit degrees of approximate membership in a set, hut not degrees of definite membership (in spite of lakofPs quite persuasive remarks about 'degrees of hirdiness' and the like). Indeed, it seems natural to say that ifJohn is 2.10 m tall and Bill is 2.05 m tall, they both definitely belong to the set of tall men, hut the first does so more than the second.
(C)
Although it may have no top element, the subset ofpositive truth degrees always has a bottom element; dually, the set of negative truth degrees always has a top element. Such degrees correspond to the truth values of the truth (t), respectively falsity (f), constants. So, being true means having a truth degree that follows the degree of t, while being false means· having a truth degree that precedes the degree of £ This appears plausible as there are propositions, such as a is at most as P as a, which are certainly true, hut in a way seem to be minimally so (for every b strictly Per than a, a is at most as P as b seems truer). A specular argument can be carried out for their negations. Negation (...,) is conceived- of as an involution on the order of degrees. In particular, ..., A is true iff A is false and false iff A is true.
(D)
2 Elementary algebraic notions that are not explicitly defined textbook of algebn of logic, e.g. Birkhoff (r940).
can be e2Sily recovered &om any
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(B) A truth degree may be positive, in which case the proposition to which it is assigned is true; or it may be negative, in which case the proposition isfalse. Negative degrees precede in the ordering positive degrees. No degree can be both positive and negative, but some of them can be neither-i.e. there may exist intermediate truth degrees. In general, the lattice of truth degrees is not
Francesco
Paoli 8 I
{F) Implication (�) is defined in terms of negation and group-theoretical disjunction: A � B is the same as -,A EB B and is therefore true iff the truth degree of A is smaller than or equal to the truth degree ofB, in accordance
with (SA) above.
·
{G) Quantifiers Ct/ �d 3) are viewed as infinitary analogues of lattice theoretical conjunction and disjunction. Hence, they are interpreted by means of generalized greatest lower bounds, resp. least upper bounds, of bounded sets of truth degrees. 4· 3
Advantages of the approach
The comparative-logical approach to comparatives solves, at least partially, many problems plaguing supervaluational, degree-semantical and fuzzy approaches to the issue. Let us examine them one by one.
{A) Unification of scales. The main edge of comparative logic over Cresswell's theory is that all comparisons are effected on the same scale of truth degrees, so that one does not need cumbersome 'grafting' devices (as in Hamann et al. 1980) to compare to each other the degrees ofPness and Qness of an individual, a form of comparison perfectly admissible in ordinary English as shown by (14) above. (B) Overcoming co�parative trichotomy. None the less, we cannot mutually compare any old pair . of propositions, as the semantical anomaly of ( I 5) suggests. But we assumed that our sets of truth degrees be partially ordered, thus failing to rule out the possibility of incomparable pairs of propositions. (C) Relationships with classical logic. Comparative logic, being a supersystem
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(E) There are two kinds of disjunction and conjunction connectives, which we respectively dub lattice-theoretical (V and A) and group-theoretical (e and ®). Other substructural logics draw a similar distinction, although resorting to a different nomenclature-linear logicians prefer to talk about .additive and multiplicative connectives, while relevance logicians use the pair extensional-intensionaL We stipulate that the truth degree of A t\ B (A V B) is the greatest lower bound Qeast upper bound) of the truth degrees of A, B, so that, in particular, A t\ B (A V B) is true (false) iff both A and B are true (false), but may be false (true) without any of its members being false (true). The truth degree of A E9 B is the 'sum' (in a sense to be specified below) of the truth degrees of A, B, whereas ·A ® B is defined as ..., (-,A E9 •B). In particular, A EB B (A ® B) is true (false) iff each one of A and B is at most as false (true) as the other one is true (false).
82 Comparative Logic
as an
Approach to Comparison in Natural Language
of subexponential linear logic without additive constants, validates most principles generally agreed upon by logicians-perhaps even too many to win a fairly unanimous consent. The law of .noncontradiction and the excluded third, among others, are rejected, but as it has been argued within the tradition of subscruaural logics such 'fallacies of contraction' are highly questionable (cp. Troelstra 1992). Like linear logic, however, comparative logic validates group-theoretical analogues of both principles.
(D) Non-borderline comparatives. In the example (13) above, john is tall and Bill is tall might both be assigned positive degrees, yet not necessarily the same degree. Hence the two propositions can be compared to each other.
(F) Nested comparatives. Comparative implication is treated as a higher degree connective, hence it Ca.n be arbitrarily nested. Moreover, propo sitions which contain it can take any truth value. Comparisons of comparative sentences are thus feasible. 4-4
Comparative logic and the semantics of adjectives
Let us now return to an issue already debated at the outset; one could wonder whether the framework just described could be suitably adapted to deal with linguistical constructions other than comparatives-in primis, positive adjectives, common nouns, and intransitive verbs, i.e. those grammatical categories that the old-fashioned tradition of logical analysis obstinately persisted in conflating within the single logical category of one-place predicates. . In our opinion, such an obstinacy has a point. In fact, it leads to a considerable simplification of our syntax: rather than being forced to introduce three classes of expressions, we · could make do with just one. Of course, we should show that this option. can be put .into practice. Given our ontology of truth degrees, one-place predicates are naturally interpreted by functions whose arguments are individuals of a specified domain, and whose values are degrees of truth: given the individual denoted byJohn, for instance, the function denoted by tall yields the degree to which John is tall. The class of one-place predicates can be trisected as following, leaving it open to debate whether common nouns and intransitive verbs should belong to the first or to the second subclass. Sharp non gradable predicates (four-legged, three years old, etc.) are interpreted by functions _
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(E) Mixed comparatives. Comparative logic is apt to formalize general sentences of the form a is at most as P as (less P than, etc.) b is Q, whence mixed comparatives ofevery size and shape can be derived as special instances.
Francesco
Paoli 8 3
(r8) ( I 9)
Dwnbo is a small elephant; Dwnbo is small, taking into account that Dwnbo is an elephant.
Some remarks about the issue of reference classes are now in order. Some occurrences of dimension adjectives have an explicitly attached reference class, in the form of a common noun; in such cases, it is obvious how to translate our natural language sentences to a sentence of the form A * B. In other cases; the appropriate reference class is implicit and must be supplied by the context, so our identification of the logical form of the sentence will proceed from, for example,John is tall toJohn is tallfor a person, toJohn is tal� taking into account thatJohn is a person. Where there is neither an explicit nor an implicit reference class, we take a is tall to mean something like a is tall by
any (plausible) standard.
In our ontology of truth degrees such a connective has to be matched by a suitable noncommutative 'product'. This requires that we broaden our set of stipulations of section 4+
(H) We assume the existence of an associative product operation on truth degrees satisfying a restricted form of distribution over sum on both sides. Both hypotheses are mainly made for the sake of algebraic niceness and in order to prove some suitable restrictions of isotony principles, though being at least not too implausible from an intuitive standpoint. (I) The product of two positive degrees is again a positive degree. For example, someone who is a child and is tall by any plausible standard is someone who is tall for 1- child. ·
(]) We have to guarantee a certain interplay between product and negation. For example, we must be able to conclude that a non-small elephant is not a small elephant. On the other hand, something which is not a small elephant
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whose range of values consists ofjust two degrees, a positive and a negative one, each one the opposite of the other. Sharp gradable predicates (acid, acute, etc.) are interpreted by functions possibly assuming any positive or negative degree, though never an intermediate one. Finally, vague dimension predicates (taU, big, etc.) correspond to functions having for a range the whole spectrum of truth degrees. As we remarked in the first two sections, .the main drawback of the predicative approach to adjectival semantics is the implausibility of a conjunctive reading of such sentences as (3) above. Suppose · now to have, beside group-theoretical and lattice-theoretical conjunctions, a third, noncommutative, conjunction connective. Read A * B as 'A, taking into account that B'. Now a 'conjunctive' reading seems plausible: the semantical equivalence of (r8) and (19) can be argued for with good reasons.
84 Comparative Logic as an Approach to Comparison in Natural Language
is not necessarily a non-small elephant (it might be a tiger or whateVer), but an elephant which is not a small elephant is certainly a non-smaJ.l elephant. (K) If a product of truth degrees . is greater than or equal to zero, so is its second member. For example, 6:om the fact that Dumbo is a small elephant we must be able to conclude that it is an elephant. . {L} 'Cancelling' a (strictly) positive degree on both sides of a disequality, the direction of the disequality is preserved. This expresses some invariance. of relative rankings across different standards. For example, ifJohn is a taller man than Bill, then John is taller than Bill Were we to accommodate value adjectives into our framework, we should probably drop this requirement. .
s OUTLINE OF A FORMAL MODEL Let us now try to give a formal clothing to the informal considerations developed so far. ·
5.1
· Ontological notions
A lattice-ordered pregroup (henceforth 1-pregroup: see Casari 1997) structure G = {G, + , -, o, $) s.t.: (GI] [G2] (G3] (G4] (G6]
IS
a
(G, + ) is an Abelian semigroup; {G, $) is a lattice; - is an involution on {G, $); o + o = o; (Gs] x + -x = o; o $ -x + y iff x $ y.
An 1-pregroup G = (G, +, o, $) is complete iff {G, $) is conditionally complete as a lattice. L-pregroups have an intrinsic algebraic interest, being non-trivial common abstractions of Abelian 1-groups and Boolean algebras. They have been thoroughly studied in Casari (1990) and Minari (unpublished), where an extension to the noncommutative case is discussed as well. -,
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Casari (1997) already attempted a 'three-level' expansion of compara tive logic containing three sorts of conjunction and disjunction con nectives: lattice-theoretical or 'static' (both commutative and idempotent), group-theoretical or 'concurrent' (commutative but not idempotent} and ring-theoretical or 'sequential' (neither commutative nor idempotent). The applications Casari had in mind concerned the fields of action logic and analogical reasoning. What we are suggesting is a similar, but not identical, extension of comp�tive logic.
Francesco
Paoli 8 s
An 1-prering is a structure R = (G, +, - , o, $, x)· s.t.:
Lastly, a regular 1-prering R = (G, +, o, $, x ) is strongly regular iff (R3], [R6] and (R7] above are replaced by the stronger: (R3 ] If x � ·a, then x x (y + z) = (x X y) + (x x z) and (x + y) X z = (x X z) + (y X z) ; (R6 ) x X y $ y; (R7') If 0 $ Z and X X Z $ y X Z, then X $ y. -,
'
'
The following lemma establishes a weak form of isotony for x . LEMMA. In
an 1-prering, if 0 $ X and y $ Z,
X $ y, X X Z $ y X Z.
X
X y $ X X z; if 0 $ Z and
Proof We prove just the first statement. By (G6], y $ z implies o $ -y + z. From this and the fact that o $ x we get, by [14]. o $ x X ( -y + z) , which in virtue of [R3) implies o $ (x x -y) + (x x z) , whence [Rs .2] o $ - (x X y) + (x x z) . By [G6] again, we conclude that x X y $ x X z. Here are some examples of 1-prerings.
(1) Since our definition of 1-prering is actually weaker than the one of I-ring, any I-ring is a suitable example of 1-prering.
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[Rx ] (G, +, - , o, $) is an 1-pregroup; [IU] (G, x ) is a _(possibly noncommutative) semigroup; (R3] If x � o, then x X (y + z) $ (x x y) + (x X z) and (x + y) X z $ (x X z) + (y X z) ; [14] If o $ x and o $ y, then o $ x X y; (Rs.x ] -x x y $ - (x x y); if o $ y, then - (x x y) = -x x y; (R5.2) X X -y $ - (x X y); if 0 $ X, then - (x X y) = X X -y. Notice that our definition of 1-prering is not identical to the one given in Casari (1997), even though we have kept the same label Of course, if we . were to develop an acceptable algebraic theory of 1-prerings, our definitions would soon turn out to be insufficient, and we should add further conditions (as Casari did) in order to get at least weak analogues of the properties enjoyed by 1-rings. An 1-prering R = (G, + , - , o, $, x) is regular iff the following conditions are satisfied: (R6) o $ x X y implies o $ y; (R7) If 0 < Z and X X Z $ y X z, then x $ y; (R8] The product of two bounded sets is bounded (where the product of sets X X Y is defined as {x X y: x E X & y E Y}).·
86 Comparative Logic as an Approach to Comparison in Natural Language
(2) Let Z be the 1-group of integers and B .be the two-element Boolean algebra, both endowed with their us� operations. Now, consider the B splitting of Z w.r.t. its subgroup { o} (see Casari I989 for these notions): In other words, add to Z an element -o s.t. - I < -o < o, -{o) = -o - ( -o) = o and x + -o = x for every x, all the rest remaining unchanged. Next, introduce a product operation X s.t. x x y :..._ xy (with -o x x = -o = x x -o) unless x, y < -o, in which �e x X y = (xy) Then ( Z U { -o}, + , - , o , $ , X ) is a complete c<>mmutative regular 1-preririg which is not an I-ring. -
.
t I 0
n
0 I 2
o 1 2 0 0 0 0 I I 0 I 2
U
0 I 2
0 I 0 I I I 2 2
2 2 2 2
+
0 I 0 0 I I I 2 2 2. 2
2 2 2 2.
X 0 I 2
0 0 0 0
I I
I I
2 0 I Z
It would be interesting to know whether there are nontrivial regular, or even strongly regular, 1-rings. In virtue of [R6'), however, there can be no nontrivial strongly regular 1-rings with unit, since for every x it would be x = x x I $ I and also -I $ -x; but { o} is the only po-group with universal bounds. 5.2
Syntactical notions
A first order extended comparative language £ is defined by: Logical alphabet: a) I -place connectives: -, b) 2-place connectives: $, e, ®, V, A, * c) propositional constants: t, f d) combinators: Z, T, I, K. N, D, C, A e) quantifiers: 'V, 3 Descriptive alphabet: · a) individual variables [VAR): v0 , v1 1 (meta variables x, y, . . . ) b) individual constants: Sue, Rob, Tom, Dumbo . . . (metavariables c, d, . . . ) c) basic predicate constants: c') SHARP (sharp non gradable predicates): four legged, 3 years old, perhaps man, walks . . . c'') GRAD {sharp gradable predicates): acid, acute, perhaps man, walks . . . •
•
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(3) The following tables (with 2 the only positive element) give an example of a finite noncommutative strongly regular 1-prering with unit which is not an I-ring. Notice that it is an integral monoid w.r.t. multiplication.
Francesco Paoli 87
c"� DIM (vague dimension predicates): tall, big, small . . . The class of predicates {PRED; metavariables P, Q, ) is inductively defined as follows: - basic predicate constants are I -place predicates; - if P is in DIM and Q is a n-place predicate, then APQ is a n + I -place predicate; - if P is a n-place predicate and Q is a m-place predicate, then NP, ZP and TP are n-place pre dicates, IP and DP are n -:- I -place: predicates, �Q and CPQ are n + m-place predicates; - nothing else is a predicate. · The class of terms (TERM; metavariables t, t', t1, t�, . } is inductively defined as follows: - individual variables and constants are terms; - nothing else is a term. The class of formulas {FOR; metavariables A. B, . . . ) is inductively defined as follows: , tn are terms, - ·if P is a n-place predicate and t1, then Pt1, , � is a formula; - t and f are formulas; - if A and B are formulas, then -.A. A $ B, A E9 B, A ® B, A V B, A A B, A * B are formulas; - if x is a variable and A is a formula, then 3xA and 'VxA are formulas; - nothing else is a formula. . . .
Predicates:
·
Formulas:
..
•
•
•
·
•
•
•
Comparative logic has been axiomatized in the logical alphabet { -., $ , E9, ®, V, A, t, f, 'V, 3} and shown to be complete w.r.t complete 1pregroups (Casari I997). The present writer has given a cut-free Gentzen formulation of its purely group-theoretical fragment. 5·3
Semantical notions
A (strong) ftame is a pair F = (D, R), where D is a nonempty set and R = (G, + , , o, $ , x) is a complete {strongly) regular 1-prering. A func G is bounded iff its range is bounded. The class of bounded tion f: on functions from on {for SOI\le n) to G is closed under the following operators (cp. Casari I997; we use· (R8] for A): -
�
Zf(du . . , dn) = f(d�, . 1f(d . , . . , dn) = f(d u .
.
. .
· · ·
, dn , d.}, . du-�, dn, dn-• ) ;
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Terms:
88 Comparative Logic as an Approach to Comparison in Natural �e
� , dn): dn) = f(dl t Nf(d" . . . , dn) = -f(dr, dn); , em): Kfg(d" . , dn , e " . . . , em) = f(d " . , dn) n g{er, + , dn) g(eu . . . , em): dn , e " , em) = - f(d " Cfg(d" Afg(du dn) X g(eit em); dn, e u . . . , em) = f(dr, dn , dn+ r)}. Df(du dn) = v� . eo{f(d A function f: on --+ G is sharp iff its range is a pair {x, -x} , with o $ x; is Jf(dlt
. . . •
. . . •
· · · ,
.
. .
. . . •
. . .
. . .
. . .
. . . •
.
. . . •
. . . •
r• . . . •
. . . •
gradable iff its range is a bounded subset of {x: x $ -o or o $ x}; is d-vague iff its range is a bounded subset of the whole of G, i.e. iff it is bounded. A realization of £ in the frame F = (D, R) is a map p s.t.:
An assignment in the realization p is a map u: VAR -+ D, and the valuation induced by q in p is the map prr : TERM -+ D s.t. prr (c) = p( c) for every constant c and pu(x) = q(x) for every variable x. Given a realization p of £ in F and an assignment q in p, a valuation in F is a homomorphism prr of the free algebra of formulas of £ to R, defined as follows (pu' is the valuation induced by u in p): ' ' - Prr (Ptu . . . , tn) = p(P)(pu (tr ), . . . , pu (tn) ); pu(A A B) = prr(A) n prr(B); - prr( -, A) = -prr(A); prr(A V B) = p''(A) U p11(B); prr(A $ B ) = -pu(A) + prr(B); prr(A e B ) = prr (A) + pu(B); prr(A ® B ) = -( -pu(A) + -pu (B)); prr(A * B ) = pu(A) x prr(B); prr(vxA) = Aci e o{ Pu(x/di (A) }; pa ( 3xA) = Vd e o {Pu(x/di (A)}; prr(t) = o; pu(f) = -o. Ifp is a n-place predicate and pu a valuation in F, the extension ofP in prr is the set { (d 1 , , dn): � E p & 0 $ prr(P)(d u . . . • dn ) }. . We say that pu satiJ:fies A (pu I= A) iff o $ p11(A); that A is p-true (p I= A) iff prr I= A for every assignment u in p; that A is F-true (F I= A) iff p I= A for t:Very realization p in F; that A is (weakly) valid { l= (w) A) iff F I= A for every {strong) frame F; that the set of formulas M is p-true (p I= M) iff •
•
•
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. - if c is an individual constant, p( c) is in D; - if P is a basic predicate constant: . if P is in SHARP, p(P) is a sharp function f: D -+ G; if p is iti GRAD, p(P) is a gradable function f: D -+ G; if P is in DIM, p(P) is a d-vague function f: D -+ G; - if P, Q are generic predicates: p(NP) = Np(P), and so forth. p(APQ) = Ap(P)p(Q );
Franc:csco Paoli 89
p F A for every formula A in M; lastly, that A is a (weak) consequence of M (M Few) A) iff for every (strong) frame F and every realization p in it, p F M implies p F A. If A is a (weak) consequence of M, we also say that the inference from M to A is (weakly) valid. 54
Antonymy and modifiers
(2o) IfJohn is taller than Bill, then Bill is shorter than John; {2I) IfJohn is tall, then he isn't short; (22) IfJohn isn't short, then he is tall. the first two ones should count as valid, while the last one should not. Analogues of (22), however, should hold true for such complementary pairs as acute-obtuse or acid-basic. To translate these ideas in formal terms, we could proceed as follows. A (strong) two-complement 1-prering is a pair R• = (R, "') where R = (G, + , -, o, ::;, x) is a complete {strongly) regular 1-prering and "' is a dual automorphism on (G, ::;) satisfying two additional properties: ,
(R9) x :$ --x; [Rio] - - x ::; x iff x ::; -o or o ::; x. For example, we might define -x = -x iff x·::; -o or o ::; x, "'X = - (x + o} otherwise. We could then add to our logical alphabet a new one-place connective, ...,• , and a new combinator, J, matched by the operatorJ on functions. They should satisfy the obvious conditions: - If A is a formula, then ...,•A is a formula; - If P is a n-place predicate, then JP is a n-place predicate;
- Jf(d,, . , dn) = - f(du . - p(JP) = jp(P); - p(...,*A) = -p(A) . .
.
•
.
, dn);
By (R9) and [G3), -x :$ -x; by [Rio), moreover, -x = -x iff x :$ -o or o ::; x. It follows then that the 'antonym' of a sharp predicate has the same
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The foregoing model could perhaps be extended in order to deal with antonym adjectival pairs and adjective modifiers. Since we are not completely satisfied with such a treatment, we prefer to consider these issues separately. Let us take a paradigmatic antonym pair, tall-short. Out ofthe following three implications:
90 Comparative Logic as an Approach to Comparison in Natural Language .meaning
Table
I
Formalization of some natural language sentences
Sentence
Formalization
Dumbo is a big elephant IABEd Micro is a sxnall elephant IAJBEm Dumbo is a four-legged IKLXd
mammal All elephants are mammals All big elephants are big mammals John is a very tall man
Arabella is at least as beautiful as John is clever John is at least as tall :as Bill John is at least as tall as fat John is at least as tall a man as Bill
John's height e:xceed.s Tom's mote than Tom's height exceeds Bill's
Meaning p17(Bd} X p17(Ed} '""'P17(Bm} X p17(Em} p17(Ld} n p17(Xd}
Vx(Ex S Xx}
A { -pa[x/dl (Ex} + pa(x/dJ (Xx) }
Vx(IABEx S IABXx} Yj S Ua
A{ -(pafx/dl (Bx) x p17fxfdl (Ex)) +(pa[x/di (Bx} X pafx/dJ (Xx}) } p17(Hj} X (p"(Hj} X p17(Mj)} -p17(Yj) + p17(Ua}
Hb .., < H�
-p17(Hb) + p17(Hj}
fj < H"�
-pC7(Fj) + pC7{Hj}
Mj A Mb A (IAHMb :5 IAHMj}
p17(Mj} n p17(Mb} n (-(p17(Hb} x p17(Mb)} +(p" (Hj} x p"(Mj))} -( -p" (Hb} + p17(Ht)} +( -p17(Ht} + p17(Hj})
nAHAHMj
(Hb S Ht) :5 (Ht :5 Hj}
Letd = �bo, m = Micro,j = John, b = Bill, a = Arabella,o = Ophidia,E = elephant, = four-legged. X = mammal, M = man, B = big, Y = clever, H = tall. F = fat. u = beautiful
L
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as its complementary adjective, a result which is perfectly in line i w th our intuitions. Modifiers such as very, rather, etc. are not easy to treat within our present framework. As regards very, however, _there might . be a cheap way out. Following Wheeler's (1972) analysis of superlatives, a very tall man co�d be considered, for example, as someone who is tall for a tall man; hence, if H and M are predicate symbols standing for the ·properties of being tall and being a man, and j is a constant symbol denoting John, we could translate John is a very taU man as IIAHAHMj, whose. meaning is p11(Hj) x (p11(Hj) X p11(Mj)). We are not quite sure, however, that such a treatment wholly captures the real essence of very, much in the same way as we believe that our account of antonymy could prove to be unsatisfactory.
Francesco Paoli 91
5·5
Examples
In Table I we formalize some natural language sentences frequendy quoted in the literature on the subject. Out ofthe several possible formalizations arising out of the interplay among combinators and connectives, we have chosen the ones that most closely mirror the superficial structure of the sentence. In Table 2 we list some other frequendy quoted sentences and inferences. The invalid ones are countersigned with a star; the weakly valid ones with a double star; the remaining ones are strongly valid. We leave it up to the reader to verify the correctness of the latter claim; we prove just a couple of examples.
Dumbo is _a big elephant => Dumbo is not a small elephant We have to prove that o :S p11(Bd * Ed) implies o :S p11(..., ( ...,*Bd * Ed)). By [R6), however, o :S p11(Bd * Ed) implies o :S p11(Ed). Moreover we saw that ""P11(Bd) :S - p11(Bd), whence by our restricted isotony lemma ;...,p11(Bd) X p11(Ed) :S - p11(Bd) X p11(Ed). Contraposing by (G3 ), - ( - p11(Bd) X p11(Ed)) :S Table 2. A list of (in)valid sentences and inferences --If Dumbo is a big elephant, then Dumbo is an elephant;
- Dumbo is a big elephant => Dumbo is an elephant; - Dumbo is a big elephant, Jumbo is a bigger elephant than Dumbo => Jumbo is a big elephant; - Micro is a small elephant, Jumbo is a big elephant => Jumbo is a bigger elephant than Micro; - All horses are mammals, Dan is a four-legged horse => Dan is a four-legged mammal; - All horses are mammals => All four-legged horses are four-legged mammals; -* All basketball players are men => All shon basketball players are shon men; -* Every small elephant is small; --Bill is a taller man than Tom => Bill is taller than .Tom; --John is a tall student, Bill is a non-tall student, John is a jockey, Bill is a tall jockey => John is a tall jockey; - John is taller than Bill => Bill is shoner than John; - Durnbo is a big elephant => Dumbo is not a small elephant; - Dumbo is a four-legged animal => Durnbo is four-legged and Dumbo is an animal; -* Dumbo is a small elephan� => Dumbo is small and Dumbo is an elephant; - John is very tall => John is tall; - John is a taller man than Bill => Bill is a man; -* John is a taller man than Bill => Bill is a tall man. ·
·
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IfDumbo is a big elephant, then Dumbo is an elephant. We have to prove that o :S - p11(IABEd) + p11(Ed) for every p11• By [G6), this is the same as p11(IABEd) :S p11(Ed), which, after carrying out the appropriate computa tions, turns out to be equivalent to p11(Bd) X p11(Ed) .$p11(Ed ). But in every complete strongly regular 1-prering, by [R6'], x X y :S y.
92. Comparative Logic as an Approach to Comparison in Natural Language
- (-p11(Bd) X p11(Ed)) . But o � p11(Bd * Ed) = - ( -p11(Bd) X p11(Ed)) (by the definitions, (G3), and [Rs.I], which applies because o � p11(Ed)) � - (-p11(Bd) X p11(Ed) ) = p11(_, (...,*Bd * Ed)) . 6 LIMITATIONS O F THE MODEL AND FURTHER DEVELOPMENTS
(23) (24) (25) (26)
I thought your yacht was longer than it is; If Ede had smoked less, he would be healthier; A polar bear could be bigger than a grizzly bear could be; I thought Plato could have been more boring.
Next, he takes into account B) a group of examples related to the logical form of comparatives: sentences and inferences where comparatives interact with quantifiers or connectives (27), unwarranted inferences (28) that must be blocked, sentences with negative polarity items (29) or with negative quantifiers (3o). (27) Konstanz is at least as nice as Diisseldorf or Stuttgart :::::? Konstanz is at least as nice as Diisseldorf and Stuttgart; (28) *Konstanz is at least as nice as DUsseldorf:::::? Konstanz is at least as nice as Diisseldorf and Stuttgart; (29) Ede is cleverer than anyone of us; (3o) *Ede is cleverer than no one of us. ·
Lasdy, he examines C) th� issue of measure phrases and differential comparatives (31, 32). (3 1) John is six inches taller than Mary; (32) .Ede is twice as fat as Angelika.
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We remarked earlier that the focus of the present paper would have been on general issues of adjectival semantics and on rather simple and standard forms of adjectival comparison, whereas more tricky issues such as, for example, those examined by von Stechow (1984) in his well-known survey of theories of comparison would have been left aside. We feel that our model, unless it were suitably refined, would prove seriously deficient if applied to problems of that sort. None the less, these puzzles could provide effective stimuli to further improve our framework. Von Stechow considers A) a group of examples related to comparatives in opaque contexts: Russell's ambiguity (23), ambiguous counterfactuals (24), sentences with possibility operators (25) or with iterated modalities (26).
Frmcesco Paoli 93 At present, there is not much to say about A-C in the context of our model But, perhaps, some hints pointing to · further developments can already be given.
B) An adequate model of comparatives should explain why (27) seems a legitimate inference whereas (28) does not One could plausibly argue that Konstanz is at least as nice as DUsseldorf and Stuttgart means nothing else than Konstanz is at least as nice as Diisseldorf and Konstanz is at least as nice as Stuttgart, i.e. its logical form is (A $ C) 1\ (B $ C). On th� other hand, Konstanz is at least as nice as Diisseldorf or Stuttgart does not mean
Konstanz is at least as nice as Diisseldorf or Konstanz is at least as nice as Stuttgart, but, rather, Diisseldorf or Stuttgart (whichever is nicer) is at most as nice as Konstanz. Its logical form seems then A V B $ C, which is comparative-logically equivalent to (A $ C) 1\ (B $ C). This would
seem to settle the matter.
C) At present, our approach has very little to say on measure phrases and differential comparatives. It would seem as though we should at least endow our models with appropriate measure functions, .in order to make some sense of expressions like 'A is twice as true as B' and the like. Still, it is hard to see how such 'sentences as (3 I ) could be formalized. Our model, moreover, is plagued by some more general inconveniences.
)
The fragment of English herewith analysed is extremely narrow: it does not even cover the whole class of relative adjectives (value adjectives and nonstandard adjectives such as Jake, former, etc. are left out of the picture), let alone other grammatical categories such as adverbs and the like. It has to be stressed that Montague grammar successfully deals with most such constructions. The propositional view must then be considerably widened if it has to become a plausible alternative to the current approaches to the analysis of natural la.nguige. I
)
The mathematical model calls for further re'finement work, both from a purely algebraic standpoint (the study of 1-prerings in themselves) and from 2
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A) As regards (23), we can in principle reproduce Russell's interpretation of the ambiguity, based on a scope distinction, in terms of degrees of truth ('The degree of truth of The size that I thought your yacht was is large is greater than the degree of truth of The size that your yacht really is large'). In any case, all this presuppose$ . a modalization of our model How could this be attained? Perhaps we should replace classical possible worlds, notoriously equivalent to valuations into the two-element Boolean algebra, with valuations into arbitrary 1-pregroups. We do not know, right now, how fruitful such a line of thought could be.
94 Comparative Logic as an Approach to Comparison in Natural Language
the point of view of its intuitive adequacy, at present far less than satisfactory. Consider for instance the following phenomenon of 'opacity'. Let John be a 1.71 m tall American, and Jim a 1.70 m tall Pigmy. It would seem sensible to assign bothJohn is an American andjim is a Pigmy the same (positive) truth degree: Jim is no more and no less a Pigmy than John is an American. Moreover,john is tall is truer thanjim is tall. By restricted isotony, then, John is a tall American should be truer than jim is a tall Pigmy, a disputable conclusion indeed. It seems, then, that sharp predicates which are used as reference classes cannot as a rule denote functions with the same range in the same modeL
In a sequel to the present paper we hope to tackle in a successful way some of the previously discussed difficulties. Acknowledgements The contents of this paper were discussed in an informal seminar at the Philosophy Department of the University of Florence. We are indebted to all those who Qffered their suggestions and comments, especially Ettore Casari and Pierluigi Minari, who provided preciow insights. We wish to thank also Timothy Williamson, with whom we had extremely helpful conversations on some of the topics covered in the present article, and an anonymou$ referee of the journal of&mantics, who helped w to improve a first draft of the paper. Francesco Paoli Diparti�nto di Filoso.fia UnivmitJi di Milano Via Ftsta tkl Pm1ono 7 20122 Milan Italy
Received: o6.o6. I 998
Final version received: 21.12..1998
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3) Our postulates on logical constants might seem rather artificiaL Some readers could point out that any natural language construction can be roughly simulated by introducing a new connective, a corresponding algebraic operation, and some ad hoc postulates governing their behaviour. We concede such a criticism as regards our treatment of antonymy, partly concede it for relative adjectives, and definitely reject it as regards comparatives. A logic having a well-understood proof theory and a natural algebraic interpretation is not ad hoc. Comparative logic has both; its three level extension might tum out to have a sufficiently plausible algebraic counterpart; the logic of antonym adjectival pairs, unfortunately, has neither, at least so far.
Fnncesco Paoli 9S
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