Types of degrees and types of event structures

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Objective of the paper
The goal of this paper is to propose a treatment of the degree structures associated with various linguistic expressions, and thus to shed light on some related aspectual phenomena. Although degrees can be ascribed to different types of objects within a linguistic ontology, either concrete (e.g., material objects, events) or abstract (propositions, propositional attitudes, speech acts), we will be mostly concerned with the former here. We will focus on the treatment of degree modifiers such as completely (in Yannig ate a cookie completely or The table is completely wooden), and their relationship with event structure, considering that the nature of the scales over which they range notably determines telicity.

Main theoretical concepts and issues
So far, gradable adjectives like long have been the main topic of interest with respect to a formal theory of scalar structures (cf. Kennedy 1999, 2001, Paradis 1997. Degrees are indeed convenient to account for the meaning of expressions such as long, two meters long, longer than a boat, extremely long, etc. Like many other authors, we take degrees to be arguments of gradable predicates (see Kennedy 1999Kennedy , 2001 for a review of the different technical strategies available). The idea is that an adjective like long takes at least two arguments, an argument x for the entity which is said to be long, and an argument d for the degree of length which is attributed to x.
One of the central issues at stake is whether only certain linguistic expressions have a degree argument (see e.g., Piñón 2000;Kennedy et al. 1999), or if all do (Ballweg and Frosch 1979). We assume that most predicates can receive a degree argument, either for ---------- * We would like to thank two anonymous reviewers for their constructive and detailed criticisms.
To appear in C. Maienborn & A. Wöllstein (eds.), Event Arguments: Foundations and Applications, Tübingen: Niemeyer inherent lexical reasons, or by virtue of their semantic and/or syntactic context. We take degree arguments to range over a discrete set (e.g., {0,1}) or an interval (e.g., [0,1]). Only certain sets of degrees can give rise to scalar readings (e.g., they can be modified by completely if they contain more than two degrees); in contrast, intervals of degrees can always do so.

Degrees and aspect
For clarity's sake, we will first examine stative and telic predicates 1 . We claim that both types of predicates can possess a degree argument, whose value is made explicit by degree modifiers such as completely, cf. for instance Yannig cooked a chicken completely.
Introducing degree arguments for verbal predicates will yield in particular a new analysis of telicity. The 'localist' analysis of telicity (cf. Jackendoff 1996, Verkuyl 1993 amounts to treating changes -of-state as changes of location, regardless of the type of telicity involved. Localists treat in this manner the following classes of telic verbs: 1. Verbs with totally affected arguments like leave; 2. Directed motion verbs like drive to Birmingham; 3. Path-argument verbs like walk the trail; 4. Verbs with incrementally affected arguments like eat; 5. Verbs expressing gradual changes of state like cook.
In our opinion, some of the crucial characteristics of these aspectual classes are obscured by the existing localist proposals, which treat them on a par. We will propose a degree analysis which makes it possible to understand both the unity of these cases of telicity, as well as their specific differences.
On top of telicity, we will also pay attention to the relationship between degrees and atomicity. Atomicity should be understood as in Dowty (1986). That is, atomic telic events are based on a holistic, 'one step' change-of-state, and reject finish and completely. 2 They involve only two degrees, i.e. a minimal degree and a maximal one, cf. (1). On the contrary, non-atomic telic events are based on a complex change-of-state, possessing intermediary degrees between the minimal and the maximal degree, and combine with finish and ----------1 By predicate we understand a specific, disambiguated use of a verbal predicate-i.e., within a particular sentence and discourse context -rather than a purely lexical, out-of-context predicate. This term will be contrasted with that of predication, which refers to the combination of a predicate and its arguments. 2 However, proportional degree modifiers (e.g., completely) seem to offer more reliable tests for atomicity than finish (the 'traditional' test inherited from Vendler 1957 (2). We consider that atomic telic events involve a one-step change-of-state, while remaining capable of forming complex degree structures (i.e., scales) with collectiondenoting argument noun phrases, cf. (3).
(1) #Yannig left completely. 3 (*finish) atomic (2) Yannig ate his pancake completely. ( OK finish) non-atomic (3) The tourists left completely. ( OK finish) non-atomic So, as said above, we assume that both telic and stative predicates can receive a degree argument. In the next section, we will see what the linguistic data tell us about the types of degree scales that are associated with various stative and telic predicates.
2. Degrees and their empirical manifestations 2.1 Some foundational elements for a treatment of scalarity As observed in Kennedy et al. (1999), degree scales can be closed or open. Intuitively, the range of degrees lexically associated with an adjective like wealthy is not bounded (there is no limit to wealth), and therefore forms an open scale. On the contrary, a predicate like destroy has a maximal degree. Once a building is completely destroyed, no further destruction of it is possible. The degree scale associated with destroy is therefore closed.
These intuitive distinctions are empirically corroborated by certain distributional facts. Thus, adjectives are endowed with closed scalar structures when they combine with adverbs such as completely, and reject very or extremely, cf. (4); conversely, they are endowed with open scales when they exhibit the opposite syntactic behavior, cf. (5).
(closed scale) b. The door is completely/*very/*extremely wooden.
(open scale) Similar tests can be proposed for verb phrases in general, by replacing very / extremely with a lot: (6) a. Yannig ate his pancake OK completely/*a lot.
(open scale) Besides scale closure, we introduce the notion of restricted accessibility (following Caudal 2000aCaudal ,b, 2002 where it is also called zoning). The degree scale associated with some predicates is such that it is not possible to access certain zones on the scale, cf. the excluded low degrees in (8) Restricted accessibility should not be confused with the notion of standard degree, 4 that is, with the fact that some gradable predicates have a 'normal', default degree value: The glass is full. = The glass is completely full.
(standard degree) ≠ The glass is half-full.
(non-standard accessible degree) (10) This pull-over is damp. = This pull-over is damp enough. (standard degree) ≠ This pull-over is entirely damp.

Types of degrees and adjectival predications
Crucially, proportional degree mo difiers can receive several types of interpretation. The 'quantity' interpretation (Caudal 2000a,b) is characterized by the inference pattern given in (11)-(12): (11) The high wall of the sitting room is half painted. → Half the high wall of the sitting room is painted.
(12) The gatehouse on the High Street is half-wooden. → Half the gatehouse on the High Street is wooden.
From the high wall of the sitting room is half painted we can deduce that half the high wall of the sitting room is painted. Such predicates will be noted [+quantity]. We call 'quantity argument' any theme or patient argument whose reference can be measured by verb phrase adverbials, following the inference pattern exemplified above. The notion is broader than that of incremental theme, which is restricted to [+quantity] changes-of-state.
In contrast, scales of degrees involving an 'intensity' interpretation do not allow for the same kind of inference pattern, as shown in (13)- (14): The hostel-guy was half drunk, and served us welcome drinks.
The man was half awake, as if under the effects of some sort of drug.
-/→ Half the man was awake.
Predicates involving this kind of degrees will be called [+intensity].

Degrees and VP reference: events
It has been observed for a fairly long time already (see e.g., Kennedy et al. 1999, Caudal 2000a that degree modifiers interact with the internal structure of events. We take such verb phrase modifiers to be event descriptor modifiers. Thus, completely and partially As is apparently the case in Hay et al. (1999).
To appear in C. Maienborn & A. Wöllstein (eds.), Event Arguments: Foundations and Applications, Tübingen: Niemeyer contribute to the internal structure of the event described in (15), whose development is measured or graded by the modifier at stake.

(15)
Yannig ate his pancake completely / partially. (non-atomic telic event) As a consequence, certain types of non-stative predications are also incompatible with certain degree modifiers, cf. (16)- (18). Being a closed-scale degree modifier, completely rules out dynamic predications which are atelic and deprived of a closed scale, or those which lack a complex degree structure (cf. the dynamic atelic event description in (16) and the atomic telic descriptions in (17)- (18), as opposed to (15) In short, scale structure and event structure are related: whenever they bear on non-stative predications, degree modifiers can either require them to be telic or to be atelic, depending on whether these modifiers require open or closed scales. Modifiers can therefore be used for purposes of aspectual classification among non-stative predications.
In contrast to the data discussed so far, certain predications do not involve events or objects that can be measured by (at least certain) degree modifiers-i.e., they offer either an inappropriate scale or are not lexically scalar. Whenever degree modifiers bear on such predications, the scalar interpretation they receive does not involve a concrete, lexicallyencoded degree scale, but one that is associated with an abstract object of discourse. In sentences such as (23)-(24), the function of half is not to measure a lexically encoded variable, but to grade the relevance of a given propositional content to describe a situation: (23) He half-ran, half-stumbled down the obsidian corridors of his home, relishing even the dim green light that permeated the place. (web corpus) (24) Vanessa moaned then and half-fainted on the couch.
(web corpus) Degree modifiers can also grade commitment for speech acts, as the French example in (25) suggests; we leave the study of such cases to future research. A complete classification of telic predications in terms of degree structures emerges from these tests (cf. Table 1). Each type of telicity involves a specific type of degree structure (cf. Table 2), depending on whether it is simple (e.g., the set {0,1}) or complex (e.g. the interval [0,1]), and depending on whether it is [+quantity] or/and [+intensity].  To appear in C. Maienborn & A. Wöllstein (eds.), Event Arguments: Foundations and Applications, Tübingen: Niemeyer Note that contrary to a widespread opinion (cf. Tenny 1994), X drive to Y is an atomic telic predication, since it rejects both finish and completely. Like all atomic telic predications, it is associated with the simplest possible degree structure, i.e., the set {0,1}.
This classification indicates that telicity is sensitive to degree structures; but this is not the only parameter of event structure which is related to scalarity. As already shown in §1.3, [ -quantity], [ -intensity] telic predications describe atomic telic events, cf. (31a); so there is also a connection between atomicity and degree structure. Furthermore, verbs which are lexically [ -quantity] can possess a complex [+quantity] degree structure (i.e., comprising more than two degrees) and describe a non-atomic event when they receive a quantity argument with a mereologically complex denotation, cf. (31b). 3.2 Event structure and degree structure: telicity, atomicity and scales The data discussed above show that event structure is related to scalarity through atomicity and telicity, which are respectively related to the complexity and closure of degree structures. The latter fact has already been largely commented on in the literature (cf. e.g., Caudal 2000a: telic predications seem to require a closed scale, i.e., a scale possessing a specified maximal degree. 5 Yet, as we will see below, this is a necessary but not a sufficient property of telic predications: there must also exist a mapping between the degrees of the scale and the internal structure of the event described (indeed, otherwise, stative predications associated with closed scales would also turn out to be telic).
Interestingly, in the case of (at least some) atelic predications, degree adverbials can bear upon an implicit quantity argument, thus rendering the predication telic: (32) a. Yannig ran (for a long time).
b. Yannig ran a lot.
(a quantity argument is required by 'a lot') (meaning: "Yannig ran a long distance/for a long time") A lot apparently requires an open scale as its input, and yields a closed one as its output (cf. the telic predication Yannig ran a lot in (*for) two hours). Its function is similar to that of temporal modifiers such as for, which require an atelic event as their input, and yield one which is temporally bounded.
In addition to this, it seems that implicit quantity arguments are ruled out with atelic predications possessing an overt, strong internal argument, cf. (33), whereas they are licensed with verbs receiving a syntactically weak internal argument, cf. Yannig listened to the radio a lot.
We should therefore carefully distinguish atelic predications without any lexical degree argument (i.e., lacking a degree scale), from those that can receive one (and require one with a degree modifier). Finally, a third class of atelic predications should be isolated, namely those associated with [+intensity] scales (cf. widen). We will come back to these issues when we propose a typology of scales.

Aspectual interest of the notion of restricted accessibility
Besides complexity and closure of degree structures, we take the notion of restricted accessibility (already evoked in §2.1) to determine some interesting aspectual constraints (cf. Caudal 2000a,b). The degree scale associated with certain predicates is such that it is impossible to access certain zones on the scale. When applied to a telic predication, restricted accessibility implies that the change-of-state is restricted to a subpart of the associated degree scale (which grades the event's degree of development). For instance, in the case of destroy, the lower end (i.e., the lowest degrees) of the degree scale is not accessible: (35) NATO destroyed Belgrade completely / #NATO barely destroyed Belgrade.
Restricted accessibility makes it possible to identify some aspectual phenomena which have gone unnoticed so far. If, in the case of destroy, the lowest degrees of the associated scale are inaccessible (e.g., barely is out), in the case of annihilate, the associated scale is even more restricted: the only possible degree is the maximal degree, although annihilate predications are not atomic (cf. The bomb completely annihilated the building). So there seems to exist more or less restricted brands of degree structures, the simplest or poorest one being associated with atomic telic predications. 6 To put it in an aspectual perspective, annihilating-events are almost atomic telic events, insofar as they admit a severely restricted range of degrees, while destroying-events are not so close to atomicity, since they exclude only the lowest degrees (i.e., any degree below partially), thus retaining a fairly large range of accessible degrees. And indeed, while destroy is fully compatible with the perfect progressive, cf. (36), annihilate does not accept it so readily, cf. (37) -a clear sign of its vicinity with verbs capable of describing atomic telic events (since they reject the perfect progressive, cf. note 2). (36) Obasanjo's arrogance and his belief that he knows it all, has been destroying the very basis of our federalism. (Web corpus) The semantics of annihilate suggests that one should not understand restricted accessibility in terms of 'gaps' on scales. Indeed, if the scale associated with this verb comprised only degrees 0 and 1 (with 'gaps' between them), then this would be tantamount to saying that annihilate is an atomic telic predicate. But it is not one, since it combines with completely. Note also that annihilate admits an 'almost completed' reading with almost, unlike atomic telic predicates. Almost precisely selects the 'threshold' of the first or last accessible zone on a scale.
To appear in C. Maienborn & A. Wöllstein (eds.), Event Arguments: Foundations and Applications, Tübingen: Niemeyer (37) The Dominion is less organized and more erratic. They've been annihilating nearby worlds (#this world) and disrupting the entire quadrant. (Web corpus) 4. Degrees at the syntax/semantics interface Now that the foundations for a detailed account of scalar structures are laid, we want to propose some elements concerning their role at the syntax/semantics interface, before moving on to a more thorough semantic account.
4.1 Event structure and the syntax/semantics interface: event templates We interpret telic predicates as changes -of-state, therefore crucially involving states. Thus, the shirt is dry should so to speak be embedded within the meaning of the shirt dried, and the cake is eaten should be embedded within the meaning of John ate the cake. To meet these requirements, we assume a compositional syntax/semantics interface in the spirit of Levin and Rappaport's (1999) 'event templates'. We use three types of event templates: (38) a. [<State> X] = « "state/property" is attributed to X » Temp late for states (The

Assumptions concerning degree structures and event templates
Moreover, we make the following assumptions: (i) Some state templates [<Stative> X] are associated with quantity arguments; (ii) Change-of-state templates introduce a mapping between a scale of degrees and the internal structure of an event variable. We assume that the above BECOME predicate is responsible for this, since it expresses a change-of-state, and possibly a gradual one. Starting from a stative predicate (e.g. to be dry), BECOME constructs a change-of-state predicate (e.g. to dry) which measures an event along a degree d.
In section 5 below, we develop further our analysis, in terms of a mapping from degrees to events, as illustrated on Figure 1. It departs substantially from Krifka (1992Krifka ( , 1998, notably because it does not introduce a mapping between objects and events, but also because it assumes a different definition of telicity, as we will see.

A semantics for degrees, events and objects
Before putting forth a formal treatment of the relationship between event and degree arguments, that is, of the aspectual effects of degree modifiers (cf. §5.3- §5.6), we need to expose ( §5.1- §5.2) the properties which degrees, objects and events have in our model.

Degree structures: an ontology and a typology
We take degrees to be elements of either Á, the set of natural integers, or Å + , the set of positive real numbers. We assume that many predicates are associated with a set of degreespossibly a scale, if that set is sufficiently complex.
When a predicate P is associated with a set of degrees, we note it S p . S P is part of the domain of degrees, U D . 7 We take the minimal element of S P to be the constant d min = 0. S p is closed if it has a specified maximal element, noted d max ; it is said to be open otherwise.
The simplest degree structure is that associated with atomic telic predications like The tourist has left: it is reduced to the discrete set {0,1}⊂Á. This simple structure excludes degree modifiers like completely (*The tourist left completely): indeed, completely can apply only if the degree structure associated with the predicate contains an element d different from 0 (the minimal degree) and d max (the maximal degree). However, definite plural arguments can ascribe complex [+quantity] degree structures to atomic telic predications, namely a set {0,…, d max }⊂Á, where d max is the cardinality of the set denoted by the definite plural noun phrase. This makes completely acceptable, as in The tourists left completely (cf. §1.3 and §3.1). 8 The degree structure associated with a predicate is often richer than {0,1}. Many closed scale predications have an associated scale that corresponds to an interval of Å + , of the form [0, d max ], with 0 < d max . This is the case of non-atomic telic predicates (e.g. cook the chicken, eat an apple), and of some stative predicates associated with a closed scale (e.g., be wooden, cover).
Interestingly, non-atomic telic predicates combined with definite plural argument noun phrases and proportional degree modifiers can receive two distinct kinds of scales, depending on whether the degree modifier receives a 'narrow scope' or a 'wide scope' interpretation. Thus, To appear in C. Maienborn & A. Wöllstein (eds.), Event Arguments: Foundations and Applications, Tübingen: Niemeyer can receive two distinct interpretations: one for which the scale is [0,1] ⊂ Å + (this is the 'narrow scope' reading of partially, which bears on each individual pancake: Yannig partially ate each pancake), and one for which the scale is {0,…,30} ⊂ Á (this is the 'wide scope' reading; partially bears on the whole collection; e.g., Yannig ate twenty pancakes).
This, however, does not happen when the argument position at stake is filled by a bare plural noun phrase. Thus, (40) can only mean that Yannig ate completely each individual apple, not that he ate an entire set of apples. Likewise, in (41), completely can only be interpreted with respect to individual fields-i.e., it only has a 'narrow scope' reading.
#A thick blanket of snow completely covered fields as far as one could see.
This is due to the fact bare noun phrases do not possess any fixed quantificational information (i.e., they do not have a fixed cardinality; cf. Verkuyl 1993). 9 Therefore, no maximal degree is specified for the quantity argument, thus blocking proportional degree modifiers.
But the most delicate part of this lexical semantic typology of scales concerns activity predicates (i.e., predicates describing dynamic, atelic events even without bare noun phrase arguments). These predicates fall into at least three broad classes with respect to scalarity: ----------9 In fact, the correct generalization should extend to semantically weak noun phrases in general, whose quantificational information seems to remain inaccessible to proportional degree modifiers, cf. the absence of 'wide scope' reading for completely in #Yannig ate thirty apples completely. 10 They can only become scalar by means of some meaning-shift operation; cf. half-ran in (23).
To appear in C. Maienborn  As argued for instance in Kennedy and Mc Nally (2002), the latter kind of predicates do not have a terminus; they describe an open-ended change-of-state. And they are atelic because their scales lack a specified maximal degree that would correspond to the endpoint of that change-of-state.
It appears then that degree structures are not a purely lexical category, but are construed at the sentence level, on the basis of lexical information combined with syntactic and semantic information. The corresponding procedure can be summarized as follows: (i) Lexical semantics determines what kind of scale is available (or not) for a given interpretation of a predicate (e.g., whether it is [+/-quantity] or [+/-intensity], whether it is a priori open or closed, whether it has a standard value and restricted accessible zones, etc.).
(ii) The syntax/semantics interface interacts with this information; e.g., degree modifiers and determiners can restructure or introduce 11 scales (for instance, a lot can turn an open scale into a closed one, or enforce the presence of an implicit quantity argument; similarly, collective argument noun phrases can introduce a complex [+quantity] scale).
(iii) Semantics provides additional information about the structure of the available scale(s) (for instance, the value of d max ), by means of certain axioms (e.g., QUANTITY in (56)).
(iv) The value of the degree d is constrained or specified by combining the semantics of degree modifiers (if any) with the scale thus construed.

Part structures for objects and events
Degrees are not the only entities in our ontology. We assume for instance that it also comprises material objects and events . They are modeled using part-structures, in the spirit of Simons (1987) and Krifka (1992Krifka ( , 1998. The crucial fact for us is that there is a part relationship among objects, and a part relationship among events.
A part structure P = 〈U P , < P , ≤ P , ⊗ P, ⊕ P 〉 is defined over a domain of entities, U P . The relation of mereological part, < P , is taken as primitive. The relations of improper mereological part, overlap and sum are then defined: Definitions: ---------- 11 We do not propose here any semantic mechanism capable of adding a degree argument to a predicate that would initially not possess one, but see Piñón (2000) for an instance of such a mechanism.
To appear in C. Now that the technical foundations of our theory are laid, let us now move to the treatment of the interaction of degrees, objects and events.

The semantics of proportional degree modifiers
Proportional degree modifiers like partially, halfway and completely can combine with a gradable predicate P only if the scale associated with P is closed and if it is not reduced to just two elements, 0 and d max . These degree modifiers then constrain or specify the value of the degree argument as indicated in (51).
(51) a. partially, completely and half(way) can apply only if the scale is not reduced to {0, d max }; then: b. partially specifies that the value of the degree argument of P is a degree d such that 0 < d < d max c. completely specifies that the value of the degree argument of P is d max d. halfway specifies that the value of the degree argument of P is d max / 2 We characterize [+quantity] verbs, and then changes -of-state (telic change-of-state verbs, in fact) in the following section. ---------- To appear in C. Maienborn & A. Wöllstein (eds.), Event Arguments: Foundations and Applications, Tübingen: Niemeyer

The semantics for [+quantity] verbs
In what follows, we use a Parsons-style 'subatomic' semantics (cf. Parsons 1990) in order to represent the semantics of complex sentences. We assume the following kind of neodavidsonian decomposition for [+quantity] stative predications, such as The snow covered the field: (52) ∃x ∃y ∃e ∃d [ Verbal-Predicate(d,e)∧Quantity(d,y)∧Role-1(x,e)∧Role -2(y,e)∧ Semantic-Type-1(x)∧ Semantic -Type-2(y) ] 13 Verbal-Predicate(d,e) stands for a stative verbal predicate describing a state e with degree d. e) and Role-2(y,e) are classic semantic roles, such as agent and patient (of course, if the predication is intransitive, rather than transitive, only one role will be present in the decomposition Note that in this case, patient argument y acts as a 'event delimiting argument' (in the sense of Tenny 1994) with respect to the verbal predicate.
----------13 This decomposition is related to Parsons' (1990) decomposition of the semantics of transitive verbs, i.e., Verbal-Predicate(e)∧Theta-role-1(x)∧Theta-role-2(y)∧Semantic-Type-1(x)∧Semantic-Type-2(y). However, Parsons' verbal predicate does not receive any degree argument d on top of an event argument e, unlike in the present account. We do not give here the complete set of compositional rules making it possible to derive such formulas, because they are rather trivial; but see e.g., Krifka (1992:36 sqq.) for such rules.
To appear in C. Maienborn & A. Wöllstein (eds.), Event Arguments: Foundations and Applications, Tübingen: Niemeyer The following axiom then characterizes [+quantity], stative and telic predicates. It establishes a morphism from degrees to quantities. 14 The axiom concerns only predicates that have an associated degree scale, with a specified maximal degree, and degrees intermediary between 0 and d max .
(56) QUANTITY: The left part of the axiom's consequent says that if the degree associated with the quantity role Quantity(d, y) is between 0 and d max , then there is a unique, greatest part y° of y such that Quantity(d max , y°) (Max acting as a maximizing function). 15 Thus, if an apple is half eaten, then there is a unique, greatest part of the apple that is completely eaten. Likewise, if a table is half wooden, then there is a unique, greatest part of the table that is completely wooden.
The right part of the axiom's consequent, d / d max = Prop (y°, y), means that y° is in the same proportion to y as d is in proportion to d max . Thus, if an apple is half eaten, then half the apple is eaten.
It is worth mentioning that the above axiom is trivially valid for atomic predicates, insofar as they are associated with a scale reduced to {0,1}. In that case, we have d = d max and y° = y in (56) -atomic predicates apply to entities 'as a whole'. It is clear that this holistic and binary nature of atomic predicates explains why they are incompatible with intermediary degree adverbials. But somewhat surprisingly, even 'maximal degree' adverbs such as completely or entirely are incompatible with atomic predicates (cf. (31a) and (51). We believe that these adverb imply the existence of intermediary degrees between 0 and d max =1. Indeed, degree adverbs seem to involve an implicational scale (in the sense of scalar implicatures) -maximal degree adverbials somehow imply lesser degree adverbials; having completely eaten an apple implies having partially eaten an apple. In other words, the validity of a certain proposition associated with the maximal degree d max implies the validity of similar propositions associated with lesser, intermediary degrees -i.e., less informative propositions. But of course, such implications are ruled out in the case of atomic predicates, and therefore, these predicates reject maximal degree adverbs.
----------14 Naumann (1996) noted that Krifka's (1992) homomorphisms between objects and events need not apply directly to the denotation of some patient noun phrase, but rather to some facet of its denotation-cf. peel the apple, which does not involve the whole apple but merely its skin. In (56), y is therefore built from the relevant facet or dimension of the quantity argument at stake. 15 For the sake of simplicity, we ignore issues related to so-called event and object uniqueness.
Indeed, Krifka (1998) uses homomorphisms like the following, where θ O shows 'uniqueness of objects', and θ E shows 'uniqueness of events': These notions are relevant to the treatment of predications such as read the novel, where several parts of the same novel can be read several times in a telic reading event.
To appear in C. Maienborn & A. Wöllstein (eds.), Event Arguments: Foundations and Applications, Tübingen: Niemeyer It should also be noted that the notion of atomicity is a lexical semantic issue, by and large; it amounts to the possibility or impossibility of a distributive/incremental reading of the verb with respect to a certain argument noun phrase. Thus, rent the car is atomic, while paint the car is not. This is reflected in the content of the Semantic-Type-1/2 predicates in e.g., (52)-(54): rent requires an object of e.g., the vehicle type, whereas paint requires its object to be of the material_surface type. Only in the latter case is the predicate nonatomic. See Caudal (1999) for a detailed discussion of these phenomena and a treatment within a formal lexical semantic framework.
Finally, we should stress that the part-of relation on objects is to be applied differently when the quantity argument is a definite plural. In that case, indeed, it can be understood in a manner similar to Link's (1998) i-part operator ('individual-part'), which returns a subset of a collection of individual entities. This reading arises if the degree expression involved has 'wide scope', i.e., if Yannig ate his pancakes partially means that Yannig ate part of the set of pancakes, and not that he ate part of each pancake-this would be the 'narrow scope' reading of the degree expression. In the latter case, the part-of relation on objects is to be understood as Link's (1998) q-part operator ('quantity of matter-part'), which returns a subpart of an individual entity. This also accounts for examples such as The tourists left partially: the agent noun phrase acts as a quantity argument, with an associated scale {0, …, d max }⊂Á, d max being the cardinality of the set of tourists.

Changes-of-state
We see changes-of-state as events developing along a degree scale. Non-atomic telic events are endowed with complex changes-of-state: they go through different intermediary degrees of development. On the contrary, atomic telic events are endowed with simple changes-ofstates, reduced to two degrees: 0 and 1. We capture the notion of change-of-state in axiom (57), which makes sure that if a verbal predicate P describes an event e with degree d, every initial part e' of e is associated with a unique lesser degree d' (cf. (57a)), and vice versa (cf. (57b)). A part is said to be initial if it satisfies the INI predicate, cf. Note that the relation « expresses strict temporal precedence; e" « e' is true if the temporal trace of e" strictly precedes that of e' (a fact which implies ¬e" ⊗ E e').
This definition guarantees that the order on degrees matches the temporal ordering of the initial subparts of a change-of-state event e. It also implies that: ---------- 16 We do not pay attention here to event connectedness-see Piñón (2000) for details about that issue.
To appear in C. Maienborn (d,e). This implies that e'=e.
To illustrate how axiom (57) functions, let us consider the treatment of (60), assuming decomposition (61). Become(cook) says that there is a strict mapping between degrees and initial parts of the event e described. As the event unfolds and mereologically 'grows', the corresponding degree steadily increases. In addition, lexical information indicates that cook has an associated [+intensity] closed scale, [0,1]⊂Å + , and that, by default, the degree in (61)  These observations indicate that there is a strong relationship between degree structures, telicity, and modes of change-of-state (cf. Caudal 2000a, Kennedy andLevin 2000). We express it through the following characterization of telicity: (67) TELICITY: A predication is telic if and only if (i) it has an associated set of degrees, with (ii) a specified maximal degree, and (iii) its verbal predicate satisfies axiom BECOME (57), i.e., it describes a change-of-state.
---------- Now, there is also an atelic reading of dry, for which it could be true that the shirt becomes dry before the event ends (cf. Yannig dried the shirt for a whole day, although it was already dry after only two hours). How are we to treat it? If we considered the scale of the atelic dry to be open, then we would loose the insight that there is a maximal degree of dryness. Let us therefore assume that, under such a reading, the scale associated with dry is closed. However, by hypothesis, the drying events described can keep on developing beyond whatever subevent is associated with the maximal degree of dryness: there are at least two drying events e and e' such that INI(e', e), P(e, d max ) and P(e', d max ). Thus, given (67), the predicate is not telic.
In addition to dry, cook and bake also admit telic and atelic readings with the same set of participants, in so far as the degree of 'cookedness'/'bakedness' of some edible thing can exceed its 'normal' maximal degree-in which case it is too cooked/baked; it becomes, e.g., burnt. Under this 'too much' reading, cook and bake are atelic (cf. Yannig cooked the chicken for three hours, and he burnt it). These are cases in which a normally closed scale becomes To appear in C. Maienborn & A. Wöllstein (eds.), Event Arguments: Foundations and Applications, Tübingen: Niemeyer open. They could be handled either by postulating two different scales for cook / bake, or by creating a special subtype of optionally closed scales.
6. Comparison with some existing approaches Before concluding, we would like to compare our approach with other theories, i.e., that of Kennedy et alii on the one hand, and that of Krifka and Verkuyl on the other hand.
6.1 Comparison with Kennedy et al. (1999) Our work differs from Kennedy et al. (1999) in two main respects. First, we do not use negative degrees. We believe that antonymy is not lexically, but contextually determined; in our view, introducing negative degrees is thus going one step too far in the lexical semantics of degree modifiers. For instance, cold can be either opposed to warm or hot, depending on whether we are talking about people or substances, and, crucially, these adjectives do not possess identical scalar properties. (Another, related difference is that Kennedy et al. model degrees as intervals, whereas we identify them with numbers.) Second, we have introduced the notion of restricted accessibility, which does not have any equivalent in Kennedy et al. (1999). The degree scale associated with some predications is such that it is impossible to access any arbitrary zone on the scale. For instance, *Yannig slightly destroyed the chair is out: the low degree specified by slightly is inaccessible on the scale associated with destroy. Krifka (1992Krifka ( , 1998 and Verkuyl (1993Verkuyl ( , 1999 The present account differs from Krifka's and Verkuyl's in the following respects.

Comparison with
First, it does not characterize telicity in terms of a relationship between events and objects, but in terms of a relationship between events and degrees and the notion of scale closure. It is neither a purely mereological approach ( like Krifka's) nor a purely quantificational one (like Verkuyl's). This makes it possible to avoid some of the shortcomings inherent to a definition of telicity in terms of quantization, by resorting to quantificational mechanisms, while retaining some of the insights of a mereological approach.
Second, our degree-based approach is capable of capturing a number of (dis)similarities between classes of verbs, which have been overlooked in Krifka's and Verkuyl's works: • A partial analogy exists between [+quantity] stative verbs and [+quantity] telic verbs: while both satisfy axiom QUANTITY (56), only the latter involve a change-of-state mapping between degrees and events (i.e. satisfy Become (57)); • A partial analogy exists between [+intensity] stative verbs and [+intensity] telic or activity verbs: while they all fail to satisfy axiom QUANTITY, only the latter involve the Become change-of-state mapping between degrees and events; To appear in C. Maienborn & A. Wöllstein (eds.), Event Arguments: Foundations and Applications, Tübingen: Niemeyer • All types of stative verbs are analogous, insofar as they lack the Become change-of-state mapping between degrees and events; • All types of change-of-state verbs ([+intensity] activity verbs / telic verbs) are analogous, insofar as they involve the Become mapping between degrees and events. Table 3 gives a synthetic overview of these results.

Conclusion
In this paper, we have proposed a treatment of adverbial modification by degree expressions like halfway and completely. In order to do so, we have introduced a degree argument within the semantics of many verbal predicates, including stative predicates.
We have shown that the scales underlying telic and atelic change-of-state predicates are strongly correlated with their aspectual properties, and we have put forward a conception of (a)telicity and event structure which is broader than the one assumed in most existing formal theories of aspect.
We have defined telicity by conjoining a mapping between degrees and events (expressed by the Become predicate) with a theory of degree structures. Our conception of telicity extends to cases such as cook the chicken or dry the shirt, without resorting to any special 'path-argument'. 19 The addition of a quantity argument with verbs such as eat causes the same scale to be mapped onto the mereological and quantificational structure of an argument noun phrase, thus capturing Dowty's (1991) notion of incremental theme. 20 ---------- 19 Unlike Krifka (1998), and 'localist' approaches generally (e.g., Tenny 1994), which assume that [+intensity] telic verbs are associated with a special path argument corresponding to the different degrees of cookedness involved in the cooking event, and playing a role similar to that of so-called incremental theme arguments. 20 As it stands, the present account is unable to address cases where more than one argument noun phrase have an impact on aspect calculus, e.g., when the agent and the patient simultaneously act as [+quantity] arguments. It thus seems to fall under some of the criticisms addressed by Verkuyl (1999:47) to Krifka (1992). However, we believe that predications always possess at most one [+quantity] argument, but that this argument needs not be a patient or a theme argument.
To appear in C. Maienborn & A. Wöllstein (eds.), Event Arguments: Foundations and Applications, Tübingen: Niemeyer Indeed, following Caudal (1999), we have identified [+quantity] predicates as predicates relating a degree scale with the mereological and quantificational structure of some specific noun phrase argument (which we call a quantity argument). This has revealed deep analogies between various types of stative and telic predicates.
Concerning scale structures, we have noted that degree modifiers act as scale modifiers. We have hypothesized in §3.2 that a lot affects the scale associated with certain intransitive verbs, cf. (32b) Yannig ran a lot: these modifiers transform an initially open scale into a closed one. Similarly, adding certain degree modifiers to an open scale, atelic predicate turns it into a telic predicate, with a specified maximal degree, cf. (69) From a more general point of view, our proposal bridges the gap between quantificational approaches to aspect such as Verkuyl's and mereological approaches such as Krifka's, leaving room for both types of mechanisms in the realm of aspect calculus.