Events as Dividuals: Aspectual Composition and

Event Semantics∗

Henk J. Verkuyl

Institute for Language and Speech OTSUniversity of Utrecht

1 Introduction

Let me first restrict the empirical domain of the present paper to the aspectu-ality of the a-sentences in (1)–(4). It will help me to clarify the terminologyand to take away possible misunderstandings:

a. Judith ate a sandwich terminativeb. #Judith ate a sandwich for an hourc. Judith ate a sandwich in an hour

(1)

a. Judith ate sandwiches durativeb. Judith ate sandwiches for an hourc. ?Judith ate sandwiches in an hour

(2)

a. Nobody ate a sandwich durativeb. For an hour nobody ate a sandwichc. ?In an hour nobody ate a sandwich

(3)

a. Judith disliked a sandwich durativeb. Judith disliked a sandwich for an hourc. ?Judith disliked a sandwich in an hour

(4)

The a-sentences illustrate the compositional nature of aspectuality: the aspec-tual distinction between (1a) and (4a) can only be attributed to a differencebetween the verbs. One could call eat a terminative and dislike a durative verb,but this would become problematic in view of the difference between (1a) and(2a): here the verb is kept constant; yet (1a) is terminative (some say ‘telic’)and (2a) is durative (some say ‘atelic’). This proves that the internal argumentplays a role in the determination of aspectuality. The same applies to the exter-nal argument: the only relevant difference between (1a) and (3a) is a differencebetween the NP Judith and the NP nobody .

∗I would like to thank Mana Kobuchi-Philip, Ralf Naumann, Bill Philip, Yoad Winter,Martijn Spaan and Joost Zwarts for their helpful comments on earlier versions.

The conclusion of this brief inspection of the data is that both the verb andits arguments contribute to the aspectuality of sentences. This conclusion wasreached in Verkuyl 1972. Although people in general seem to adhere to theidea of a compositional approach, many of them do not take the consequencesthat in my view should be drawn: to find out which basic semantic materialunderlies aspectual composition and how the composition proceeds at higherphrasal levels. In the present paper, I shall discuss the mathematical principlesunderlying aspectual composition in the a-sentences from the point of view ofanalyzing the notion of event as it plays a role in event semantics, viz. asa primitive notion. I will argue that this notion is as complex as aspectualcomposition requires it to be.

The adverbials in the b- and c-sentences serve the purpose of identifyingthe a-sentences as being terminative or durative. I will call (1a) terminative.The other three a-sentences are durative. An adverbial like for an hour appliedto (1a) excludes the single event interpretation of (1a) the result being someother form of aspectuality (a queer sort of repetition), but it is compatible withthe durative aspectuality of the other a-sentences. The reverse holds for thein-adverbials.

It would not be appropriate to say that the b- or c-sentences or sentences likeJudith ate a sandwich yesterday or Judith ate sandwiches twice are terminative,at least not in the same sense as (1a). The b- and c-sentences in (1)–(4) consistof their a-sentence plus an adverbial. In a way, (1a) can be seen as the aspectual“kernel” on which adverbials and other adjuncts may operate. If we extend(1a) to e.g. Yesterday Judith ate a sandwich in the bathroom at midnight thiskernel remains intact and it should be the point of departure, so to say, forthe determination of higher forms of aspectuality. This analysis of aspectualitydiffers considerably from what follows from event semantic approaches. In thepresent paper, I will characterize the aspectual kernel consisting of the verb andits internal argument (a sandwich in (1a)) and its external argument (Judithin (1a)). I will use the term ‘inner aspectuality’ for the aspectual informationexpressed by this kernel structure. The term ‘outer aspectuality’ will be usedfor the result of applying modifying adverbials to inner aspectual information.

The idea behind this position is that if one takes (Fregean) interpretation se-riously, the proper way to go is from the bottom of phrase structure to top of it.The syntactic structure suitable for a bottom-to-top interpretation construingthe aspectuality of sentences like (1a)–(4) will be Figure 1.I will not go into the question of how fine-grained the phrasal structure of (1a)–(4a) should be from the syntactic point of view. It is easy to extend Figure 1with all sorts of functional projections of the generative kind, but this would notsubstantially alter the syntax involved in the composition of aspectuality andpresented below. Therefore, Figure 1 reduces a number of equivalent alternativesyntactic representations to their bare minimum. In current generative theory,the lower S would correspond with the top of a V-projection (= IP), S′ withCP, and NP will be read as DP, etc.1 But for the purpose of the present paper

1In Verkuyl 1993 more structure than shown in Figure 1 is involved in the construal ofinner aspectuality, but as it does not play a role in the argument of the present paper, I will

2

S′

@@

��� . . .

@@

S

NPext VP

V NPint

outer aspectuality

inner aspectuality ��

�

@@@

��

�

@@

@

6

?

Figure 1: Basic structure of aspect construal.

the relabeling of syntactic nodes is not important.There is no separate AspectPhrase at this stage of the analysis. This is

a matter of strategy and reasoning. Such a phrase is often assumed by thosewho take events as primitive individuals. I will argue that if one needs such aprojection, it should be allowed only to mark the transition from inner to outeraspectuality. Only in this way, certain aspectual phenomena can be explainedthat cannot be explained otherwise. On the assumption that there are goodlinguistic reasons for assuming that the NP VP partition of Figure 1 in sentenceslike (1)–(4) is motivated, the question arises whether or not this structure maybe the input of an interpretation function yielding values from the domain ofdiscourse.

There are two positions. The first one is the standard first order positiondeveloped on the Russell-Quine line in mathematical and philosophical logic:Figure 1 is translated into the logical form of sentences like (1)–(4). This is whatRussell, Quine, Chomsky and Davidson have in common: they translate sen-tences into the format given in (5), with P an n-place predicate and a1, . . . , an

as its arguments.2

Pn(a1, . . . , an) (n ≥ 1)(5)

At this point there are again two options: to stop at (5) or to interpret (5) asa subset of the Cartesian product over the domain of discourse, as in (6).

[[Pn]] ⊆ D × . . .×D︸ ︷︷ ︸n

(n ≥ 1)(6)

Although (6) represents the standard semantics of (5), neither Quine nor Chom-sky nor Davidson seem to be very happy about really employing it.3 Chomsky

not discuss it here.2Whenever I refer to (5) this will also pertain to the quantified version of (5)

Qx1 . . . Qxm.P n(a1, . . . , an), with 1 ≤ m ≤ n. In this way, generative logical forms obtainedby Quantifier Raising as developed in May 1977;1985 also fall under its heading.

3Certainly, Davidson’s T-convention seems to place him on the model-theoretic side, butin his analysis of events he never employs the available model-theoretic machinery and ratheroperates on the syntax side.

3

places the interpretation of (1)–(4) outside his grammar in LF′, which makesit hard to expect any theory formation about aspectuality from his side if thisturns out to be a semantic phenomenon. Davidson also restricts himself to (5)extending it so as to make it possible for him to yield the right inferences. Forhim it is important to show that D may contain events as individuals due tovalid inferences drawn from the extended (5) and not because he really is in-terested in how the relation between the structure of (1)–(4) and the structureof the domain D is.

Turning now to the second position with respect to Figure 1, we find it inMontague’s work. For example, Montague 1974a explicitly defends the viewthat a natural language may be seen as a formal language. This means thatstructures like Figure 1 may be interpreted in the Fregean way. The no-tion of Fregean interpretation based on type-logic in the sense explained byHalvorsen and Ladusaw 1979 comes to mind here. The general idea is not un-known in generative circles because Katz, Fodor and Postal used it informallyin the sixties (without paying debt to Frege). The slogan roughly is that themeaning of a sentence is composed of the meanings of its parts. With respect toFigure 1 a Fregean interpretation means that (1)–(4) are interpreted bottom-up,starting with the lexical items and yielding phrasal meanings.

Montague’s model-theoretic work respects (6), though the scheme is ex-tended so as to account for intensionality, but it is primarily his use of type-logicthat makes it possible to do away with the format of (5) as a fixed syntacticpattern of interpretation and to leave the predicate and its arguments “in situ”,as in Figure 1. That is, the V can be seen as a predicate of a certain functionaltype asking for an internal argument NPint and forming a VP which combineswith the NPext so as to form an S. I will restrict myself here to this aspect ofthe Montagovian outlook on interpretation of syntactic structure, but I wouldlike to add that his semantics may be put on “stable” trees, as HPSG and re-lated grammars have shown, i.e trees as we know them from generative syntax.In other words, it is possible to consider the tree in Figure 1 as a generativetree, which rather than being converted into an LF receives its interpretationeither directly as in Montague 1974a, or more indirectly as in Montague 1974b,in both cases by translating the tree into an equivalent type-logical structure.For the purpose of the present paper it is not necessary to execute this, becausefor convenience and for the benefit of the exposition of the machinery involved,the semantics of the sentences in (1)–(4) will mainly be given in terms of thecorresponding domain structure, i.e. in terms of the mathematical functionsunderlying sentences like (1)–(4) rather than their representations.

I think that the bifurcation into two sorts of representations of sentencesexplains a lot about the opposition between certain branches of event semanticsand theories based on the compositionality of aspectuality, such as Verkuyl 1972;1993.It is the purpose of this paper to analyze some of the differences between thetwo approaches. Let me therefore identify the problems that will be discussed.

First of all, the differences between (1)–(4) can not be satisfactorily dealtwith in the philosophical branch of event semantics, that is, along the lines ofDavidson and Parsons. This can be shown quite easily. Before the Davidsonian

4

extension of the standard format (5), (1a) would be represented as (7):

∃x[Sandwich(x) ∧ Eat(j, x)](7)

This is to be interpreted as being true when there is a sandwich x and the pair〈j, x〉 ∈ [[Eat]]. The verb is a set of pairs. This makes it impossible to explainthe aspectual differences between the a-sentences of (2)–(4). For example, thestandard logical form of (4a) is (8):

∃x[Sandwich(x) ∧Dislike(j, x)](8)

This means that the aspectual difference between (1a) and (4a) must be foundsomewhere in the difference between 〈j, x〉 ∈ [[Eat]] and 〈j, x〉 ∈ [[Dislike]]. Thisdoes not help very much, so an extension of (5) was called for anyhow.

Not in Davidson 1985 but e.g. in Parsons 1990 an appeal was made torepresentations like (9).

a. ∃e[. . .Cul(e)]b. ∃e[. . .Hold(e)]

(9)

That is, new predicates are postulated which may express that an event holdsor culminates. This seems to improve on (7) and (8):

∃e∃x[Sandwich(x) ∧ Eat(j, x, e) ∧ Cul(e)](10)

∃e∃x[Sandwich(x) ∧Dislike(j, x, e) ∧Hold(e)](11)

One could then point out that ‘Cul’ be incompatible with for an hour andcompatible with in an hour and that the reverse holds for ‘Hold’. It seems thenthat aspectuality comes within the reach of the extended first order logic.

Apart from the problem that the introduction of predicates like these hardlyexceeds the level of observational adequacy by just making the representationlook like the semantic difference between (1a) and (4a) without giving anyhope for an explanation, Parsons’ version of (9) turns out to be inadequate andunilluminating given what is known about aspectuality in the compositionaltradition, as I will point out in section 3.3. The general question arising iswhether or not event semantics may be considered explanatorily adequate bytreating aspectual phenomena in terms of ‘Cul’ and ‘Hold’. Thus I will have alook at event semantics from the point of view of its treatment of aspectuality.

There is a second problematic area that I want to explore. In Figure 1 theVP plays a very crucial role. It carries the information of what I will call thePath of the external argument denotation. So, aspectually the VP is taken asa unit, as is the (lower) S but with quite different semantic properties, as I willshow.

Now recall that Davidson’s famous 1967-paper started with considering theanaphoric reference of it in sentences like (12) as applied to (1a).

Judith did it slowly deliberately in the bathroom at midnight(12)

What does it refer to in (12)? This question led to the Davidsonian eventsemantics, because Davidson could not live with the idea that the it in (12)

5

would pick out the VP of sentences like (1a). Why not? Because for him underthe standard format in (5) a VP is not a semantic unit that can be referred to.But as I shall point out, if a phrase may refer to a function there is no needto postulate an event as a primitive inside the area of inner aspectuality. Theevent is there because there is a VP which denotes it but this VP is there onlygiven more elementary information contributed by the verb and its internalargument. The presence of the VP for eventhood is only a necessary condition:there must be an appropriate mapping between the external argument and theVP, which is absent in (3a).

From this problem of anaphoric reference, which turns out to be connectedwith a particular choice between two sorts of logical languages, it is a small stepto get into another problem that may be solved if stated correctly. Vendler’swell-known aspectual classes proposed in his Verbs and Times have been de-fined on the verb. Dowty clearly took the notion of verb linguistically and inparticular, lexically and many followed him on this route.4 In my critical anal-ysis of his essay in Verkuyl 1989 I also assumed that Vendler’s notion of verbis linguistic. However, it occurs to me that Vendler may also be interpretedas taking the notion of verb much in the way Montague did, namely as corre-sponding with the VP, expressing a sometimes complex unanalyzed one-placepredicate. This still does not solve the problems for Vendler but this ratherbenevolent new interpretation turns out to make more sense than trying to pinVendler down on four different lexical verb classes. On this line of interpreta-tion the VP gets its own place. More strongly, I will show that it is possible tobring Vendler’s notion of Verb as VP and Davidson’s it into the compositionaltradition by making clear that they intuitively must have referred to what canbe made explicit as the Path in the localistic tradition that gave rise to thescheme in Figure 1.

So, this will be the theme of the present paper: to focus on the role of theVP in the making of terminative aspectuality and to show that only due to thepresence of certain information in the sentence itself may one conclude that thesentence pertains to an event or a process or a state. In this way, the notion ofevent is saved but it will be understood quite differently from what nowadaysseems to be the case in the (neo-) Davidsonian tradition.

The remainder of the paper will be structured as follows. In § 2, a briefsketch will be given of the compositional machinery of Verkuyl 1993, calledplug+. In § 3 I will go into four proposals concerning the issues discussed above—proposals by Davidson, Vendler, Parsons and Krifka— and I will compare theplug+-solutions with these proposals.

4The Edinburgh coercion tradition can be seen as a result from it.

6

2 The plug+-framework

2.1 Introduction

It is very informative to see that, Krifka 1989a excepted, there is no systematicevent semantic analysis available of sentences like (13).

a. Judith ate three sandwichesb. Two girls ate three sandwichesc. Several girls ate at least three sandwichesd. Many girls ate three sandwichese. Some girls ate all sandwiches

(13)

Some of the quantifying NPs cannot be treated in first-order logics as pointedout by Barwise and Cooper 1981. Their article, together with an interest inthe behaviour of plural NPs generated by e.g. Scha 1981 resulted in seriousattention for important issues in generalized quantification. All sentences in(13) are terminative, so from the point of view of temporal structure these sen-tences have also been under investigation. Joint work of Jaap van der Doesand myself led to work in which we set out (a) to reduce the number of read-ings of the sentences in (13): Scha 1981 assigned (at least) ten readings to e.g.(13b), Link 1991 about eight; (b) to clean up the notions of distributivity andcollectivity as they were used in the literature.5 This work resulted in a gram-mar called plug, extended in Verkuyl 1993 to plug+ to form the basis for theanalysis of aspectuality.

I give this information here just to point out how deep the connection be-tween an aspectual theory and a theory of quantification should be. In my view,this bears directly on the criteria by which event semantics should be evalu-ated: it can only be taken as a serious enterprise in the study of aspectualityof sentences if it deals with quantification.6 However, it is also important tosee that by developing a framework in which the account of plurality receives acentral place, certain problems arise in the treatment of singularity, as discussedin § 3.4.

5Distributivity is often treated as synonymous with atomicity, and collectivity is oftencontaminated with the notion of spatiotemporal contiguity or joint agency, etc. The relevantpublications are Verkuyl and van der Does 1991, Van der Does 1992, Verkuyl 1993 and con-cluding joint work on multiple quantification Van der Does and Verkuyl 1995, among others.The discussion involves: Scha 1981, Link 1987, Lønning 1987, Landman 1989, Gillon 1987,Lasersohn 1989;1990;1995, among others

6This judgment does not hold for Hinrichs 1986b or Hinrichs 1986a which are focussed onthe role of aspectuality in construing discourse structure. In general, event semantic toolsare very handy in the study of macro-structures. But for the study of which elements in asentence determine aspectuality, the judgment seems fair enough.

7

In the present section, I shall account for the compositionally formed ter-minative aspectuality of the sentences in (14).7

a. Mary lifted four tablesb. Mary lifted a tablec. Three girls lifted four tablesd. Three girls lifted a table

(14)

It will be shown that their terminative aspectuality is systematically predicted.8

Moreover, the framework in which the sentences in (13) and (14) are treated,contains a complete theory about distributivity and collectivity and subsumesScha’s cumulativity.

In brief, there are three functions involved in the composition of the aspec-tuality of the sentences in (13) and (14):

1. the function s contributed by the verb lift and providing for the sense ofprogress in verbs expressing change;

2. the function ` amalgamating the V and its internal argument into theVP-denotation [[V P ]].

3. the function π relating the members of the external argument denotationto [[V P ]].

These functions will now be defined and explained in some detail in the followingthree subsections.

2.2 The successor function s

There is an aspectually important lexical distinction between verbs express-ing change and verbs expressing a state. This well-known distinction is oftenreferred to as an opposition between non-statives and statives. In order todistinguish them, one may also use the feature opposition [±stative]. Byemploying features one follows a long linguistic tradition, the idea being thatfeatures should have a proper interpretation. In my work on aspectuality, I havebeen using the opposition pair [±add to] rather than [±stative], in order tounderscore that verbs of change are responsible for the property of additivityin nonstative verbs: they yield temporal structure construed from some pointof origin. This treatment of verbs belongs to the localistic tradition in whichchange has been given a dynamic treatment.

The localism involved in the account of change can be tied up very naturallywith the number systems: each [+add to]-verb is interpreted as introducinga well-ordered set I of indices i. Let I := N. That is: I is taken as the set ofnatural numbers, which by Definition 1 are the endpoints of intervals in R, theset of real numbers.

7The reason for shifting from sandwiches to tables is has to do with the introduction ofindices.

8As shown in Verkuyl 1993 this does not only hold for numerals, but for all generalizedquantifiers. It is simpler to demonstrate the system in this way.

8

Definition 1 IV := {(0, k) ⊆ R |k ∈ N}

The successor-function s : I −→ I is defined over I by ∀k ∈ I : s(k) =k + 1.9 The connection between I and IV is made by a function succ :IV −→ IV defined such that for all ∀k ∈ N : succ((0, k)) = (0, s(k)) In thisway [+add to]-verbs meet the Peano axioms characterizing natural numbers(cf.Partee et al. 1990:75–78 for details). Indices are introduced as part of theinformation expressed by a [+add to]-verb. Ignoring at this stage the infor-mation about the external argument X, the verb lift in (15) introduces indicesof which the information about the internal argument Y is made dependent:

. . . λiλY.[[lift]](i)(Y )(X)(15)

Type-logically, lift is here of type 〈i, 〈〈e, t〉, t〉〉〉, the type of functions taking anindex and yielding a collection of sets. This is the type the function ` discussedin detail in § 2.3.10

The indices i contributed by the verb constitute a well-ordered set and sothere is a point of origin which is the starting point of an enumerative device.We are quite familiar with this sort of use of natural numbers when we in factoperate in the reals. We apply it when we put the daily odometer of a caron zero and start to drive. Its natural numbers give us a means to experiencethe sense of progress. If we take the train we make progress in the reals butwe speak about the progress in terms of the stations, discrete entities in thenaturals. Also the picture in Figure 2 shows that whenever we can we usenatural numbers as representatives of stretches in the reals.

month

week

day1 2 3 4 5 6 7

1 2 3 4

n

�����*

HHHHHY

��

�3Q

QQk

Figure 2: Days, weeks, months as numbers and stretches

In natural language shrinking and expansion play a prominent role: we are usedto treat days, weeks, months both in N and in R as in Figure 2. It is importantto see that there is no absolute hierarchy: a month is a set of weeks and a week

9The reason for distinguishing I from N is that I might be taken as a set isomorphic to Nexcept for the property of equidistance, as argued in Verkuyl 1987.

10The full representation of lift is (i) λIλXλiλY.[[lift]](I)(i)(Y )(X). The verb expresses arelation between two atemporal sets Y and X, indices i and the set of indices I construed fromthe indices i. The treatment of tense requires a set I collecting the indices i and discussedin § 2.6. In the plug+-grammar the λIλX-part of (i) is treated syncategorematically to keepderivations simple (see Verkuyl 1993:292–297). The important point demonstrated in (15) isthat the dependency structure in the VP is mirrorred in the lexical information. One mightsay that the i in (15) corresponds with Davidson’s e, but it should be observed that the indicesi are taken as just part of an event rather than being events themselves.

9

is a set of days. Yet, a month is both a collection of sets of days, but it is alsosimply a set of days. I consider this shifting from one number system to theother as crucial for our use of language, especially when we see that the effectof these shifts is to use the naturals as the most simple way of counting.11

2.3 The Path-function `

The general picture for VP’s containing a [+add to]-verb as in (1a)–(3a), (13)and (14) is given in Figure 3. The VP is [+TV P ] (terminative) if the internalNP is [+sqa], and [−TV P ] (durative) otherwise.

VP[±TV P ]

V[+add to] NP[±sqa]-s⊂

��

��

@@

@@

` -

Figure 3: The Path-function `

In (14a) Mary lifted four tables the internal argument NP is labeled [+sqa]. TheNP pertains to what I call a Specified Quantity of A where A is the denotationof the head noun. The label will be made more precise shortly.12 At thispoint it is sufficient to observe that by the presence of the subscript [+sqa] theco-domain of ` is finite. The subscript [−sqa] in (2a) Judith ate sandwichesindicates that the co-domain of ` is not finite.13 Dependent on the plus- orminus-values of the NP, the VP will become VP-terminative or VP-durative.

The domain of ` is the set I, its co-domain DL is set of “positions” p makingup the internal argument denotation.

Definition 2 `x : Ii−→ DL with `x = {〈i, p〉 : [[at(p)(x)]]m,i = 1}

As the definition of the Path-function contains a place for information concern-ing the external argument, ` will be subscripted with x, when necessary for theexposition. Definition 2 characterizes the injective function ` as a set of pairs〈i, p〉 such that x is in the position p at i, given a model M. The term “position”is used to indicate that ` defines a Path in the sense of the so-called localistic

11To conclude with another metaphor. A clarinet-player playing a score observes discreterepresentations (notes) and performs some structure in the reals. The s-function of a [+addto]-verb offers a score getting its actualization in the reals by appying tense. Don’t worryabout metaphors: they will be replaced by a formal machinery below.

12In Verkuyl 1972 [+sqa] is purported to generalize over such NPs as a sandwich, theconcerto, that John was ill in John heard that John was ill, a piece of/from X, three concertos,some of Boccherini’s cello concertos, the whisky, a draught of/from X, a Norwegian sweater,a house, many things, etc. That is, it covers the same ground as the theory of generalizedquantification which did not exist at the time, one of the reasons why it took so long before Icould give a formal treatment of this intuitively clear notion.

13I avoid the term infinite on purpose: non-finite may mean ‘indeterminable’ as in (2a).

10

tradition in the linguistic analysis of verbs expressing change.14 The localisticelement in Definition 2 is the at-predicate which localizes the external argu-ment x in a certain position with respect to the predication involved. Thus,`x constitutes a ‘Path of x’ keeping track of how x relates to the members ofthe internal argument denotation as far as satisfaction of the predicate is con-cerned. This is important for the analysis of plurality, because in (14c) Threegirls lifted four tables there are three x-s involved each obtaining a Path, i.e. away in which they are involved in the predication.

Still stated in terms of positions, for each actualization of a Path `, the pic-ture in Figure 4 holds. It illustrates the way in which the localistic heritage is

p1 p2 p3 . . .

• • • •i i + 1 i + 2 . . .[[V ]]:

`x :

[[NP ]]:6 6 6 6

Figure 4: ` forming a set of pairs, i.e. a ‘Path’

integrated in the model-theoretic machinery. It shows how the relationship be-tween an external argument x (Mary in our lifting examples) and the membersof the internal argument NP denotation (a set of tables) may be structured.

The information expressed by the VP is built up cumulatively. Let medemonstrate this with the help of (14a) Mary lifted four tables. Let the setof tables T = {t1, t2, t3, t4}. The situation in Figure 5 would make (14a) true,with three liftings involved.

p1 p2 p3

{{t1},{t2}} {{t1},{t2},{t3}} {{t1},{t2},{t3},{t4}}

`Mary :

• • •i i + 1 i + 2Verb:

NP:6 6 6

Figure 5: The information of a cell is carried over to its successor.

Each box is a collection of subsets Y of T made dependent on an index, asdiscussed in the type-logical explanation of the verb lift in (15). The last box

14That ` is defined in terms of positions is formally not so crucial as we shall see below:it is rather a tribute to the body of ideas called localism as we find it in e.g. Gruber 1976and in Jackendoff 1976; 1983; 1990, among many others and going back to Latin grammar.An attractive linguistic feature of the localistic perspective on change and statehood is thatby the use of the term ‘position’ a rather natural generalization is expressed over differentsemantic fields such as positional change (Mary lifted four tables), change of possession (Shebought four tables), identificational change (She melted four tables) and other forms of change.

11

is formed by successive steps. In this particular situation, the Path of Mary`Mary would be: {〈1, p1〉, 〈2, p2〉, 〈3, p3〉}.

〈Mary, 〈1, p1〉〉 〈Mary, 〈1, {{t1, t2}}〉〉〈Mary, 〈2, p2〉〉 〈Mary, 〈2, {{t1, t2}, {t3}}〉〉〈Mary, 〈3, p3〉〉 〈Mary, 〈3, {{t1, t2}, {t3}, {t4}}〉〉〈Mary, 〈3, {p1, p2, p3}〉〉 〈Mary, 〈3,T〉〉

(16)

The left-hand side of (16) says that `Mary distinguishes three positions at eachof which the model satisfies the Lift-predication applied to Mary and subsetsof T . Given a model M in which (14a) is true, Mary reached the position atwhich at({{t1, t2}})(Mary) = 1 at index 1. At index 2 a new collection wasadded and so at({{t1, t2}, {t3}})(Mary) = 1, etc.15 In this way, ` keeps trackof the way in which the individual members of the external argument of thepredicate are involved in the predication. It is also explained why the functions comes to a stop: after p3 there is no table from T available for being lifted.This accounts for the terminativity of (14a) – (14d) as well as for the sentencesin (13) and for (1a).16

In general, one may say that (14a) is made true by a set of possible con-figurations given cardinality 4, and three positions, among which (16). Thesituation in (16) and Figure 5 is just one of the possible combinatorial configu-rations that may arise due to the presence of the cardinality information in theNP. In the plug-representation of the internal argument NP, this is providedfor by the presence of an existential quantifier introducing a partition of the setwhich is the denotation of the head noun of the internal argument. Leaving outsome details, the NP four tables is represented as (17).

λP∃W [W ⊆ [[table]] ∧ |W | = 4 ∧ ∃QpsW [Q = P]](17)

This says that there is a set W of four tables having a cardinality of 4 structuredas a partition Q such that Q is the collection of sets of things being lifteddenoted by the predicate P.17 The existential quantifier over Q purports toexpress the degree of uncertainty about which of the possible actualizationshas been the case, but it warrants that there is one. This concurs with theempirical fact that none of the sentences in (13) and (14) give away how thesandwiches were eaten or how the tables were lifted: one-by-one, as a group orin intermediate configurations.

By taking walk as a [+add to]-verb and defining it as in (18), one explainswhy Zeus in Zeus walked may have walked indefinitely.

λXλi[[walk]](i)(X)(18)

15The formal machinery contains a Union-operator to reduce the collection p3 to the set T.16In my own work, this idea was of course present in Verkuyl 1972, but it did not get a

formal semantic treatment until the Groningen conference on Tense and Aspect in 1983 andthe Amsterdam Colloquium in 1984 at which I presented Verkuyl 1987. Though worked outin different ways this idea has also been adopted by Krifka 1989a and by Tenny 1994, whereit is called ‘measuring out’.

17ps means that W is partitioned by Q. P is to be restricted to lifted tables, so it should bewritten as P[[table]] but I will not bring this into the formula here; cf. Verkuyl 1993:143–167

for the details.

12

The difference between lift in (15) and the unergative verb walk is that thereare no internal argument sets Y to bring s to a stop. We assume that in thiscase the co-domain of ` yields the empty set for every application of s.18

The presence of cardinality information in (17) can be seen as the [+sqa]-property of the NP four tables in (14a). For NPs like many tables, few tables,at least 3 tables the [+sqa]-information adopts the form |W | = k where aprecise value of k is not conveyed. In the case of the bare plural sandwiches,the representation does not contain cardinality information of the form |W | =k. Therefore, the co-domain of ` cannot be bounded, so the partitioning isunbounded and s does not come to a stop. This explains the durativity of (2a)Judith ate sandwiches.19

2.4 The participancy function π

A third function relating the external argument NP and the VP is requiredto complete the basic internal aspect construal. This participancy function πtakes as its domain the external NP-denotation assigning to each element x init a unique `. In order to relate an x to its own Path, we shall write π(x) = `x.

Definition 3 π : [[NPext]] −→ (I −→ DL) is defined as: for all (singletons)x ∈ [[NP ]]ext : π(x) = `x, where `x := {〈i, p〉 : [[at(p)(x)]]m,i = 1}

What π does in (14a) is to assign the VP-denotation to the individual Mary,that is π(Mary) = `Mary. The same applies to Judith in (1a): π(j) = `j .20

In Verkuyl 1988 a distinction was made between two constraints on π: πis either injective or π is constant. This turns out to be a natural divisionbased on the intuition that in many languages some sort of plural (all differ-ent)/singular(all one)-distinction is called for. Two familiar types of functionare taken into account. Given f : A −→ B:

1. f is injective: ∀a, a′ ∈ A : a 6= a′ ⇒ f(a) 6= f(a′)

2. f is constant: ∀a, a′ ∈ A : f(a) = f(a′)

Applied to (14c) Three girls lifted four tables, this means that the externaldomain [[NPext]] contains three elements to which π will be applied and thatthere are two different “modes”:

a. π is injective : `1 6= `2 6= `3

b. π is constant : `1 = `2 = `3

(19)

18As argued in Verkuyl 1995 this explains why Mary began to walk is terminative: beginputs a restriction on ` yielding a truncated ` : {〈1, ∅〉} which is of the same type as `. Therange of the function is bounded. The argument shows that in the case of walk the `-functionis present, indicating that walk at the lexical level is different from walk at the VP-level.

19The general idea is that a bare plural introduces a collection of sets and that the cardinalityinformation is at the collection level (≥ 1) and not at the set level. For an extensive analysisof bareness of NPs see Verkuyl 1993:129–140.

20In Verkuyl and van der Does 1991 and Van der Does and Verkuyl 1995 the elements ofdom(π) are singletons, but nothing hinges on that. Def 3 can be adapted to take individualsof type e.

13

On the injective “mode” each of the girls is given a unique value in the co-domain of π. So there are three different images of π. In terms of predicationthis means that the predicate ‘lift four tables’ applies fully to each of the girls.It is called the distributive mode because it closely resembles the situation wefind in simple arithmetical equations like:

3× 4 = (2× 4) + (1× 4) = (1× 4) + (1× 4) + (1× 4)(20)

The distributive mode lets the function π distribute in the same way as the lawof distributivity does in (20). Note that the number 3 is split up in 2 plus 1and 1 plus 1 plus 1 and that 4 is repeated in all splits. This is what happensin (21), where the VP is repeated dependent on the number of elements in theexternal argument domain.21

3 girls lifted 4 tables = . . . =girl1 lifted 4 tables + girl2 lifted 4 tables + girl3 lifted 4 tables

(21)

This makes it possible to recast the notion of distributivity in terms of Figure 6.

︷ ︸︸ ︷{a, b, c, d}︸ ︷︷ ︸ → . . .

︷ ︸︸ ︷{a, b, c} → . . .

,{d}︸︷︷︸ → . . .

. . .

︷︸︸︷{a} → . . .

,

{b} → . . .,

{c, d}︸ ︷︷ ︸ → . . .

︷︸︸︷{a} → . . .

,

{b} → . . .,

{c} → . . .,

{d}︸︷︷︸ → . . .

Unit set AtomsDistributivity� -

Figure 6: Distributivity in the new sense.

Possible actualizations of (14c) along the lines of Figure 6 are (22)–(24), amongmany other configurations:

{girl1} 7→ {〈1, {t1, t2, t3, t4}〉}{girl2} 7→ {〈2, {t3, t4, t5, t6}〉}{girl3} 7→ {〈3, {t3, t4, t6, t7}〉}

(22)

{girl1} 7→ {〈1, {t1}〉, 〈j, {t2, t3}〉, 〈k, {t4}〉}{girl2} 7→ {〈2, {t3, t4, t5, t6}〉}{girl3} 7→ {〈4, {t3, t4, t6, t7}〉}

(23)

{girl1} 7→ {〈1, {t1}〉, 〈j, {t2, t3}〉, 〈k, {t4}〉}{girl2} 7→ {〈2, {t1, t2, t3, t4}〉}{girl3} 7→ {〈3, {t1, t2, t3, t4}〉}

(24)

21Contrary to (20) the law of distributivity in (21) works only one way, i.e. asymmetrically.This is due to the fact that the combinatorial algebra underlying Figure 1 is not associative.See Verkuyl 1993;1996 for a more detailed analysis.

14

An important advantage of the index-dependency of the tables in (21) is that theindices create the room for varying over all combinatorial possibilities. Thus,there are two ways of distinguishing between two tables:22

〈i, {t1}〉 6= 〈j, {t1}〉 vs. 〈i, {t1}〉 6= 〈i, {t2}〉

Definition 4 captures the range of combinatorial possibilities of NP-denotationsillustrated in Figure 6.23

Definition 4 X is distributive iff for all X, Y ⊆ D :

(i) if X ∈ X & Y ⊆ X, then Y ∈ X;(ii) if X ∈ X & Y ∈ X, then (X ∪ Y ) ∈ X

When π is a constant function we obtain for (14c) the situation in (25).

π(g1) = `g1 = π(g2) = `g2 = π(g3) = `g3(25)

The most natural (terminative) sentences for obtaining this interpretation aregiven in (26), although they have a distributive interpretation in the senseexplained above.

a. The twelve passengers killed that horrible villainb. Hans and Uwe wrote that introductory book about DRT

(26)

In both cases none of the individuals involved may claim the VP-predicateitself. So, the passenger Sean may not say ‘I have killed that horrible villain.Likewise, Uwe may not say that he wrote that introductory book about DRT. Itis exactly this what I call kolchoz-collective or totalizing sense of the predicatethat is accounted for by making use of πconstant.24 In the plug-framework, forany collection of sets X, given a domain D kolchoz-collectivity is defined as:

Definition 5 X is totalizing with respect to the collections A,B iff :

(i) A ∩B ⊆ X;(ii) there is no X ⊆ D : X ⊂

⋃(A ∩B) & X ∈ X

This says that the unit set⋃

(A ∩ B) satisfies the predicate denoting X and itprecludes the predicate from being applied to any proper subset of

⋃(A ∩ B).

Note that the distinction between the notions of distributivity and kolchoz-collectivity concurs with the old distinction between count nouns like child ,passenger , book etc. and group nouns like herd and committee. Although theset denoted by herd contains individuals the use of the noun herd prevents that

22This makes it possible to formulate the first part of a “law”: in all the configurations basedon the injective (19a) the total number of tables is always k × m, where k is the cardinalityof the external argument NP and m the cardinality expressed by the internal argument: ineach of the configurations (22)–(24) the total number of token-dependent tables is 12. Fromthe preceding footnote it follows that the total number of girls remains 3.

23Def. 4 holds trivially for (1a): the function π is applied to a single argument and yieldsjust one ` and so Def. 4 applies. This is because the grammar is based on the treatment ofplurality.

24A related approach to the notion of collectivity can be found in Lasersohn 1995, but it isworked out quite differently.

15

any member of a herd be called a herd. The lowest level at which we mayquantify is the group level, whereas regular count nouns allow us to “enter”the set of children and get at its subsets. Similar considerations apply to thedifference between Def 4 and Def 5.25

2.5 The three functions working together

The three functions s, ` and π do not only account for terminativity, they alsoaccount properly for the distributive and collective facts. The result of theinteraction between s, ` and π is given in Figure 7. It comprises all the different

�

�

�

�

�

�

�

�

�

- -

s `x1

I I DL. . .

. . .

�

�

�

�

�

�

�

�

�

- -

s `xn

I I DL

�� � x1 xn[[NPext]]

6 6

π π

Figure 7: The relation between s, ` and π.

sorts of terminativity discussed so far:

1. n = 1 as in (1a), (13a), (14a) and (14b). Here the difference between thedistributive mode and the totalizing mode is irrelevant: they collapse.

2. n ≥ 1 in dom(π) as in (13b)–(13e), (14c) and (14d). Here there are twopossibilities:

(a) n = 1 in ran(π) : totalization defined in Def. 5;

(b) n ≥ 1 in ran(π): distributivity defined in Def. 4.

If Figure 7 is the right picture of the compositional process involved, it followsrather straightforwardly that events are not individuals. Which part of thesystem making up the information should be called an event seems to be amatter of convention. In (14c) Three girls lifted four tables one might call eachs + ` + π-combination an event, even though the indices provided by s are thesame from the temporal point of view. Note also that parts of a Path may becalled an event. But one might also call the VP and event. If so, then oneconclusion should be that mereological approaches to event structure tend toblur the differences between the levels at which we speak about events.

25For a more detailed discussion see Verkuyl 1994; see also Landman 1989.

16

2.6 Tense and Progressive Form

Tense is syntactically defined as taking a tenseless S of type 〈〈〈i, t〉, t〉, t〉 to forma tensed S′: S′ = infl(S).26 Its semantics is given in (27), where α varies overthe different tenses.

[[infl]] = λS∃I∃IV [S(I) ∧ I = Ent+(IV ) ∧ Tenseα(IV )(i∗)](27)

In (14a) Mary lifted four tables α = <, because of its Past Tense. When (27)applies to it, it yields (28):

∃I∃IV [Mary lift four tables(I) ∧ I = Ent+(IV ) ∧ Tense<(IV )(i∗)](28)

This says that there is a set of indices I (in the naturals) and an interval IV

in the Reals such that I is associated with Mary’s lifting of four tables andI represents the stretch of intervals IV and this stretch is before the point ofspeech i∗.27 This means that the actualization of the abstract I-structure intoa real time event takes place by Tense.

The Progressive Form Prog applies before infl and is syntactically definedas taking a tenseless S to form a tenseless S: S′ = prog(S).28

[[prog]] = λSλJ ′∃J [S(J) ∧ J ′ ⊂ J ](29)

For (30a) application of prog to the tenseless [Mary lift four tables] yields(30b), after which infl is applied, yielding (30c).

a. Mary was lifting four tablesb. λSλJ ′∃J [S(J) ∧ J ′ ⊂ J ](Mary lift four tables)

= λJ ′∃J [Mary lift four tables(J) ∧ J ′ ⊂ J ]c. ∃I∃IV ∃J [Mary lift four tables(J) ∧ I ⊂ J

∧ I = Ent+(IV ) ∧ Tense<(IV )(i∗)]

(30)

This says that the Mary-lift-four-tables-predication associated with J can onlybe warranted as to its truth as far as its proper subset I is concerned. That is,only the I-part of J is given as actualized in the reals and Tense commits itselfonly for I and not for its complement in J .

This is a non-modal treatment of the Progressive Form as far as the appli-cation of a rounding-off function is non-modal. It assumes that a tenseless Scontains the information that supplies a full Path. Tense takes only part of itas the Progressive demands.

26Recall from footnote 10 that at the level of S the sentential structure of (14a) is (roughly):λI.[[lift]](I)(i)([[four tables]])([[Mary]]), where I, the collection of i’s associated with the verb,is of type 〈i, t〉. So, this makes Tense as taking an S of type 〈〈i, t〉, t〉 to form an S′ of type t.For details: Verkuyl 1993:318–327.

27Ent+ is defined as a rounding-off function from the Reals into the Naturals. I = Ent+(IV )says that I is the set of representatives from the intervals in IV .

28This means that it is of type 〈〈〈i, t〉, t〉, 〈〈i, t〉, t〉〉, the modifying type.

17

2.7 Some ontological and representational conclusions

From the point of view in which the terminative aspectuality of sentences like(1a), (13) and (14) is formed on the basis of Frege’s principle of composition-ality, there is hardly room for the notion of event as an individual, i.e. as anexplanatory notion. As pointed out above, the term ‘event’ can be tied upwith different phrasal levels such as VP or S. This is actually exactly whatcorresponds with normal intuitions about the notion of event: it is as flexibleas we want to have it. One might say that in distributive sentences like (14c)Three girls lifted four tables and (14d) Three girls lifted a table the notion ofevent applies minimally to `. But it could also pertain to the level at which πapplies. Let us take (14d): it underinforms us with respect about what reallyhappened. On its totalizing interpretation three girls as a group lifted one ta-ble. On its distributive interpretation we distinguish between three events andnow the notion of event seems to shift to each individual Path. So far so good:one might say that π provides for eventhood. But shifting to (14c) one shouldobserve that it could be the result of a check on a list in which it is found outthat in the past year in different months each of the girls lifted four tables, sothat say the 12 liftings are distributed over twelve months. Does that count asone event? Maybe, maybe not.

It seems to me that we have to do here with the same phenomenon observedin Verkuyl 1972 with respect to the frequency adverbials in (31):

a. Piet warned him three timesb. What Piet did three times was warn himc. What Piet did was to warn him three times . . .

(31)

Sentence (31a) can be paraphrased as (31b) and in that case it seems as if wetalk about three different events, whereas the paraphrase in (31c) presents (31a)as pertaining to one event consisting of three warnings. Is it proper to say that(31a) is ambiguous and that we have to look for the minimal event in each ofthe two readings? Or should we simply say that the notion of event is not soappropriate for the analysis of (31a)? I have not a precise analysis to offer forthese sentences because (31a) is a case of outer aspectuality, but in the presentframework a solution would be to let three times modify π in (31b) and ` in(31c). In both cases we do not gain any insight if we use the term ‘event’. It isa handy descriptive term that we employ as language users, but as linguists itmakes less sense than we are inclined to think. The real work in the building oftemporal structure is done by the function s which creates the counting pointsfor the structuring of the internal argument domain into Path-structure.

The localistic notion of Path in the formalization given above is really es-sential for an understanding of the notion of event. There is an event as soonas there is an `x with a proper value of the variable x. Thus it turns out tobe the case that the VP is a very central element in the analysis of eventhood.Syntactically, the internal argument has closer ties with the verb than the ex-ternal argument and so a unit is formed. Semantically, it provides the secondfactor of a multiplication of the form (32) as demonstrated in (20):

|[[NP ]]| × [[V P ]](32)

18

It should be possible to refer to this VP by linguistic means, e.g. by the itin (12. Moreover, it should be possible to make the VP central in the theoryof aspectuality in the sense that some systematic form of aspectuality can beexpressed by it because in sentences with a plural external argument it is nec-essary to copy this form; and finally, it should be possible to treat the VP as afactor in a well-organized pattern made visible in Figure 7, i.e. as dependent onthe external argument for the calculation of the values involved. These threerequirements rephrase the main problems raised in § 1. In the next section, Iwill discuss what event semanticists have to say about these questions.

3 Event Semantics on aspectuality

In the present section, I will show in § 3.1 that Davidson’s problem of anaphoricreference to events is dissolved as soon as one allows for reference to functions,in § 3.2 that Vendler’s proposal can be recast so as to fit in the localistic schemeof Figure 1 and in § 3.3 that Parsons’ attempts to use the aspectual predicatesin (9) have nothing to do with aspectuality. In section § 3.4, I will discussKrifka’s event semantic approach to aspectuality.

3.1 Davidson’s anaphoric reference

Recall that the original problem in Davidson 1980 was the use of it in (33b):

a. Jones buttered the toast slowly, deliberately, with a knife,in the bathroom, at midnight

b. He did it slowly, deliberately, in the bathroom . . .

(33)

Davidson argued that it is not sufficient to have something like ‘There is someaction x such that Jones did x slowly, etc.’, because butter the toast is not anappropriate singular term. What he did was (after throwing out slowly anddeliberately) to assign to (33a) the logical form in (34a).29

a. ∃e.Butter(jones, the-toast, e) ∧With(a-knife, e)∧In(the-bathroom, e) ∧At(midnight, e)

b. ∃e.Buttering(e) ∧Agent(jones, e) ∧ Patient(the-toast, e)∧With(a-knife, e) ∧ In(the-bathroom, e)

(34)

The it in (33b) can now be said to pertain to the e. In (34b), the neo-David-sonian format of (34a) is given: the thematic roles of Agent and Patient aretaken as two-place predicates relating the Agent and Patient to the event inwhich they have that role.

29Neo-Davidsonians tend to throw out the temporal adverbial as well. This sounds reason-able, also in view of the fact observed in Verkuyl 1993:245–251 that Davidson can only handlethe inferences properly if the Agent-NP in sentences like (33a) denotes a monotone increasingquantifier. Thus, if one replaces Jones by At most three students one cannot infer to At mostthree students buttered the toast α, where α = at midnight . The problem is more general: in-ferences are no longer valid under monotone decreasing quantification in cases where α =witha knife or in the bathroom or deliberately or slowly . Event semanticists remain quite silentabout this disturbing fact. Not many of them like to face the problems of quantification, soit is Jones and Sebastian rather than three girls and few boys.

19

How would the anaphoric reference behave in sentences in which no actionis involved, such as (35a).

a. Jones died in the bathroom at midnightb. It happened in the bathroom at midnight

(35)

It is clear that one cannot continue (35a) with He did it in . . ., but that onehas to appeal to something like (35b). What is this it and should one expect,like Davidson, a singular term for it (and not find it)? No, of course not, oneshould expect sloppy identity as becomes clear by sentences like (36) and (37).

a. Three girls buttered their toast in the bathroomb. They did it at midnight

(36)

a. Judith and Jessica buttered their toast in the bathroomb. They did it at midnight

(37)

In (36) the anaphoric reference of it is sloppy. Of course, one could declare(36a) ambiguous, but as pointed out there are at least as many arguments forassuming that (36) underdetermines what really happened: the girls may ormay not have been in the bathroom together and they may or may not haveshared the toast.30 What we need in the case of (36) is that it may pertain tothree or two events or to just one. It must indeed be sloppy. The same appliesto sentences like (37): here it is clear that for a situation in which the two girlseach buttered their toast, the it is the sloppy identity it well-known from theliterature on do so-constructions, for which an analysis in terms of lambda-abstraction turns out to be the proper sort of solution. In Verkuyl 1972:142ff.,I proposed such a solution for sentences like Arie ate a herring and Piet didso too in order to account for the terminativity of this sentence. In terms ofthe Plug-framework of § 2, this amounts to saying that the `-function is theinterpretation of the it in sentences like (33), (35), (36) and (37). In this waywe may explain why it is that sentences like (38a) pertains to six sandwichesand why (38b) pertains to twelve tables.

a. Judith ate three sandwiches at twelve o’clock and Jessica did ithalf an hour later

b. Three girls lifted four tables and each of them did it in twentyseconds

(38)

It is not difficult to think of ` in terms of lambda-abstraction: `x contains avariable x, which needs to receive its value before the function can be applied.This is built in in Definition 2 and it is part of the definition of the VP in thePlug-grammar (cf. Verkuyl 1993:350). So, it turns out to be quite natural torefer to the `x by the sloppy anaphoric it being able to vary over the differentvalues of x. Therefore, it seems to me that if we let it in (33) pertain to `(Jones)in the compositional framework of § 2, Davidson’s problem would have been

30In particular the fact that we could speak about three bathrooms is telling in this re-spect. For event semanticists there is no way of getting more than one bathroom out of theinformation provided by the bathroom.

20

solved in principle, that is, if he had allowed quantification over functions inthe way proposed by Reichenbach 1947.

Recall that Reichenbach had analyzed (39a) as in (39b), which was para-phrased by Davidson as (39c).

a. Amundsen flew to the North Poleb. ∃e[f(a, n)]∗(e)c. ∃e[e consists in the fact that Amundsen flew to the North Pole]d. ∃e[a flying e by Amundsen to the North Pole took place]

(39)

Apart from the question of whether or not (39c) is the appropriate paraphraserather than e.g. (39d), it seems to me that (39b) raises the problem of whyDavidson ignored the fact that the [f(a, n)]∗-part of the formula is clearlytreated as a mathematical function that can be referred to and quantified over,as e.g. in Reichenbach 1947:307. So, the question does not seem to be ofwhether the particular format of (39c) meets certain logical difficulties, butwhether Reichenbach’s main point, namely that events are entities (p.272) car-ries over to functions. In my view, the Reichenbachian position is that mathe-matical functions are entities as well: you may assert their existence and so youmay quantify over them. This is exactly what happens in the plug-frameworkwithout having the (extensional substitution) problem that Davidson assignedto (39c). In short, the real issue involved in Davidson’s treatment and rejectionof (39c) is whether or not we can treat reference to events in terms of referenceto functions.

My conclusion is that Davidson’s problem with finding no singular term forthe anaphoric it in (33b) is an artefact of his wish to restrict his formalismto a first-order language. I also conclude that Reichenbach’s original idea canbe given a better expression in a higher order language and that the strengthof it can be made concrete by assuming that his fact function by and largecorresponds with the VP taken as a Path, i.e. as a lambda-function whichapplied to an argument yields an eventuality.31

3.2 Vendler’s Verbs and Times

In Verkuyl 1989 I raised a series of objections against Vendler 1957, when takenas a linguistic analysis and I showed that Vendlerians when it comes to makegeneralizations about aspectuality never use the four verb classes distinguishedby him.

It may be helpful to put in some history at this point so as to be able toexplain why there is so much confusion about Vendler classes. The main sourceof trouble seems to me to be Dowty 1979. Dowty interpreted Vendler’s notionof verb linguistically, i.e. as a lexical notion. This move was very influential andwe see it back in what can be called the Vendler-Dowty tradition. Part of thistradition is the Edinburgh-approach in which again some lip service is paid tothe compositional approach, e.g. in Moens 1987, Moens and Steedman 1987.

31Not an actualized eventuality in the sense of § 2.6, i.e. we are talking about the formationof S before Tense applies.

21

Moens and Steedman try to maintain the Vendler classes but postulate someoperators in order to get compositional. Consider, for example, (40).

a. Mary walkedb. Mary walked thirty milesc. Mary walked miles

(40)

The leading idea is that walk in (40a) is an Activity verb and that this verbis coerced into an Accomplishment verb in (40b), which is coerced back againinto the activity verb in (40c) and (40a). The only reason for not saying thatwalk in all three sentences of (40) has a constant meaning and that the effect ofcombining thirty miles with this walk is different than the effect of combiningmiles with walk , is the assumption that the Vendler-quadripartition is lexical.This sort of coercion is in fact a rather forced notational variant of regularFregean interpretation with the help of the features [±add to] and [±sqa] onthe basis of Figure 1.

Suppose that we interpret Vendler’s notion of Verb at the VP level, muchin the way Montague used the notion of intransitive verb. Vendler can be takenthen as treating walk and walk a mile as VPs. Historically this interpreta-tion would explain the discussion about homogeneity in the seventies in whichBennett and Partee 1975 played an important role. Bennett and Partee alsofollowed Montague in treating walk and walk home as intransitive verbs, butrather than acting as linguists trying to break down a VP into its constituentparts, they simply took walk home as an unanalyzed predicate of a certainVendlerian type: walk as an Activity and walk home as an accomplishment.So, a couple of things were getting mixed up: the logical Montagovian way oftreating walk and walk home as being of type IV was accepted. Then a shiftwas made to acting as a linguist by taking walk as a lexical aspectual class andthen again the logician returned by declaring that walk home should be treatedas an unanalyzed verb after which it could be put in a different lexical aspectualclass than walk . After which the coercion operators appeared . . ..

Indeed, if Vendler’s notion of aspectual classes is not lexical but ratherphrasal, it would mean that the coercion-operators in (40) may be abolished.There is no need for them. But it would also mean that Vendlerian Verbswhether complex (as in (40b,c)) or not (as in (40a)) should be analyzed alongthe lines of Figure 1. This is another way of saying that, at best, Vendler’sproposal is an unanalyzed compositional approach, not paying attention to theinformation contributed by NPs in- and outside the VP. I rather consider ita throwback with respect to proposals in the early decennia of this centuryin the mainly German literature on aspectuality, where his criteria like thefor an hour -test were well-established. For the present purpose, it suffices toobserve that on a reinterpretation of Vendler’s proposal as a proposal aboutVerb Phrases and Times, the localistic approach in § 2 would be compatiblewith its rearticulation, but it would be more appropriate because it accountsfor the contribution of the arguments of the verb. And this would mean thatthere would not be foour VP-classes.

As a final remark, it should be noted that if Vendler can be reanalyzedalong these lines, Vendler’s notion of event comes closer to the notion of event

22

developed in the framework of § 2 than to the Davidsonian notion. It is nocoincidence that event semanticists cannot express any of the Vendler classesin their representations: they only use his terms pre-theoretically.

3.3 Parsons’ Hold- and Cul-predicates

In this section, I want to focus on Parsons’ predicates Hold and Cul in orderto see how and how much they contribute to a better understanding of thedifferences between durative and terminative sentences. The relevant rules ofhis grammar are given in (41):

R1. If α ∈ Vintr, then α ∈ VPR2. If α ∈ Vtrans & β ∈ Name, then αβ ∈ VPR3. If β ∈ Name & α ∈ VP, then βα ∈ S &

a.

{λeλt[α′(e) ∧ Agent(β′, e) ∧Hold(e, t)], if α State, Process VPλeλt[α′(e) ∧ Agent(β′, e) ∧ Cul(e, t)] if α Event VP

or:

b.

{λeλt[α′(e) ∧ Agent(β′, e) ∧Hold(e, t)], if α State VPλeλt[α′(e) ∧ Agent(β′, e) ∧ Cul(e, t)] if α Process, Event VP

. . . . . .R10. If α ∈ Adv & β ∈ VP, then βα = λe[β′(e) ∧ α′(e)]

(41)

Both Hold and Cul in R3 are treated as two-place predicates between eventsand times. R3 is split up into two different options. The a-option is chosenbefore 1984, the b-option after 1984. There is an intriguing replication of thetransition from R3a to R3b in Parson’s work. The transition from R3a to R3btook place around 1984 and so one expects R3b to be established from thattime on. But in Parsons 1990 the rules R3a reappear at the beginning of thebook and it is only at the close of the book that they are rejected and that R3bis reintroduced. This makes it necessary to have a look at the two options.32

3.3.1 Holda and Cula

First consider the definition of the two predicates:

Definition 6 (Parsons 1989: 220; but also Parsons 1990:25) Holda: “meansthat an eventuality e holds at time t, which means that e is either a state ande’s object is in state e or e is an event which is in development at t”. Cula:means “that e is an event that culminates at time t”.

Applied to the sentences in (42) this yields the primed representations.

a. Mary dislikes Freda′. ∃e[Disliking(e) ∧ Subj(e,m) ∧Obj(e, f) ∧Hold(e, t)]b. Mary was crossing the streetb′. ∃e[Crossing(e) ∧ Subj(e,m) ∧Obj(e, the street) ∧Hold(e, t)]c. Mary crossed the streetc′. ∃e[Crossing(e) ∧ Subj(e,m) ∧Obj(e, the street) ∧ Cul(e, t)]

(42)

32Preda is Pred from R3a, Predb from R3b, etc. Parsons sometimes writes simply Hold(e)or Cul(e) but this is only for convenience.

23

Hold and Cul can only be seen as being added to the stock of Davidsonian pred-icates in order to distinguish between durative and terminative aspectuality.Parsons absorbed the Moens-idea of attributing to an event a development-partand a culmination-part. Thus, Holda in (42b′) covers the development part ofthe crossing and expresses that the event did not (necessarily) culminate. Culaexpresses the sense of culmination that Parsons attributes to (42c).33

In the absence of any treatment of quantification in Parsons’ work, we mustassume that the difference between (1a) and (2a) would be accounted for as in(43).

a. ∃e[Eating(e) ∧ Subj(e, j) ∧Obj(e, 3 sandwiches) ∧ Cul(e, t)]b. ∃e[Eating(e) ∧ Subj(e, j) ∧Obj(e, sandwiches) ∧Hold(e, t)]

(43)

The question arises: does Parsons present here a theory about aspectuality?If so, the theory does not explain anything: Hold/Cula are at best descriptivepredicates. That is, some sort of meta-rules may be formulated:

if [α . . . Holda(e)], then α durativeif [α . . . Cula(e)], then α terminative

However, this does not work properly. Firstly, Lascarides 1988 pointed out dev-astatingly that there is no logical relationship between (43a) and (43b). Thatis, there is no Hold-part in Cula. The two predicates are totally unrelated, sothere is no way to provide for the one-way inference relation between sentenceswith a Hold-predicate and a Cul-predicate, i.e. between (43b) and (43a). Sec-ondly, from the purely aspectual point of view it should be obvious that theaspectual difference between (43a) and (43b) must be found in the differencebetween three sandwiches and sandwiches and it is very hard to see what Holdand Cul could possibly contribute to explaining that difference, the more so be-cause in (42) it seems to account both for the difference between the Progressiveform and the simple Past, and for the difference between stative and certainnonstative predications. At best, the addition of the Hold- and Cul-predicateis a way of saying that the two sentences in (43) differ, but this is what we al-ready knew by looking at them. For Parsons there is no way of expressing thatHold sometimes has to look at the subject NP—Nobody in (3a)—, sometimesto the object NP —three sandwiches, sandwiches—and sometimes to the verb(dislike in (4a)) in order to establish durativity. In short, Hold and Cul do notdo anything to connect themselves to the spots where the relevant aspectual

33Moens’ distinction between development-part and culmination-part looses the advantageof the distinction on which it was partly based, namely the distinction between [+add to]and [+sqa]. The [+add to] is indeed a development part but the [+sqa]-information shouldnot be taken as culmination, even though, of course, a sense of culmination follows fromthe presence of [+sqa]. In the Plug-framework, the [+sqa]-information is systematicallyrelated to the development part throughout the application of the function s as part of thefunction `. All sorts of misunderstanding, most recently in Pustejovsky 1995 arise if onefollows Moens’ track, such as the need to characterize the culmination point by a propositionexpressing a state. One of the problems with this is that the culmination-proposition is notat all determined by information given by the sentence itself. In fact, the split between Holdand Cul brings back Von Wright’s problem of a truth value gap between two states involvedin a transition.

24

information is. Not the slightest hint about which direction should be takenhas been given in any of Parsons’ articles nor in Parsons 1990.

3.3.2 Holdb and Culb

The definitions of Holda and Cula in Def. 6 reappear in Part 1 of Parsons 1990.So, one should expect a reaction to Lascarides 1988. In spite of the change inposition from Hold/Cula to Hold/Culb in Parsons 1985, Parsons 1990: 23-25simply gives the Hold/Cula definitions and carries them with him to far beyondp.142. Chapter 9 of Parsons 1990 is a slightly revised version of Parsons 1989.So, at the very last pages the quite drastic change from Hold/Cula to Hold/Culbtakes place (at p. 184/185 of the book!). The differences between Hold/Culaand Hold/Culb are minimized: “The effect of this proposal is to broaden thescope of the earlier analysis without affecting its formulation. We just treatprocess verbs as a special kind of event verb and apply the theory as before”(p. 184). From then on we have:

a. Mary rana′. ∃e[Running(e) ∧ Subj(e,m) ∧ Cul(e, t)]b. Mary ran to the storeb′. ∃e[Running(e) ∧ Subj(e,m) ∧ To(e, the store) ∧ Cul(e, t)]c. Mary was running to the storec′. ∃e[Running(e) ∧ Subj(e,m) ∧ To(e, the store) ∧Hold(e, t)]

(44)

And we may add:

a. Mary dislikes Freda′. ∃e[Disliking(s) ∧ Subj(s,m) ∧Obj(s, f) ∧Hold(s, t)]b. Mary was crossing the streetb′. ∃e[Crossing(e) ∧ Subj(e,m) ∧Obj(e, the street) ∧Hold(e, t)]c. Mary crossed the streetc′. ∃e[Crossing(e) ∧ Subj(e,m) ∧Obj(e, the street) ∧ Cul(e, t)]

(45)

It is clear that some additional notions are required. It is also clear that thedistinction between Hold and Cul now is just the lexical distinction betweenstative and non-stative verbs, or rather the way the events corresponding withthem actualize in time. This raises a problem in (44c′): Hold cannot hold forevents any longer. This is solved by:

a. Mary be running to the storea′. ∃e[Running(e) ∧ Subj(e,m) ∧ To(e, the store) ∧Hold(In-prog(e), t)]

(46)

Operator In-prog takes an e and makes an s out of it: Parsons 1990:234 de-scribes this operator as follows: “whenever an event e is in progress, there is acorresponding state of affairs, ‘that e is in progress’. This is a state that holdswhile e is in progress (and at no other time). Call this the ‘in-progress state’ ofthe event e, or ‘e’s IP-state’ ”.

In my view the change of position can only mean that for Parsons Holdb

and Culb are no longer part of an aspectual theory: they are now part ofthe Tense system in the sense that they can be considered as accounting for

25

the connection between States and Events with real time. They have nothing(structurally) to do in making a sentence (in-)compatible with for an hour andin an hour. Parsons distinguishes between states and events and for each of thetwo ontological categories he needs a term to express two types of actualizationin real time: Cul for non-stative predications; Hold for stative ones. Then anaspectual notion comes in: IP-state. The reasoning must be the following: Culbrings us in the domain of non-stative semantic objects—events—and thesehave a development part, a culmination part and a consequent state part, inParsons 1990:235 called Resultant State. In-prog always takes the developmentpart and yields a state.34

3.3.3 Conclusion

From the aspectual point of view there is a lot more to say against Parsons’analyses but I will not pursue my discussion of his Hold/Cul-analysis here anyfurther, because most of what I have to say follows from the framework pre-sented in § 2. However, there is one general point that I would like to raiseagainst the neo-Davidsonian event semantic format which grew out of (5):

p1 ∧ p2 ∧ . . . ∧ pn(47)

The format in (47) allows for inferences based on the scheme in (48).

a. p ∧ q ⇒ pb. p ∧ q ⇒ q

(48)

Parsons 1990:13–15 made some restrictions on this scheme for it to work prop-erly but these restrictions do not bear on the point I want to make here. Thatpoint is that p and q in (48) should be independent: q may not pertain infor-mation that is necessary for p to be asserted. This condition is not fulfilledin representations like (44), (45), (46), etc. This can be made visible by re-turning to problems already raised by Davidson himself when he pointed outthat from sentences like He sank the Bismarck you may not infer to He sank .In Dowty 1989 this problem returned in examples like John devoured his lunchfrom which one cannot infer John devoured . Likewise one may not derive Hewatered from He watered the plants. Some event semanticists seem very happyin taking p1 in (47) as ‘there is some sinking’ or ‘there is some devouring’ and‘there is some watering’, and they are inclined to solve this problem in termsof ambiguity or polysemy. However, polysemy makes it impossible to cut thedependency ties between p1 and the other members of the conjunction. Andif ambiguity is the deus ex machina for dealing with polysemy, then the sim-ple answer to that move is to point out that in that case one needs a specialgrammar for the exclusion of unwanted inferences.

34This is a problem for Parsons: there turns out to be no way to distinguish processes asthey can only be presented as states. One odd consequence: the resultant state R(e) andprogress state are treated alike: the first argument of Hold in Hold(In-prog(e),t) is a stateand the same applies to Hold(R(e),t). However, the R-state is not aspectual at all: it is usedto define the Present Perfect (p. 234ff.). A lot remains unclear in the Hold/Culb-part ofParsons 1990 because it is presented very informally.

26

3.4 Aspectual Postulates

3.4.1 Introduction

Krifka looks like an event semanticist, but if he is, he is quite different fromneo-Davidsonians like Parsons and Higginbotham. I will argue that Krifka infact uses two different notions of event. His first notion concurs with my notionof index and it is visible in his event first-order semantic representation basedon mereology. His second notion is a composite one and it is expressed byhigher order postulates. I have criticized Krifka in some detail in Verkuyl 1993and will not repeat most of the points raised there. A new set of problems forKrifka is discussed in much detail in Naumann 1995.35 I will first focus on whatseparates Krifka’s framework from the plug+-framework and then discuss someof the problems that arise in the present context.

3.4.2 Lattices

Following Link 1983, Krifka 1989a;1989b attributes to the domain of discourseD the structure 〈D,t〉 of a join semilattice, i.e. a lattice with only the join-operator t as binary operation on its elements. D is partitioned into a setO of objects and a set of events E and so there are two join-operations inthese subdomains: 〈O,E,tO,tE〉. He also assumes the (mereological) part ofrelation v. Thus, the resulting model structure is: 〈O,E,tO,tE ,vO,vE〉.This move has clear ontological consequences: it helps to retain the type logicinherent to standard format (5) in cases like (49).

a. Mary diedb. John and Mary died

(49)

Standardly one has Die(et)(me) for (49a), which is true iff I(me) ∈ I(Die(et)).This analysis leads to problems for (49b) because if John and Mary are treatedas denoting a set, a type clash results. There are two ways out to save thegeneralization over the two sentences in (49): (a) to treat Mary as a singletonand take NPs as being of type ((et)t), which is the set-theoretical solution;(b) to take John and Mary as an individual of type e. This latter is obtainedby allowing for the extension of the predicate logic with an operator ⊕, whichis interpreted as the join t of two individuals. Thus, on a lattice-theoreticalapproach (49b) can be represented as Die(John ⊕ Mary), where [[John⊕Mary]]is considered an individual of type e.36

[[Die(John⊕Mary)]] = 1 iff (I(j) t I(m)) ∈ I(Die)

Lattice theory can be used for the account of the relation between mass andcount although it should be observed that there are other ways to relate them,one of them being the extension of set theory to ensemble theory in Bunt 1985.

35This impressive work gives very detailed and thorough analyses of three aspectual theories(Dowty, Krifka, Verkuyl) and results in a dynamic theory of aspectuality combining ingredientsof the three theories. In the present section I will sometimes draw on him.

36This strategy also in e.g. Jackendoff 1990 and formal work based on this, e.g. White 1994.More in general, one can see the flattening of ontology in computational circles. An interestingmathematical example of this strategy is Touretzky 1986.

27

In Krifka’s lattice-theoretical approach heavy emphasis is put on the parallelstructure one may attribute to the verbal (event) domain and the nominal(thing) domain: in both domains the distinction between mass and count showsup and therefore Krifka partitions the model structure so as to contain eventsand individuals and assumes that the operations on these entities occur in thetwo subdomains DO and DE . Krifka capitalizes on the parallelism between thetwo domains and thus he gives his aspectual theory a rich ontological flavour.Or put differently, his system underscores the naive physics of the domain ofdiscourse. We have to get interested in how Mary’s dying is built up and how thethree sandwiches are eaten in sentences like (13a). Something of the structureon the event-side must be found in the structure on the nominal side.

3.4.3 A derivation and some postulates

Let us now first consider the derivation of (13a) Judith ate three sandwiches inthe detail necessary for a comparison:

a. eat λe[Eat(e) ∧Ag(e, x) ∧ Pat(e, y)]b. three λRλPλe∃y[P(e) ∧ Pat(e, y) ∧ R(y, 3)]c. sandwich λnλxSandwich(x, n)d. three sandwiches λPλe∃y[P(e) ∧ Pat(e, y) ∧ Sandwich(y, 3)]e. eat 3 sandwiches λe∃y[Eat(y) ∧Ag(e, x) ∧ Pat(e, y) ∧ Sandwich(y, 3)]f. Judith λPλe∃x[P(e) ∧Ag(e, x) ∧ x = j]g. (13a) λe∃x∃y[Eat(e) ∧ Pat(e, y) ∧ Sandwich(y, 3)

∧Ag(e, x) ∧ x = j]

(50)

This derivation shows that Krifka chooses the neo-Davidsonian format by iso-lating in (50a) the verb eat from its thematic roles: there is an eating and thereshould be an Agent x and a Patient y. The NP three sandwiches is formed soas to take the verb eat as its argument and the numeral expresses a relation Rbetween the set of sandwiches and a number. The final stage of the derivationsays that there is an Agent Judith and that there is an complex individual yconsisting of three sandwiches.37

In one important respect (50) deviates from Parsons’ derivations: the lackof a Cul-predicate. The question is: what makes (1a) and (13a) terminativefor Krifka? And what makes (2a) durative for him? Krifka’s last line for (2a)Judith ate sandwiches is (51g):

. . .g. λe∃x∃y[Eat(e) ∧ Pat(e, y) ∧ ∃n[Sandwich(y, n) ∧Ag(e, x) ∧ x = j]]

(51)

in which there is no Hold-predicate. So, where does the aspectual informationcome from. The answer to this question is that Krifka does not put this infor-mation in his representations but in “meta-representations”. That is, he makes

37As pointed out in Verkuyl 1993 the derivation does not work: there is a doubling of theAg(e, x), visible in applying (50f) to (50e) in which the free variable x of (50a) must be boundby the ∃x in (50g), which is, of course, not allowed. This problem cannot be solved by puttinglambda’s in front of (50a), because there is no way to get rid of them. Krifka acknowledgedthis problem but his solution (p.c.) does not convince (cf. Verkuyl 1993:261).

28

use of a set of postulates by which one can determine whether a certain logicalform satisfies the proper conditions for terminativity or for durativity. Thepostulates are cast in a higher order language and in this way the formulas in(50) may retain their first order nature.38 Sentences like (1a) and (2a) receivetheir aspectual labels by matching their logical forms with the scheme in (52).

ϕ ; λe∃y[α(e) ∧Θ(e, y) ∧ δ(y)](52)

Immediately one must take one step back: Krifka does not account for theaspectual composition above the level of ϕ, i.e. above the level of the VP. So, noaccount is given of what is covered by the π-function in the plug-framework.With this reservation in mind, we continue to observe that α stands for theverb, that Θ is the Patient-role and that δ stands for the internal argument NP.Given (52), Krifka says that sentences like (2a) are durative just in case α andδ are cumulative and Θ is summative. Sentence (1a) is terminative because α iscumulative, δ has quantized reference and Θ satisfies the constraints Uniquenessof Object and Mapping to Object.

Before discussing the merits of this approach, let us have a more detailedlook into the postulates:

Definition 7 Cumulativity (CUM):

∀P(CUM(P) ⇔ ∀x∀y[P(x) ∧ P(y) → P(x t y)])

Definition 8 Summativity (SUM):

∀R(SUM(R) ⇔ ∀e∀e′∀z∀z′[R(e, z) ∧ R(e′, z′) → R(e tE e′, z tO z′)])

Definition 9 Quantized reference (QUA):

∀P(QUAO(P) ⇔ ∀z∀z′[P(z) ∧ P(z′) → ¬z′ v z])

Definition 10 Uniqueness of Objects (UoO):

∀R(UNIO(R) ⇔ ∀e∀z∀z′[R(e, z) ∧ R(e, z′) → z = z′])

Definition 11 Mapping to Objects (MtO):

∀R(MAPO(R) ⇔ ∀e∀e′∀z[R(e, z) ∧ e′ vE e → ∃z′[z′ vO z ∧ R(e′, z′)]])

Definition 12 Mapping to Events (MtE):

∀R(MAPE(R) ⇔ ∀e∀e′∀z[R(e, z) ∧ x′ vO x → ∃e′[e′ vE e ∧ R(e′, z′)]])

38In general, it seems to me that an important disadvantage of Krifka’s approach is thatthe equilibrium between what is expressed by the representations and what is determined bythe postulates is out of balance. And unnecessarily so. There are many more postulates thangiven here. So, Krifka seems to operate on the side of the ontologists: describing the worldrather than the language. The question is again: why are people so afraid of using higherorder representations, especially when it can be shown that they work well empirically?

29

Cumulativity says for α that if one has [[Eating]](e) and [[Eating]](e′), one mayjoin them into [[Eating]](et e′) and for δ it says that if you have [[Nice]](j) and[[Nice]](m), you also have [[Nice]](j t m). Summativity accounts for the Θ-part:it joins two thematic relations.

As far as (2a) is concerned, a couple of problems are showing up. Firstof all, Naumann observed that the cumulativity requirement on α is empty.There is no verb that is not cumulative. This means that any hope for makinga distinction between stative and nonstative verbs in terms of the notion ofcumulativity must be given up. It also means that all information about therole of the verb is relegated to the Θ(e, y)-part. I would like to add that thisholds also for the information expressed by δ(y) because in derivations like (50)and (51) there is no δ at all: the information about the Patient is only expressedby Pat(e, y). R(y, 3) or R(y, n) do not have the form δ(y). If δ(y) is in fact anabbreviatory device for the “sandwich-part” of R(y, 3), i.e. its R(y, . . .)-part,then a new problem arises: the δ(y) is the same in R(y, 3) and R(y, n), so ifδ(y) is cumulative in the latter it should also be in the former. And this meansthat the distinction between (2a) and (1a) cannot be made on the basis of δ(y).Note that the “sandwich-part” of R(y, 3) and R(y, n) is already expressed by theSUM-postulate in Def. (8) applying to Pat(e, y): its cumulativity is accountedfor by y t y′. So, all things in (52) really boil down to Pat(e, y) and here Ireturn to Naumann who raises the same objection to Θ(e, y) as to α(y): allthematic relations are cumulative. This means that the only place to look forthe difference between the durative (2a) and the terminative (1a) is the secondargument of R(y, 3) and R(y, n). But here we meet an objection raised inVerkuyl 1993: ∃n.Sandwich(y, n) assumes that there is a specific unidentifiednumber. But the difference between an unknown number and the number 1cannot be the basis for a difference between unbounded and bounded. If wesay John killed some men we know that there is a finite number of men killedby John, but we cannot identify the number n of men. The sentence itself isterminative. Basically Krifka treats bare plurals in the same way as quantizedNPs: in both cases there is a specific number n. Recall that in plug+ thesolution was to assume that in bare plurals no cardinality information is given,so that it cannot be determined. This seems to me a more correct approach.

There is another problem on top of the ones treated above. Krifka says that(1a) is terminative just in case δ has quantized reference and Pat(e, y) meetsthe conditions UoO and MtO.39 Note that Def. 10 and 11 do not follow fromany linguistic material inside the sentence: they just stipulate that the condi-tions hold for terminative cases. In my view, this keeps Krifka’s theory fromexplaining the nature of terminativity. Naumann shows that the conditionshold good for sentences like (1a) containing verbs like eat , but that e.g. UoOdoes not work properly in sentences like (53).

John read two books(53)

39QUA defines roughly our [+sqa], but it should be observed that part of the diffulties inKrifka’s system arise from the fact that his definitions are focussed on the N rather than onthe NP. So, the role of the determiner in the determination of whether an NP is [+sqa] or[–sqa] is ignored. This point has also been made by Naumann.

30

UoO requires that if some reads a book, the event e has this book as the onlyobject to which it relates by the Pat-role. But one may read two books simulta-neously, and if humans cannot do that, computers can. One may also describe(53) as saying that there are two events in each of one a book was read. But thesame situation can also be described in terms of two events meeting summativ-ity. According to Naumann this makes clear that whether or not the e relatesto one or to two (complex) individuals depends on underlying individuationcriteria for events and these determine whether or not UoO holds good. Butthis amounts to saying that UoO looses its explanatory value, because as soonas one try to solve this problem by pointing out that UoO may vary dependenton the verbs involved, one is doing away with the constancy in the Θ-relation.40

As to MtE and MtO Naumann also raises the problem of sentences like (54).

a. John drove his car from Boston to Chicagob. John peeled the apple

(54)

In (54a) halfway the Path is not halfway the car and in (54b) it is clear thatonly a part of the apple is affected. The same applies to sentences such asMary built a house: the building of a house involves a lot of planning, erectingscaffolds, etc. and these are not part of the mapping between the house andthe event.

More in general it seems to me that MtO and MtE are not appropriate fordealing with sentences like (13) and (14). For their proper analysis it is essentialthat we capture the underdeterminedness of which of the possible combinatorialconfigurations is actualized. Note that for each of the actualizations the inter-play between s and ` warrants that the structure of I and the internal argumentdenotation are in a mapping-relation. But this relation is not a naive physicalone but it can be formulated at the set-theoretical level at which quantificationis dealt with.

3.5 Conclusion

These objections against Krifka’s lattice-theoretical approach in an event se-mantic framework strengthen my wish (bias, some will say) to retain the posi-tion in which the cardinality information and, for mass NPs, information aboutsome appropriate measure along the lines of Bunt 1985 are taken as primordialfor the aspectuality of a sentence. As said before, this is due to the fact thatin the plug-framework quantificational information is considered an essentialingredient. Focussing on cardinality means that one abstracts from (naive)physics. I must admit that this results in some unnaturalness in the treatmentof (1a) because the partitioning of the NP-denotation [[a sandwich]] yields justa singleton and so there is little room left for what seems at first sight so niceand natural in Krifka’s treatment of this sentence in terms of MtO and MtE.In my treatment the eating is a one swoop-event so to say and one may wishthat there should be more to it. Actually, I am not yet sure whether plug+

should be “mereologized” in this direction, because from the point of view of40Note in passing that the individuation problem for Krifka raised by Naumann is also

raised by sentences like (31).

31

aspectual composition the [+sqa]-information interpreted model-theoreticallyas the presence of cardinality information (or some measure function informa-tion), as discussed e.g. in § 2.1 could be judged sufficient. But in my view,Krifka’s lattice-theoretical apporach has to solve a lot of problems before it canexplain the facts explained by the plug+-framework. In this sense, I prefer thestrategy of a step-by-step extension of set theory to ensemble theory as a basisfor capturing the relation between Mass and Count.

This raises the question of how Krifka’s e’s relate to the indices in theplug+-framework. If he would follow Naumann’s advice to simply recast MtOand MtE in terms of the monotonicity property, then it would follow that theconditions come closer to what is inherent to the interpretation of nonstativeverbs in terms of the s-function. It would be another way of saying that Krifka’se’s may in fact be taken as indices. Evidently, that would make Krifka’s gram-mar part of the plug+-framework as far as the verbal information is concerned.

This brings us finally to the point raised earlier, namely that Krifka actuallyuses two notions of event. From the fact that α(e) in (52) does not differentiateaspectually between verbs, if follows more or less straightforwardly that theaspectual differences are to be located at the level of ϕ itself. That is, at thelevel of VP. It is at that level that Krifka may distinguish between a terminativeevent and a durative event But this means that Krifka looks at ϕ-events asdividuals. This is actually not what he says. But it seems the only plausibleway to interpret him. It should be added that ϕ does not capture the role ofthe subject and that (52 does not say anything about the aspectual relationbetween VP and S.

In Verkuyl 1993 I have argued that aspectuality can only be understood ifwe do not assume that events exist as primitives. However, I think that theyexist as soon as we allow them to exist as the result of some Fregean procedure.I used the analogy of macro- and micro-economy to make this point: in myview the notion of event is a typical macro-construct. Figure 8 illustrates thisposition:

S⇐= e

NPext VP

V NPint

��

�

@@@

��

�

@@

@

Figure 8: Asp as an eventuality node

The notion of existential closure seems Expressions like ∃e[. . . e . . .]? can beunderstood as being allowed or disallowed on the basis of certain informationbeing present or absent. The best way to think about it is in terms of Figure 1,which reappears here in an extended form as Figure 9.What we are looking for is evidence that the Asp-node is there as a sign that

32

S′

@@

��� . . .

@@

��Asp S

NPext VP

V NPint

��

�

@@@

��

�

@@

@

Figure 9: Asp as an eventuality node

the aspectual information expressed by S is of a certain event type. In otherwords, one is looking for a situation in which Asp is lexicalized as αterm if andonly if S is terminative and as αdur otherwise. In the former case one could saythat Asp is a (terminative) event-place holder.

Now, firm evidence for the existence of an αterm alternating with an αdur

comes from Scottish Gaelic. In Ramchand 1993;1996 it is shown that the mor-pheme air is αterm and that the morpheme ag is αdur. That is, air can onlybe selected if the proper conditions for terminativity are fulfilled at the level ofS, otherwise ag must be selected. I am aware that from the syntactic point ofview in the relation between Asp and the S could or would be stated differently,but in terms of bottom to top interpretation it is necessary to state it the wayI did. 41

This leads to the conclusion that certain event semantic assumptions, suchas the existence of an AspPhrase projection may receive support from the sideof plug+, but from that side it is adduced that the notion of event is no longerprimitive.

As said before, I will investigate Event Semantics critically from the pointof view of the contribution it makes to the study of inner aspectuality. Thisdoes not imply that I am against Event Semantics, but why should linguisticsnot have the distinction common to economics: the difference between macro-economics and micro-economics. In that sense, inner aspectuality is a form ofmicro-semantics and event semantics should be seen as macro-semantics. Thiswould evade a lot of unnecessary discussion about things that are otherwisevery clear.

41In Verkuyl 1996 I have given a more elaborate analysis of Ramchand’s analysis in thecontext of the question in how far the notions of definiteness and strength interfere with the[+sqa]-notion.

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References

Barwise, Jon, and Robin Cooper. 1981. Generalized Quantifiers and NaturalLanguage. Linguistics and Philosophy 4:159–219.

Bennett, Michael, and Barbara Partee. 1975. Some Extensions of a MontagueFragment of English. Indiana University Linguistics Club. Bloomington.

Bunt, Harry C. 1985. Mass Terms and Model-theoretic Semantics. csil, Vol. 42.Cambridge: cup. Originally: The Formal semantics of Mass Terms Diss.Amsterdam 1981.

Davidson, Donald. 1980. The Logical Form of Action Sentences. In Essayson Actions and Events. 105–148. Oxford: Clarendon Press, 3rd edition.[1967].

Davidson, Donald. 1985. Essays on Actions and Events. Oxford: ClarendonPress. 3rd edition.

Dowty, David. 1979. Word Meaning and Montague Grammar. The Semanticsof Verbs and Times in Generative Semantics and in Montague’s PTQ. Syn-these Language Library, Vol. 7. Dordrecht: D. Reidel Publishing Company.

Dowty, David. 1989. On the Semantic Content of the Notion of Thematic Role.In Properties, Types and Meaning, ed. Gennaro Chierchia, Barbara H. Par-tee, and Raymond Turner. 69–130. Dordrecht: Kluwer Academic Publish-ers.

Gillon, Brendan S. 1987. The Readings of Plural Noun Phrases in English.Linguistics and Philosophy 10:199–219.

Gruber, Jeffrey S. 1976. Lexical Structures in Syntax and Semantics. North-Holland Linguistic Series, Vol. 25. Amsterdam, etc.: North-Holland. PartI. Studies in Lexical Relations (1964); Part II. Functions of the Lexicon inFormal Descriptive Grammars (1967).

Halvorsen, Per-Kristian, and William A. Ladusaw. 1979. Montague’s ‘UniversalGrammar’. An Introduction for the Linguist. Linguistics and Philosophy3:185–224.

Hinrichs, Erhard. 1981. Temporale Anaphora im Englischen. Master’s thesis,University of Tubingen, Tubingen.

Hinrichs, Erhard. 1986a. A Compositional Semantics for Aktionsarten andNP Reference in ENglish. Doctoral dissertation, Ohio State University,Columbus, Ohio.

Hinrichs, Erhard. 1986b. Temporal Anaphora in Discourses of English. Lin-guistics and Philosophy 9:63–82.

Jackendoff, Ray S. 1976. Toward an Explanatory Semantic Interpretation inGenerative Grammar. Linguistic Inquiry 7:89–150.

34

Jackendoff, Ray S. 1983. Semantics and Cognition. Current Studies in Linguis-tics Series, Vol. 8. Cambridge (Mass.): The M.I.T. Press.

Jackendoff, Ray S. 1990. Semantic Structures. Current Studies in LinguisticsSeries, Vol. 18. Cambridge (Mass.): The M.I.T. Press.

Krifka, Manfred. 1989a. Nominal Reference, Temporal Constitution and Quan-tification in Event Semantics. In Semantics and Contextual Expression, ed.Renate Bartsch, Johan van Benthem, and Peter van Emde Boas. Groningen-Amsterdam Studies in Semantics, Vol. 11, 75–115. Dordrecht: Foris Publi-cations.

Krifka, Manfred. 1989b. Nominalreferenze und Zeitkonstitution. Studien zurTheoretischen Linguistik, Vol. 10. Muenchen: Wilhelm Fink Verlag.

Landman, Fred. 1989. Groups I & II. Linguistics and Philosophy 12:559–605and 723–744.

Landman, Fred. 1992. The Progressive. Natural Language Semantics 1:1–32.

Lascarides, Alex. 1988. A Formal Semantics of the Progressive. Doctoral dis-sertation, University of Edinburgh, Edinburgh.

Lasersohn, Peter S. 1989. On the Readings of Plural Noun Phrases in English.Linguistic Inquiry 20:130–134.

Lasersohn, Peter S. 1995. Plurality, Conjunction and Events. Studies in Lin-guistics and Philosophy, Vol. 55. Dordrecht: Kluwer Academic Publishers.

Link, Godehard. 1983. The Logical Analysis of Plurals and Mass Terms.In Meaning, Use and Interpretation of Language, ed. Rainer Bauerle,Christoph Schwarze, and Arnim von Stechow. 302–323. Berlin and NewYork: De Gruyter.

Link, Godehard. 1987. Generalized Quantifiers and Plurals. In GeneralizedQuantifiers and Logical Approaches, ed. Peter Gardenfors. 151–180. Studiesin Linguistics and Philosophy. Dordrecht: D. Reidel Publishing Company.

Link, Godehard. 1991. Plural. In Semantics. An International Handbook ofContemporary Research, ed. Dieter Wunderlich and Arnim von Stechow.Berlin and New York: De Gruyter. Better known as: Link 1984.

Lønning, Jan Tore. 1987. Collective Readings of Definite and Indefinite NounPhrases. In Generalized Quantifiers, ed. Peter G’ardenfors. 203–235. Dor-drecht: D. Reidel Publishing Company.

May, Robert. 1977. The Grammar of Quantification. Doctoral dissertation,MIT, Cambridge, Mass.

Moens, Marc. 1987. Tense, Aspect and Temporal Reference. Doctoral disserta-tion, University of Edinburgh, Edinburgh.

35

Moens, Marc, and Marc Steedman. 1987. Temporal Ontology in Natural Lan-guage. In Proceedings of the 25th Annual Meeting of the Association forComputational Linguistics, 1–7. Stanford University, July 1987. ACL.

Montague, Richard. 1974a. English as a Formal Language. In Formal Philos-ophy. Selected Papers of Richard Montague, ed. Richmond H. Thomason.188–221. New Haven: Yale University Press. [1970].

Montague, Richard. 1974b. The Proper Treatment of Quantification in Ordi-nary English. In Formal Philosophy. Selected Papers of Richard Montague,ed. Richmond H. Thomason. 247–270. New Haven: Yale University Press.[1973].

Naumann, Ralf. 1995. Aspectual Composition and Dynamic Logic. Doctoraldissertation, Heinrich Heine Universitat, Dusseldorf. Habilitationsschrift.

Parsons, Terence. 1980. Modifiers and Quantifiers in Natural Language. Cana-dian Journal of Philosophy Supplementary Volume VI:29–60.

Parsons, Terence. 1985. Underlying Events in the Logical Analysis of English.In Actions and Events. Perspectives on the Philosophy of Donald Davidson,ed. Ernest LePore and Brian P. McLaughlin. Chap. 16, 235–267. Oxford:Basil Blackwell.

Parsons, Terence. 1989. The progressive in English: Events, states and pro-cesses. Linguistics and Philosophy 12:213–241.

Parsons, Terence. 1990. Events in the Semantics of English. A Study of Sub-atomic Semantics. Current Studies in Linguistics Series, Vol. 19. Cam-bridge, MA: The M.I.T. Press.

Partee, Barbara H., Alice ter Meulen, and Robert E. Wall. 1990. Mathemati-cal Methods in Linguistics. Studies in Linguistics and Philosophy, Vol. 30.Dordrecht: Kluwer Academic Publishers.

Pustejovsky, James. 1995. The Generative Lexicon. Current Studies in Lin-guistics Series, Vol. 2x. Cambridge, MA: The M.I.T. Press.

Ramchand, Gillian Catriona. 1993. Aspect and Argument Structure in ModernScottish Gaelic. Doctoral dissertation, Stanford, Stanford.

Ramchand, Gillian Catriona. 1996. Aspect and Predication: The Semantics ofArgument Structure. Oxford: Oxford University Press. to appear edition.

Reichenbach, Hans. 1947. Elements of Symbolic Logic. New York: The Macmil-lan Company. First free press paperback edition 1966 edition.

Scha, Remko. 1981. Distributive, Collective and Cumulative Quantification. InFormal Methods in the Study of Language, ed. Jeroen A. G. Groenendijk,Theo M.V. Janssen, and Martin J.B. Stokhof. 483–512. Amsterdam: Math-ematical Center Tracts 136.

36

Schwarzschild, Roger. 1992. Together as a Non-Distributivity Marker. In Pro-ceedings of the 8th Amsterdam Colloquium, ed. Paul Dekker and MartinStokhof.

Smith, Carlota S. 1991. The parameter of Aspect. Studies in Linguistics andPhilosophy, Vol. 43. Dordrecht: Kluwer Academic Publisher.

Tenny, Carol L. 1994. Aspectual Roles and the Syntax-Semantics Interface.Studies in Linguistics and Philosophy, Vol. 52. Dordrecht: D. Reidel Pub-lishing Company.

Touretzky, D. 1986. The Mathematics of Inheritance Systems. Palo Alto, CA:Morgan Kaufmann.

Van der Does, Jaap M. 1992. Applied Quantifier Logics. Collectives, NakedInfinitives. Doctoral dissertation, University of Amsterdam, Amsterdam.

Van der Does, Jaap M., and Henk J. Verkuyl. 1995. Quantification and Pred-ication. In Ambiguity, ed. Kees van Deemter and Stanley Peters. xx–xx.Stanford, California: CSLI Publications.

Vendler, Zeno. 1957. Verbs and Times. The Philosophical Review 66:143–160.

Verkuyl, Henk J. 1972. On the Compositional Nature of the Aspects. FLSS,Vol. 15. Dordrecht: D. Reidel Publishing Company.

Verkuyl, Henk J. 1987. Nondurative Closure of Events. In Studies in Dis-course Representation Theory and the Theory of Generalized Quantifiers,ed. Jeroen Groenendijk, Dick de Jongh, and Martin Stokhof. Groningen-Amsterdam Studies in Semantics, Vol. 8, 87–114. Dordrecht: Foris Publi-cations.

Verkuyl, Henk J. 1988. Aspectual Asymmetry and Quantification. In Tempo-ralsemantik. Beitrage zur Linguistik der Zeitreferenz, ed. Veronika Ehrichand Heinz Vater. 220–259. Tuebingen: Niemeyer Verlag.

Verkuyl, Henk J. 1989. Aspectual Classes and Aspectual Composition. Lin-guistics and Philosophy 12:39–94.

Verkuyl, Henk J. 1993. A Theory of Aspectuality. The Interaction between Tem-poral and Atemporal Structure. Cambridge Studies in Linguistics, Vol. 64.Cambridge: Cambridge University Press.

Verkuyl, Henk J. 1994. Distributivity and Collectivity: a Couple at Odds. InDynamics, Polarity, and Quantification, ed. Makoto Kanazawa and Chris Pinon. 49–80. Stanford, California: CSLI Publications.

Verkuyl, Henk J. 1995. Aspectualizers and Event Structure. In Time, Spaceand Movement. Meaning and Knowledge in the Sensible World., ed. PascalAmsili, Mario Borillo, and Laure Vieu. 31–48. Toulouse/Chateau de Bonas,23-27 June 1995.

37

Verkuyl, Henk J. 1996. Syntactic and Semantic Aspects of Aspectuality. InProceedings of the First Conference on Formal Approaches to South SlavicLanguages Plovdiv October 1995, 1–26. Trondheim Working Papers in Lin-guistics. to appear.

Verkuyl, Henk J., and Jaap M. van der Does. 1991. The Semantics of Plu-ral Noun Phrases. In Generalized Quantifier Theory and Applications, ed.Jaap M. van der Does and Jan van Eijck. 403–441. Amsterdam: DutchNetwork for Language, Logic and Information. A revised version to appearin CSLI Publications.

White, Michael. 1994. A Computational Approach to Aspectual Composition.Doctoral dissertation, University of Pennsylvania. The Institute For Re-search in Cognitive Science, Philadelphia, Pa.

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