Event Structure and Non-Culminating Readings in Turkic

Abstract

Since the seminal work by David Dowty, much inspired by the earlier ideas of Generative Semantics, a number of proposals have been developed accounting for the internal constitution and interpretation of accomplishment event predicates like ‘open the door’ or ‘break the window’. Current theories of accomplishment event structure vary along a number of dimensions, including the subevental makeup of accomplishments and semantic relations connecting components of a complex eventuality description. The goal of this paper is twofold. First, I take into account evidence from non-culminating readings of accomplishment predicates in Turkic languages and argue that this evidence supports the following generalizations about the structure and interpretation of accomplishments: (i) the activity subevent is to be represented independently from the change of state; (ii) different accomplishment predicates constrain the relation between subevents in different ways; (iii) accomplishments differ as to the internal constitution of the activity subevent. Second, I will suggest that restrictions on non-culminating readings observed with different types of accomplishments support a specific view of how non-culminating accomplishments are derived. I will propose that at least in languages like Turkic, a necessary condition for non-culminating predicates is as follows: the activity component of a complex eventuality description has to have temporal parts that make no substantial contribution into bringing the culmination about. What I will say about Turkic does not presuppose that all non-culmination phenomena cross-linguistically warrant a uniform analysis. Even though the Turkic pattern shows strict semantic parallelism in other languages, it is not unlikely that there is more than one way in which non-culminating accomplishments can be derived. But whether a variety of other cases discussed in the literature reduce to the same pattern is a separate empirical question I am not trying to answer. The paper is organized as follows. In Section one, I introduce relevant material from three Turkic languages, Karachay–Balkar, Chuvash, and Tuba Altai, and observe that accomplishments in these languages fall into three types. Some yield the failed attempt interpretation, others the partial success interpretation, yet others do not license non-culminating readings at all. Section two argues for a decompositional analysis of the accomplishment event structure, whereby activity and change of state subevents are kept representationally distinct. Two types of relations between these components of the accomplishment structure are identified; the failed attempt and partial success readings are reduced to the properties of these relations. Section three approaches the problem of why non-culminating interpretations are available for some but not for all accomplishments. It reviews a recent theory suggesting that the (un)availability of non-culminating readings is accounted for by the unique temporal arrangement of contextually salient subevents of the activity component, either lexically or contextually entailed. The concluding subsection of Section three presents a number of problematic cases for this view. Section four outlines an alternative to the unique temporal arrangement. It argues that non-culminating accomplishments describe a proper non-final part e of the activity component of an event description such that the distance to the culmination between the initial and final bounds of e is insignificant in the current context. This approach makes more accurate predictions about the attested distribution of non-culminating interpretations and successfully avoids the complications associated with the unique temporal arrangement hypothesis. After making the notion of distance to the culmination more formally explicit, in the concluding section I address a few related issues concerning the eventuality type of non-culminating accomplishments and their interaction with aspectual operators.

1. Non-Culminating Accomplishments in Turkic Languages

The goal of this section is to introduce the phenomenon of non-culminating accomplishments (NCAs), which have mostly been studied outside of the Turkic family (see, e.g., Singh (1998), Arunachalam and Kothari (2012), and Altshuler (2014) on Hindi; Koenig and Muansuwan (2001) on Thai; Bar-el et al. (2005) on Salish; Soh and Kuo (2005), Koenig and Chief (2008), Martin et al. (2021), and Martin et al. (2023) on Mandarin Chinese; Paul et al. (2020) on Malagasy; Beavers and Lee (2020) on Korean; Persohn (2022) on Bantu; some Turkic material can be found in Tatevosov and Ivanov (2009), Tatevosov (2020)).

Specifically, we will see that:

◾

NCAs are perfective and atelic,

◾

they should be kept distinct from defeasible causatives,

◾

they fall into two semantic types, and

◾

not all accomplishments allow for a non-culminating construal.

To begin with, consider (1)–(3) from Karachay–Balkar, Chuvash, and Tuba Altai:

(1)	Karachay-Balkar
	a.	kerim	on	sekunt-xa	xalı-nı	zırt-tı.
		K.	ten	second-dat	thread-acc	tear-pfv.pst.3sg1
		‘Kerim tore a thread in ten seconds.’
	b.	kerim	on	minut	xalı-nı	zırt-tı.
		K.	ten	minut	thread-acc	tear-pfv.pst.3sg
		‘Kerim tried to tear a thread for ten minutes.’

(2)	Chuvash
	a.	vaɕa	eki	minut	xɯɕɘntɕe	petuk-n-a	vərat-r-ɘ.
		V.	two	minute	within	P.-obl-acc	wake.up-pfv-pst
		‘Vasja woke up Peter in two minutes.’
	b.	vaɕa	ɕirɘm	minut	petuk-n-a	vərat-r-ɘ.
		V.	twenty	minute	P.-obl-acc	wake.up-pfv-pst
		‘Vasja tried to wake up Peter for twenty minutes.’

(3)	Tuba Altai
	a.	amɨr	on	minut-xa	eʒik-ti	ač-tɨ.
		A.	ten	minute-dat	door-acc	open-pfv.pst3sg
		‘Amyr opened the door in ten minutes.’
	b.	amɨr	on	minut	eʒik-ti	ač-tɨ.
		A.	ten	minute	door-acc	open-pfv.pst3sg
		‘Amyr tried to open the door for ten minutes.’

The accomplishment predicates in (1)–(3) allow for both culminating and non-culminating interpretations. In all the (a) sentences, the event culminates and immediately after the culmination a result state holds: the thread is torn (1a), the person is awake (2a), and the door is open (3a). These sentences are telic, as the test on co-occurrence with time-span adverbials indicates.

The (b) sentences are different in that the eventuality does not reach the culmination. They only entail that some amount of the agent’s activity has been performed at a certain time interval, but the activity stops before a result state is attained. All these sentences accept measure adverbials like ‘for forty minutes’ and are therefore atelic. The meaning of the (b) sentences is approximated as “X tried to VP for Y time” in the translations. This approximation should not be taken literally, however, since the meaning of the try-construction is not fully identical to that of NCAs, see fn. 6 for discussion. Another way of rendering NCAs like (1)–(3) used below is “X spent Y time VP-ing”.

Being atelic, the (b) sentences are perfective (cf. Bar-el et al. 2005), i.e., they involve the topic time that includes the event time (Klein (1994) and much further literature; see also the discussion in Section 2.1). This is easy to see by combining examples like (1b)–(3b) with a temporal adverbial. In (4), this is shown for Karachay–Balkar; for the sake of space, I leave out corresponding examples from the other two languages:

(4)	Karachay–Balkar
	alim	kel-gen-de	kerim	on	minut	xalı-nı	zırt-tı.
	A.	come-pfct-temp	K.	ten	minute	thread-acc	tear-pfv.pst.3sg
	‘When Alim walked in, Kerim tried to tear a thread for ten minutes.’
	1. walking in «T tearing				2. *walking in ⊆T tearing

Example (4) does not support the reading where the time of Alim’s walking in is included (“⊆_T”) into the time of tearing; the only available temporal relation is that of precedence (“«_T”). This type of temporal relationship is a signature of the perfective.

Therefore, in the languages exemplified above, perfective sentences built on accomplishment predicates like ‘tear a thread’ do not entail culmination and allow for an atelic construal, in a sharp contrast with languages like English:

(5)	a. Kerim tore a thread in 10 s.
	b. *Kerim tore a thread for 10 min.

English possesses another type of phenomenon not infrequently discussed in the context of the lack of culmination entailments, which is subsumed under the label of “defeasible causatives” (Martin and Schäfer 2012; Martin 2015), or “partitive telic accomplishments” (Martin and Demirdache 2020). For many speakers of English, sentences like (6) are not a contradiction:

(6)	David explained the puzzle to the student in twenty minutes. But the student did not understand it.

If part of what defines the culmination of ‘explain’ is the addressee’s entering a result state of understanding the puzzle, it is clear that in (6) the culmination has not been attained. In this sense, (6) can be said to be non-culminating.

However, I believe that defeasible causatives should be kept apart from non-culminating accomplishments like (1)–(3). The crucial difference between (6) and (1b)–(3b) is that the former is telic and takes an in-adverbial, not a for-adverbial. Telicity of (6) is naturally understood in the following way: whatever it takes for the agent to provide a complete explanation for the puzzle has been done in twenty minutes, even though the change of state of the addressee has not been brought about.

With a for-adverbial, explain the puzzle gets considerably degraded, even if, for some speakers, a bit less degraded than tear a thread in (5b).

(7)	??David explained the puzzle to the student for twenty minutes.

Examples (6) and (7) together show that it is possible to be a defeasible causative without being an atelic non-culminating accomplishment of the Turkic type in (1)–(3)2.

Reversely, non-culminating accomplishments do not have to be defeasible causatives. On the telic construal in (8), for example, the existence of the result state in the evaluation world is entailed, unlike in (6):

(8)	Chuvash
	#vaɕa	eki	minut	xɯɕɘntɕe	petuk-na	vərat-r-ɘ,	ɕapax ta
	V.	two	minute	within	P.-acc	wake.up-pfv-pst	but
	petuk	vəra-ma-r-ɘ.
	P.	wake.up-neg-pfv-pst
	‘Vasja woke up Peter in two minutes, but Peter didn’t wake up.’

In (8), an explicit indication that the change of state of the theme does not occur yields a contradiction. ‘Wake up’ is thus not a defeasible causative. But it is a non-culminating accomplishment, as (2a–b) show. Whereas (2b) is atelic and conveys that the activity stops before the culmination, in (8) it culminates. Once this happens, the result state of being awake must occur in the evaluation world.

This suggests that the two types of non-culmination are distinct3, as also discussed in Martin and Demirdache (2020), who separate atelic uses of accomplishments, like the ones addressed in this paper, from “telic partitive” ones, i.e., defeasible causatives like (6) and its cross-linguistic counterparts. In what follows I will not address the derivation of defeasible causatives; see Martin (2015), Demirdache and Martin (2015), and Martin and Schäfer (2012, 2017) for more data and analysis.

The next observation about the meaning of Turkic NCAs is that non-culminating readings can be different for different types of verbal predicates, see Tatevosov and Ivanov (2009). The (b) examples in (1)–(3) describe a failed attempt (FA). The agent aims at changing a state of the theme, but her attempts do not yield the desired result, and the theme retains its initial state without undergoing any detectable change. The thread in (1b) stays intact, the person in (2b) keeps on sleeping, and the door in (3b) remains closed.

The examples in (9b)–(11b), in contrast, report on a partial success (PS). In these examples, the event does not culminate either, but in a perceivably different way: each of (9b)–(11b) entails that the affected participant undergoes at least some change.

(9)	Karachay–Balkar
	a.	išci	eki	kün-ge	baxca-nı	qaz-dı.
		worker	two	day-dat	vegetable.patch-acc	dig-pfv.pst3sg
		‘The worker dug up the vegetable patch in two days.’
	b.	išci	eki	saʁat	baxca-nı	qaz-dı.
		worker	two	hour	vegetable.patch-acc	dig-pfv.pst3sg
		‘The worker spent two hours digging up the vegetable patch.’

(10)	Chuvash
	a.	vaɕa	ɕirɘm	minut	xɯɕɘntɕe	samalot	matel-l-a	puɕtar-tɕ-ɘ.
		V.	twenty	minute	within	plane	model-3.obl-acc	assemble-pfv-pst
		‘Vasja assembled a model of a plane in twenty minutes.’
	b.	vaɕa	ɕirɘm	minut	samalot	matel-l-a	puɕtar-tɕ-ɘ.
		V.	twenty	minute	plane	model-3.obl-acc	assemble-pfv-pst
		‘Vasja spent twenty minutes assembling a model of a plane.’

(11)	Tuba Altai
	a.	amɨr	eki	minut-xa	suù-nɨ	soùt-tɨ.
		A.	two	minute-dat	water-acc	cool-pfv.pst3sg
		‘Amyr cooled the water down in two minutes.’
	b.	amɨr	eki	minut	suù-nɨ	soùt-tɨ.
		A.	two	minute	water-acc	cool-pfv.pst3sg
		‘Amyr spent two minutes cooling the water down.’

One of the objectives of any theory of NCAs should be to understand what makes accomplishments like (1)–(3) different from those in (9)–(11).

Before going to the next observation, I would like to emphasize that Turkic FA data seem to diverge from the pattern discussed in Koenig and Chief (2008, p. 243). According to these authors, there exist languages where NCAs cannot describe failed attempts. As an anonymous reviewer points out, “in the languages discussed in Koenig and Chief (2008) (from 3 distinct language families), one argument has to be affected and change state (even if partially) for the NCA reading to be possible”. The following example is thus a contradiction.

(12)

#Ta

sha

le

Lisi,

danshi

Lisi

mei

shou

bandian

shang

he

kill

PERF

Lisi,

but

Lisi

not

receive

little.bit

injury

‘He killed Lisi, but Lisi was not even hurt a little bit.’ (Koenig and Chief 2008, p. 243)

However, parallel Turkic examples suggest that no change in the theme is entailed by an NCA under a FA reading.

(13)	Karachay–Balkar
	kerim	on	minut	ešik-ni	ac-tı,	alaj	ešik	cırt
	K.	ten	minute	door-acc	open-pfv.pst.3sg	but	door	at.all
	ac-ıl-ma-dı
	open-pass-neg-pfv.pst3sg
	‘He spent ten minutes opening the door, but the door did not open at all.’

Examples like (13) reassure that, if Koenig and Chief’s generalization about Mandarin and similar languages is correct, their analysis is not easily extendable to the Turkic data discussed here4. This difference between Turkic and Mandarin patterns will be discussed in more detail in Section 2.4 and Section 2.5.

The last observation is: whereas some of accomplishment predicates allow for both culminating and non-culminating interpretations, as in (1)–(3) and (9)–(11) alike, others do not. This type is illustrated in (14)–(16):

(14)	Karachay–Balkar
	a.	marat	beš	minut-xa	šatır-ny	quru-du.
		M.	five	minute-dat	tent-acc	set.up-pfv.pst3sg
		‘Marat set up a tent in five minutes.’
	b.	*marat	beš	minut	šatır-ny	quru-du.
		M.	two	minute	tent-acc	set.up-pfv.pst3sg
		‘Marat tried to set up a tent for five minutes.’

(15)	Chuvash
	a.	vaɕa	eki	ɕekunt	xɯɕɘntɕe	ɕutə	synter-tɕ-ɘ.
		V.	two	second	within	light	turn.off-pfv-pst
		‘Vasja turned the light off in two seconds.’
	b.	*vaɕa	eki	minut	ɕutə	synter-tɕ-ɘ.
		V.	two	minute	light	turn.off-pfv-pst
		‘Vasja tried to turn the light off for two minutes.’

(16)	Tuba Altai
	a.	amɨr	eki	čas-xa	baːtɨr-nɨ	köm-di.
		A.	two	hour-dat	B.-acc	bury-pfv.pst3sg
		‘Amyr buried Batyr in two hours.’
	b.	*amɨr	eki	čas	baːtɨr -nɨ	köm-di.
		A.	two	hour	B.-acc	bury-pfv.pst3sg
		‘Amyr tried to bury Batyr for two hours.’

Predicates like ‘tear a thread’ in (1) and ‘set up a tent’ in (14) in Karachay–Balkar do not look strikingly different. They occur in the same past perfective form in an episodic non-iterative sentence with a durative adverbial. Both are based on a predicate that describes a change of state of the theme brought about by an activity performed by the agent. In both cases, the change of state occurs at the final part of the activity. It may appear that (14) should have a con-culminating reading parallel to (1), which is a failed attempt reading. However, (14) are judged unacceptable5. The same holds for the pair of predicates in (2) and (15) in Chuvash, as well as for (3) and (16) from Tuba6.

The observations from (1)–(3), (9)–(11) and (14)–(16) are summarized in (17):

(17)	Basic observations about NCAs:
	NCAs occur in perfective atelic sentences. Non-culminating interpretations, if available, come in two varieties: failed attempt and partial success. NCAs can be derived from some verbal predicates, but not from others.

My goal in the next section will be to focus on (17b) and argue that it leads us to very specific assumptions about the structure of semantic representations of accomplishment predicates. I will argue, building on Tatevosov and Ivanov (2009), for a decompositional analysis of accomplishments where the agent’s activity is representationally distinct from the change of state subevent. The two subevents, then, can enter different types of relations, and the partial success and failed attempt readings are reduced to the properties of such relations.

Once an activity subevent is separated from a change of state subevent, in Section 3 and Section 4 I will suggest that its characteristics are crucial for predicting whether a non-culminating derivation is available for a predicate at all (17c). I will explore two possible ways of approaching these characteristics: through a temporal notion of inherent orderedness and through a modal notion of the distance to the culmination. I will conclude that a theory based on the latter comes with a number of empirical advantages.

2. Non-Culmination, Decomposition, and Two Relations

In this section, I take a closer look at failed attempt NCAs like ‘tear a thread’, ‘wake up a person’, and ‘open the door’ (1b)–(3b), on the one hand, and partial success NCAs like ‘dig up the vegetable patch’, ‘assemble a model’, and ‘cool down the water’ (9b)–(11b), on the other. Relying on Tatevosov and Ivanov (2009), I will build up a set of minimal assumptions necessary to account for the difference between the two classes of interpretations.

But before going there, I will briefly outline the theory of non-culmination assumed in what follows. This is what FA and PS accomplishments have in common.

2.1. Non-Culmination

As we have already seen, the non-culminating predicates in the (b) examples in (1)–(3) and (9)–(11) form a natural class: all of them accept measure adverbials and are atelic. Another observation is that they all present the same imperfective paradox as the progressive (Dowty (1977) and subsequent literature): propositions in the (b) examples can be true in the evaluation world without the corresponding proposition in the (a) examples being true. For instance, (1b) entails that there happens some agent’s activity, which, if culminates, would make a state described as ‘the thread is torn’ part of the evaluation world. Nevertheless, in (1b) the activity stops before the culmination has been reached.

This observation leads to what is called a partitive theory of non-culmination, which I assume in what follows: on a non-culminating reading, a part or a stage of an eventuality from the extension of a predicate occurs in the evaluation world, but the culmination does not.

I believe that nothing in what follows relies on any specific assumptions about the precise meaning of the partitive operator, call it PART, which maps a predicate of complete eventualities to a predicate of partial eventualities7. PART will be represented as in (18), without further commitments about its semantic content. Technically, (18) should be thought of as a function that takes an eventuality description P as an argument and returns a predicate of parts or stages of an event from its original extension.

(18)	Partitivity
	λP: P ∈ D<v,t>. λe: e ∈ Dv. PART(λe′. P(e′))(e)

PART, as represented by (18) is thus of the modifier type <<v,t>, <v,t>>. For the reasons that will become clearer shortly, I will assume proper partitivity in (19) as the only relevant condition on the meaning of PART:

(19)	Proper partitivity
	PART returns a predicate of proper non-final parts/stages of an eventuality from the extension of the event predicate it applies to.

A significant property of any predicate that denotes parts of some entity is strong homogeneity (e.g., Rothstein (2004)). Any proper part of a part of an entity is a part of that entity. Strongly homogeneous predicates are atelic. The partitive theory thus correctly predicts atelicity of NCAs.

On the morphosyntactic side, following Bar-el et al. (2005) and Tatevosov (2020), I assume that the source of partitivity, Part, comes on top of vP. PartP then merges with the perfective (PFV) creating a non-culminating perfective sentence8. I will be assuming that PFV minimally contains some or other variant of Klein’s (1994) semantics in (20), which says that the event time is a subinterval of the topic time9.

(20)	Klein’s perfectivity
	|| PFV || = λP: P ∈ D<v,t>.λt: t ∈ Di.∃e [P(e) ∧ τ(e) ⊆ t]
	where τ is the temporal trace function mapping events to their running times.

The full derivation of an NCA is shown in (21):

For [_vP John tear a thread] from (1b), for example, a derivation based on (21) outputs a proposition true of a world just in case there is a time that includes the running time of some non-final part of tearing a thread. This is the non-culminating meaning we are after.

I am assuming that the mechanism just outlined is responsible for the derivation of all non-culminating predicates introduced in Section 1, failed attempt and partial success alike. This position has been questioned by an anonymous reviewer who indicates that “the failed attempt and partial success readings are assumed to be due to a single constraint… But... there are quite a few differences between the two and…. languages can have partial success NCA readings but not failed attempt NCA readings, and possibly the reverse. So, I am not sure trying to model both kinds of NCA readings as due to a single constraint is the right move”.

I believe it is a viable analytical option to consider two distinct mechanisms for FA and PS readings. Here are a few reasons why I did not choose this option for Turkic NCAs.

First, I do not think that PS and FA readings are too different. Under the partitive view laid out above, both can receive a very simple treatment, elaborated on in the subsequent sections. For both PS and FA readings, PART extracts a non-final temporal part of a vP-eventuality. The difference has to do with whether this part only contains the agent’s activity or includes some change in the theme as well. In other words, the source of the PS/FA difference is to be found in the meaning of individual predicates at the vP level or lower, (21), not in the mechanism that comes on top of it.

Second, if the above reasoning is correct, it is not surprising that the PS/FA distinction does not have any consequences for the way the structure is spelled out. It has no morphological manifestations, not just in the languages like Turkic, where non-culminating and culminating variants of the same accomplishment are phonologically identical, but also in the languages where NCAs receive a designated spell-out (Tatevosov (2020)). In such languages, PS and FA derivations are not phonologically distinguished. If we are not dealing with “a single constraint”, this may look unexpected.

Third, imagine, to the contrary, that instead of (21), PS and FA readings are derived by two distinct mechanisms, call them M_PS and M_FA. If M_PS and M_FA are lexically unselective, and both apply freely, one would predict that any accomplishment can have both readings, contrary to the fact. If M_PS and M_FA are somehow restricted to the right classes of predicates, one would end up with a system where both M_PS and M_FA make reference to the properties identifying each class. The information which is already represented in the meaning of a predicate will be duplicated in the description of M_PS and M_FA. But the empirically identical result is achieved by (21) in a much simpler and more straightforward way.

Having made the PART and PFV components of (21) explicit, in the next section we will zoom into the denotation of vP.

2.2. A Neo-Davidsonian Representation

If tenseless and aspectless vPs are analyzed as predicates of eventualities, the denotation of the vP in (2a–b) can be first approximated in the neo-Davidsonian way as in (22).

(22)	‘Wake up Peter’: a predicate of complete eventualities
	λe.wake.up(e) ∧ agent(Vasja)(e) ∧ theme(Peter)(e)

Example (22) offers a non-decompositional analysis of ‘wake up a person’: a waking up event is regarded as an indivisible whole, from which PART would extract its proper non-final parts/stages. This is where a complication lies.

To see this, let us take a closer look at the failed attempt (FA) and partial success (PS) sentences. Whereas the FA sentences in (1b)–(3b) describe an event in which the theme retains its initial state, their PS counterparts in (9b)–(11b) convey that the theme undergoes at least some change. In other words, FA and PS accomplishments differ as to whether a process in the theme occurs in the evaluation world. They share two other characteristics: the agent’s activity does occur in the evaluation world, but the culmination does not. This is summarized in

Table 1. Culminating and non-culminating readings.

	Culminating	Non-culminating
		Partial Success	Failed Attempt
Agent’s activity	In the evaluation world	In the evaluation world	In the evaluation world
Process in the Theme	In the evaluation world	In the evaluation world	Not in the evaluation world
Culmination of the process	In the evaluation world	Not in the evaluation world	Not in the evaluation world

.

Intuitively, what makes failed attempts like ‘wake up a person’ in (2b) in Chuvash different from partially successful actions like ‘assemble a model’ in (10b) is how the agent’s activity is related to the change of state induced by that activity. PS accomplishments are construed in such a way that any (contextually relevant) part of the activity produces some change in the theme and, the other way around, any part of the change is brought about by some activity.

In contrast, in case of FA accomplishments non-final parts of the activity do not produce any change at all. If the waking up event in (2b) had culminated, the entire change of state would have occurred at the very final part of the activity. Since the activity stops before the culmination, at that point the theme is still in its initial state. The same observation extends to FA and PS accomplishments in Karachay–Balkar, (1b) and (9b), and Tuba Altai, (3b) and (11b).

Therefore, when PART extracts a non-final part/stage of an assembling eventuality, some change must have already happened to the theme. No change in the theme can occur in a non-final part/stage of a waking up eventuality, however.

The problem, then, is that this intuition cannot be captured within a non-decompositional representation like (22), since waking up and assembling events will inevitably be treated in the same way. Compare (22) with (23):

(23)	‘Assemble a model’: another predicate of complete eventualities
	λe.assemble(e) ∧ agent(Vasja)(e) ∧ theme(model)(e)]

Examples (22) and (23) cannot not impose any explicit restrictions on how the activity is related to the change of state. Given the parallelism between (22) and (23), we expect that if the event predicate in (22) has waking up events in its extension in which the activity immediately precedes the change of state, the same temporal constitution should be available for events in (23). Example (23) should describe assembling events in which the entire agent’s activity temporally precedes change of state of the model. In our world, this would not be a possible assembling-a-model event, but that does not fall out from (23), given that (23) and (22) are representationally identical up to the individual constants.

Therefore, since the temporal relations between the activity and change of state in (22)–(23) are not distinguished explicitly by the semantic representations, the distribution of subevental components between actual and non-actual worlds from Table 1 is impossible to capture. Non-decompositional representations do not provide us with enough subevental structure: activity and change of state are lumped together and represented as a single eventuality. Thus, a necessary condition for accounting for the difference between ‘wake up a person’ in (2b) and ‘assemble a model’ in (10b) is decomposing event predicates into more articulated event structures consisting of distinct subevental components. This would open up a way of making explicit that the activity subevent can occur in the evaluation world either with or without the change of state it brings about. A decompositional theory of accomplishment event structure will be outlined in the next section.

2.3. Predicate Decomposition

As a first try, let us assume that our predicates are represented as in (24)–(25), where a complex eventuality consists of the activity and change of state components, causally related10. The individual arguments of ‘wake up’ and ‘assemble’ are distributed between these components: the external argument appears as the agent of the activity and the internal argument as the theme of change of state.

(24)	‘Wake up’: two components
	λe.∃e′[wakeA(e) ∧ agent(Vasja)(e) ∧ wakeCS(e′) ∧ theme(peter)(e′) ∧ cause(e′)(e)]
	where wakeA denotes waking up activities and wakeCS events of getting awake.

(25)	‘Assemble’: two components
	λe.∃e′[assembleA(e) ∧ agent(Vasja)(e) ∧ assembleCS(e′) ∧ theme(thread)(e′) ∧ cause(e′)(e)]
	where assembleA denotes assembling activities and assembleCS events of getting assembled.

Eventualities in (24)–(25) are now composed of two elements: an activity performed by the agent (e) and a change of state of the theme (e′). However, it immediately becomes clear that (24)–(25) do not give us much. Apart from other problems with the causative analysis of accomplishments (e.g., Rothstein (2004, p. 104); Tatevosov and Ivanov (2009)), (24)–(25) present the same complication as before. Up to predicate constants, ‘wake up a person’ and ‘assemble a model’ denote the same event structure, and the source for their difference with respect to non-culminating interpretations is still not made explicit. Separating activity and process components of an eventuality is a necessary condition for accounting for the FA and PS interpretations, but is not a sufficient one.

To build up a theory of FA and PS accomplishments, one has to say more about the relation between the activity and change of state subevents. In (24)–(25), this relation is conceived of as that of (immediate) causation, which seems to be the most commonly accepted option in the literature from Dowty (1979) on. But cause is too weak for the difference between FAs and PSs to be made explicit. As Table 1 suggests, with ‘wake up a person’ the entire change of state occurs at the minimal final part of the activity. With ‘assemble a model’, every part of the change of state receives a temporally coextensive causal input from the activity. However, cause in (24)–(25) does not say anything about temporal relations between subevents, except that, trivially, the cause cannot occur after the effect.

To capture the FA and PS entailments, I propose that the causal relation in (24)–(25) can be further strengthened by two additional requirements, INCR (“incrementality”) and MMFP (“mapping to the minimal final part”), approximated as in (26a-b):

(26)	INCR and MMFP
	Any eventualities e and e′ are INCR-related iff there is a one-to-one mapping from the set of parts of e′ to the set of temporally co-extensive parts of e. Any eventualities e and e′ are MMFP-related iff the e′ is mapped to the minimal final part of e.

The incremental relation in (26a) is inspired by Rothstein (2004). Rothstein identifies a similar relation for a class of lexical accomplishments like ‘read a novel’ or ‘plow a field’. For instance, if one reads a novel, the novel undergoes change, and every part of the change requires an input of reading activity.

Rothstein’s INCR(emental) relation is defined relatively to the incremental chain that consists of parts of the become subevent such that any two elements of a chain stand in the part-of relation. The incremental relation involves a contextually salient function that establishes a one-to-one correspondence between parts of the incremental chain and parts of the activity.

Retaining the overall idea behind Rothstein’s definition, I modify it the following way. Incrementality is defined as a second order property of relations on events. This property holds of a relation R, (27), if it satisfies the three conditions in (28)–(30) that can be called mapping to subordinate subevents (MSbS), mapping to superordinate subevents (MSoS), and temporal co-extensiveness (TC)11:

(27)	Incrementality
	∀R[Incremental(R) ↔ MSoSE(R) ∧ MSbSE(R) ∧ TC(R)]

According to (27), any relation R is incremental iff it has the three properties in (28)–(30).

(28)	The relation R on events is a mapping to subordinate subevents, MSbS(R), iff
	∀e∀e′∀e″ [R(e′)(e) ∧ e″ < e → ∃e′′′ [ e′′′ < e′ ∧ R(e′′′)(e′′))]]

(29)	The relation R on events is a mapping to superordinate subevents, MSoS(R), iff
	∀e∀e′∀e″ [R(e′)(e) ∧ e″ < e′ → ∃e′′′ [ e′′′ < e ∧ R(e′′)(e′′′)]]

(30)	The relation R on events shows temporal co-extensiveness, TC(R), iff
	∀e∀e′ [R(e′)(e) → τ(e)= τ(e′)]]

Definition (28) says that a relation R shows mapping to subordinate subevents iff the following holds: whenever the two events e and e′ are in R, eRe′, any proper part of e stands in R to some proper part of e′. Definition (29) is the mirror image of (28): it requires for any proper part of e′ be R-related to some part of e. Definition (30) makes any R-related eventualities have identical running times.

According to (27)–(30), a relation R is incremental iff whenever it holds of two temporally co-extensive events, it also holds of their temporally co-extensive proper parts. From incrementality as a second order property, one can define INCR, a two-place relation on events. Two events are in INCR just in case they fall under a contextually available incremental relation:

(31)	INCR
	∀e∀e′ [ || INCR(e′)(e) ||C = 1 ↔ ∃R [RC(e′)(e) ∧ Incremental(R) ]

Making R dependent on the context would allow it to disregard irrelevant parts of related eventualities, just as under the original Rothstein’s proposal. However, nothing in what follows hinges on this refinement.

Given (27)–(31), ‘read’ and other classic incremental theme predicates come out incremental. They satisfy MSbS, (28), since any relevant proper part of a reading activity must bring about some change of state. Reversely, any proper part of the change in which the theme is being read must be induced by some reading activity, so ‘read’ also satisfies MSoS, (29). Finally, any related parts of the activity and change of state have to have the same running time, (30). ‘Open’ is not incremental since, as we have seen above, it at least does not satisfy MSbS: it allows opening activities to have proper parts that do not bring about any change.

It should be emphasized that Rothstein’s incrementality is distinct from Krifka’s incrementality Krifka (1992, 1998), which only comprises incremental theme predicates. The former is a property of relations between eventualities. The latter is a property of relations between eventualities and individuals. A predicate can therefore be Rothstein-incremental without being Krifka-incremental. ‘Push the cart into the garage’, a paradigmatic example of a predicate that lacks incremental theme entailments, is incremental in the sense of (27)–(30): it entails the incremental relation between a pushing activity and an event of change of location of the cart. INCR is thus characteristic not only of incremental theme predicates, but also of at least some of Krifka’s movements (Krifka 1998), comprising both predicates that describe movement in the physical space (‘push the cart into the garage’, ‘walk from the university to the capitol’) as well as movements in other dimensions (‘heat the water from 30 °C to 90 °C’). Note as well that ‘push the cart’ is not incremental, since one can push a (very heavy) cart without the cart moving along a path.

MMFP, mapping to the minimal final part, is defined in (32), where “MFP” and “FP” stand for “minimal final part” and “final part”, respectively.

(32)	Mapping to the Minimal Final Part
	a.	∀e∀e′ [ || MMFP(e′)(e) ||C = 1 ↔ ∃f [fC(e′)=MFP(e) ]
	b.	∀e∀e′ [ MFP(e)= e′ iff FP(e) = e′ ∧ ¬∃e″ [FP(e) = e″ ∧ e″ < e′]
	c.	∀e∀e′ [ FP(e) = e′ ↔ e′ ≤ e ∧ ¬∃e″ [e″ ≤ e ∧ e′ « e′′]
		where « is the precedence relation on events (Krifka 1998, p. 207)

Definition (32a) says that two events e and e′ are in MMFP just in case there is a contextually available function that assigns e′ to e as its minimal final part. Definitions (32b-c) define the notions of minimal final part and final part, respectively. As with INCR, in (32a) the key semantic ingredient is associated with the contextual function f_C rather than made part of the definition directly, since, intuitively, what counts as the minimal part of an eventuality can be context-dependent.

Due to MMFP, nothing in the change of state subevent is mapped to non-final parts of the activity subevent, and this is exactly what we need to capture the intuition that non-final parts of the activity do not induce a change.

Semantics in (24) and (25) can now be refined as in (33)–(34), where the causal relation is strengthened by INCR and MMPF, respectively:

(33)	‘Wake up’: two components, refined
	λe.∃e′[ wakeA(e) ∧ … ∧ wakeCS(e′) ∧ cause(e′)(e) ∧ mmfp(e′)(e)]

(34)	‘Assemble’: two components, refined
	λe.∃e′[ assembleA(e) ∧ … ∧ assembleCS(e′) ∧ cause(e′)(e) ∧ incr(e′)(e)]

Representations (33)–(34) involve a relation on (sub)events which is, set-theoretically, an intersection of cause and incr or cause and mmfp. ’Wake up’ in (33) is a predicate of waking up activities that bring about a change at the minimal final subevent. ’Assemble’ in (34) is a predicate of assembling activities that cause a change incrementally.

Essentially, the difference between the two types of accomplishments is reduced to the relation between activity and become subevents, exactly as the intuitive characterization above suggests. PS accomplishments involve Rothstein’s incremental relation, whereas FA accomplishments are constructed by mapping the whole change of state to the minimal final part of the activity12.

One more possible class of predicates, discussed in Tatevosov (2008), would be those that are lexically underspecified for mmfp/incr, which is evidenced by their compatibility with both FA and PS scenarios, as in (35)–(36):

(35)	Karachay-Balkar
	išci	eki	kün	üj-nü	oj-du.
	worker	two	day	house-acc	demolish-pfv.pst3sg
	‘The worker was involved in taking down the house for two days.’ (lit. ‘The worker took down the house for two days.’)

(36)	Scenarios for (35):
	Failed attempt: For two days, the worker was trying to take the house down, but the house was so firm that he gave up, not being able to remove a single brick. Partial success: For two days, the worker was taking down the house; he removed the roof and one of the walls, but then was asked to stop.

Given (35)–(36), one can suggest that the verb oj ‘take down, demolish’ does not specify whether the agent’s activity induces a temporally coextensive change in the theme. While at least some change of state is happening in (36b), (36a) shows that this does not have to be the case. The easiest way of analyzing such predicates would be to assume that they denote just the causal relation without further MMFP or INCR entailments, unlike the predicates in (33)–(34). If this is correct and verbs like oj ‘take down, demolish’ have indeed a weaker meaning than ‘wake up’ or ‘assemble’, one can assume that what has to be said about the former would extend to the latter without substantial qualifications13.

It should be further noted that predicates discussed throughout this paper are accomplishments based on morphologically simplex lexical verbs. It is in this environment that we normally find the general causal relation further strengthened by additional entailments. Turkic languages, including those discussed in this paper, also possess derived accomplishments illustrated in (37):

(37)	murat	butaq-nı	sın-dır-dı.
	M.	branch-acc	break.intr-caus- pfv.pst3sg
	‘Murat broke the branch.’

Accomplishments like (37), derived with the causative morpheme -dır-, typically pattern together with verb like oj ‘demolish, take down’ in being only specified for the general causal relation without further strengthening. Given the view advanced above, this comes as no surprise. The causative morpheme in Turkic can combine with any verb stem in a fully compositional manner (putting a few causative idioms aside). This suggests that it cannot come with too particular information about the way a caused eventuality is brought about, i.e., is not specified for incr or mmfp. The natural habitat of incr, mmfp, and other specific relations strengthening CAUSE, if any, are thus accomplishments lexicalized by a single lexical verb rather than causatives like (37).

Representations (33)–(34) are predicates that have complete waking-up and assembling eventualities in their extensions. Such eventualities occurring in the evaluation world are described by culminating variants of FA and PS accomplishments. The non-culminating ones, according to (21), are derived by applying the PART operator:

(38)	λe.PART(λe′. ∃e″ [ wakeA(e′) ∧ … ∧ wakeCS(e′′) ∧ cause(e′′)(e′) ∧ mmfp(e′′)(e′)])(e)

(39)	λe.PART(λe′. ∃e″ [assembleA(e′) ∧ … ∧ assembleCS(e′′) ∧ cause(e′′)(e′) ∧ incr(e′′)(e′)])(e)

The predicate in (38) denotes events that are proper non-final parts of complete waking up events. Due to MMFP, such events are activities that do not yield any change of state. In this way, the FA reading obtains.

If the INCR-accomplishment ‘assemble a model’ undergoes the same derivation, the predicate in (39) results, where a change of state subevent is incrementally related to the activity. The predicate in (39) denotes not fully developed assembling-the-model events, and any such event involves some change of state. This accounts for the PS interpretation of ‘assemble’ and other PS accomplishments.

In sum, the above analysis, a combination of a partitive theory of non-culmination and a decompositional theory of accomplishment event structure, if correct, offers a plausible account for both similarities and differences between FA and PS accomplishments. All non-culminating readings are uniformly derived by the same PART operator, which forces the culmination out of the evaluation world. Assuming different relations between subevents, INCR vs. MMFP, opens a way of capturing the distinct behavior of FA and PS predicates. It is these relations that are responsible for different distribution of subevental material between our world and the worlds where the culmination occurs, Table 1.

With this result, we are almost ready to return to the contrast between (1b)–(3b) and (9b)–(11b), on the one hand, and (12b)–(14b), on the other, and address the final, and the most difficult question: why are non-culminating interpretations (either FA or PS) are available for some accomplishments but not for all accomplishments? This will be the topic of Section 3 and Section 4. But before going there, I will make two notes on the issues taken up by anonymous reviewers. One of them discusses the related theoretical proposal of Koenig and Chief (2008), Section 2.4. The other addressees cross-linguistic variation outside of the Turkic family, Section 2.5.

2.4. Another View of Event Structure: Koenig and Chief 2008

Decompositional representations in (33)–(34) can be further refined, since for a proper semantic characterization of accomplishments one more ingredient may be relevant: a result state. Recognizing it as a separate meaning ingredient can thus lead to a three-way decomposition (see also fn. 10), involving an activity subevent, change of state/become subevent and a result state, as in Ramchand (2008) or Beavers and Koontz Garboden (2020), for example.

If result states are part of the accomplishment event structure, this may lead to a different perspective on incrementality and the lack thereof. Koenig and Chief (2008), following ideas circulating in the degree semantic literature (Hay et al. 1999; Kennedy and Levin 2002, 2008; Caudal and Nicolas 2005; Beavers 2008; among others) define incrementality in terms of a relationship between part structure of an eventuality and a degree structure of its result state, construed as (being equivalent to) a gradable property. Informally, a description of change is incremental if there is a one-to-one mapping between parts of events and degrees to which the result state gradable property holds at their final subintervals. Depending on the type of argument of a gradable property, incremental changes are further divided into object-part, path, and dimensional subtypes. Thus, for example, the more an entity is getting lengthened, the greater the degree of its length is (the dimensional subtype); the more an entity is being read, the greater the degree representing its part that has undergone reading is (the object-part subtype), and so on.

The lack of the incremental relation, so construed, between events and gradable properties results in non-incremental gradable changes, an example being ‘kill’. Finally, if a result state is not a gradable property at all, but a binary property, e.g., as in the case of ‘cast a vote’, a corresponding change of state description qualifies non-gradable.

This classification into non-gradable changes (‘cast a vote’), gradable non-incremental changes (‘kill’) and three types of incremental changes is intended to predict whether a predicate can be non-culminating, or, to use the original authors’ terminology, whether it shows the incompleteness effect. The article argues that only predicates that describe induced scalar changes (‘kill’, ‘write’, ‘wash’, ‘land’, but not ‘cast a vote’) allow for this effect. For the proposal it is crucial therefore that ‘kill’ and similar verbs form a natural class with incremental predicates but are kept distinct from predicates describing non-gradable changes.

The system outlined in Section 2.1, Section 2.2 and Section 2.3 resembles this proposal in two respects (putting representational differences aside). First, all Koenig and Chief’s descriptions of non-gradable changes would fall under MMFP-accomplishments. Second, all of their descriptions of incremental changes would qualify as INCR-accomplishments. It differs, however, in several other respects.

Most significantly, Koenig and Chief’s classification is intended to isolate predicates that allow for non-culmination (“incompleteness effect”), but the MMFP and INCR relations are not. The MMFP/INCR distinction is established to account for PS and FA readings of NCAs when an NCA construal is available; it is not designed to predict whether it is available.

The data in (1)–(3) and (9)–(11) suggest that both INCR and MMFP accomplishments can be NCAs. Examples like (14)–(16), where an NCA construal is unavailable, show that not all MMFP predicates allow for non-culminating readings. In Section 3.1 we will see examples demonstrating that being an INCR accomplishment is not a sufficient condition for non-culmination either. Therefore, whether an NCA derivation is available for an event predicate cuts across the INCR/MMFP distinction. The factor that separates NCAs in (1)–(3) and (9)–(11) from non-NCAs in (14)–(16) should be found elsewhere; this will be the topic of Section 3 and Section 4.

The other difference is that CAUSE, optionally strengthened by INCR or MMFP, as in (33)–(34), does not give rise to non-incremental gradable changes as a separate cell in the partition. ‘Kill’ and similar predicates would be classified as MMFP-accomplishments, where the transition from ‘alive’ to ‘dead’ occurs at the final part of the agent’s action. They would fall under the same class with non-gradable changes.

I believe this is not a weakness of the above analysis, since, crucially, Koenig and Chief’s classification into incremental gradable changes, non-incremental gradable changes, and non-gradable changes, aimed at accounting for the NCAhood in Mandarin and Thai (Koenig and Chief 2008), does not predict the Turkic pattern anyway. This is so because we find NCAs in each of these three Koenig and Chief’s classes.

Consider examples like (1)–(3) with ‘tear a thread’, ‘wake up a person’, or ‘open the door’ again. These examples convey that the thread stays intact, the patient asleep, and the door closed. ‘Wake up’ and ‘open’ would fall under Koenig and Chief’s gradable non-incremental changes, since awakeness and openess can manifest themselves to different degrees. But ‘tear a thread’ represents a binary change: a state of being torn can hardly be conceived of as being realized to different degrees. Finally, PS-accomplishments in (9)–(11) represent different types of Koenig and Chief’s incremental changes. These data, taken together, show that Koenig and Chief’s classification, designed for capturing NCAhood for Mandarin and Thai, cannot be used to isolate NCAs in Turkic14.

This is one of the main reasons why I believe that the Mandarin/Thai pattern and the Turkic pattern are genuinely distinct. Another reason is that the “incompleteness effect” in Mandarin and Thai comes with the inference that at least some change has occurred and the lack thereof in Turkic, as mentioned in Section 1. Therefore, one should not expect these patterns to be accounted for by the same theory.

I will address cross-linguistic variation in the realm of NCAs in more detail in the next section.

2.5. Implications for Wider Cross-Linguistic Variation

The derivation on NCAs outlined in Section 2.1 is intended to capture a fragment of the grammar of a language that does have NCAs to begin with. For a language that lacks NCAs altogether, this derivation should be made unavailable. The simplest way of achieving that would presumably be to restrict the distribution of PFV and make it come straight on top of vP, with no PART coming in between. Therefore, given the design of the non-culminating derivation in (21), one parameter of cross-linguistic variation can be formulated as in (40):

(40)	NCA vs. non-NCA languages
	In NCA languages PFV is allowed to merge outside of PART

However, in many languages, English among them, the picture is more complicated than (40). NCA readings are not completely unattested, but the borderline between available and unavailable NCAs is different from Turkic.

Consider the English examples in (41) from Tatevosov and Ivanov (2009), which are due to Rothstein, p.c.

(41)	a. Ali plowed the field for two hours (and then went home for lunch).
	b. I read this book for two weeks (before giving up half way through).
	c. I sewed this dress for two days.
	d. *I built the house for two weeks.
	e. *I opened the door for five minutes (and then gave up).
	f. *I turned the light off for two minutes.

This paradigm shows the two fragments of the NCA landscape where English is like Turkic. One is provided by the examples in (41a–c), where an NCA realization of an accomplishment predicate is allowed in English just as in Karachay–Balkar, Chuvash, and Tuba Altai. The other is (41f), where NCAs are equally impossible in English and Turkic, cf. (42f) and (15b) above. Finally, (41d–e) are examples of predicates that can allow for non-culmination in Turkic, but not in English. The set of environments where English licenses NCAs is thus a proper subset of environments where Turkic NCAs are available.

Does that mean that the INCR/MMFP distinction is relevant for one type of languages but not for the other? I believe the answer is no. As was discussed in the previous sections, whether a predicate is MMFP or INCR by itself makes no predictions as to whether it licenses an NCA derivation in (21). Rather, the prediction is: if a derivation of NCA is independently licensed, then with an INCR-accomplishment a PS-reading obtains, and MMFP accomplishments produce a FA-reading.

Moreover, neither in Turkic nor in English are the set of predicates that allow for the derivation of NCAs co-extensive with INCR or MMFP predicates. In English, a culmination is entailed for all MMFP predicates and some INCR predicates like ‘build a house’. In Turkic, the line is drawn within MMFP predicates, separating, e.g., ‘open the door’ and ‘turn the light off’ (but see Section 3.1 for a number of qualifications about INCR predicates).

In Mandarin and Thai, discussed in Koenig and Chief 2008 (see Section 2.4), the picture is by no means less complex. Moreover, it does not seem to be easily reducible to what is observed in Turkic or English.

Some of the culminating and non-culminating predicates in these languages are reproduced in (42)–(43) (Koenig and Chief 2008, p. 249):

(42)	a.	Mandarin: incompleteness effect allowed
		‘cut’, ‘repair’, ‘persuade’, ‘kill’, ‘close’, ‘read’, ‘eat’, ‘dry (clothes)’, ‘wash’,	‘cook’
	b.	Mandarin: incompleteness effect not allowed
		‘turn’, ‘deep fry’, ‘pickle’, ‘bake’, ‘pay’, ‘soak/immerse in liquid’

(43)	a.	Thai: incompleteness effect allowed
		‘read’, ‘kill’, ‘eat’, ‘write’, ‘cut’, ‘open’, ‘repair’
	b.	Thai: incompleteness effect not allowed
		‘pay’, ‘hire’, ‘distribute’, ‘vote’

Koenig and Chief argue (see the discussion in Section 2.4) that for the incompleteness effect to obtain the result state has to be a gradable property. But gradability can hardly be necessary for NCAs in Turkic, since the latter are found among gradable and non-gradable predicates alike. Neither does it seem to be sufficient: at least some of Koenig and Chief’s gradable predicates do not accept non-culminating interpretations. Thus, ‘kill’, representing the gradable non-incremental type, in the Turkic languages under discussion is not an NCA, unlike in Mandarin in Thai15. This is illustrated in (44a–c):

(44)	a.	Karachay–Balkar
		*Marat	eki	minut	Kerim-ni	öltür-dü.
		M.	two	minute	K.-acc	kill-pfv.pst.3sg
		‘Marat spent two minutes killing Kerim.’
	b.	Chuvash
		*Vaɕa	ike	minut	Petuk-na	vəler-tɕ-ə.
		V.	two	minute	P.-acc	kill-pfv-pst
		‘Vasja spent two minutes killing Peter.’
	c.	Tuba Altai
		*Amɨr	eki	minut	Baːtɨr -nɨ	öltür-di.
		A.	two	minute	B.-acc	kill-pfv.pst3sg
		‘Amyr spent two minutes killing Batyr.’

Moreover, Koenig and Chief themselves admit that gradability cannot provide a complete theory of incompleteness, since the requirement that the result state be a gradable property seems to be a necessary, but not a sufficient condition even in Mandarin and Thai. For example, among the predicates that do not show incompleteness effect in (42b) and (43b) one finds ‘deep fry’, ‘pickle’, ‘bake’, and ‘distribute’, which intuitively look much like many other gradable items16.

What these data seem to suggest is that English and Turkic, where the distribution of NCAs stand in the subset–superset relation, may share the overall way of deriving NCAs, e.g., its specific variant in (21). Particular restrictions on the derivation, however, are bound to be subject to variation. Whether the Mandarin and Thai systems, where sets of predicates that allow for non-culminating readings seem to cross-cut those in Turkic and English, can be subject to the same analysis requires further examination.

Therefore, I do not claim that the view outlined in this paper should account for all NCAs cross-linguistically. Nor do I assume, given the diversity of non-culminating phenomena, that such an account must exist. However, I would like to emphasize that the pattern discussed above is by no means unique for the Turkic family. It replicates at least in Slavic languages, as discussed extensively in Tatevosov (2020) on Russian material. Example (45) shows a parallel example from Polish, kindly provided by a reviewer:

(45)	Jan	po-czyta-ł	książkę	#w /	przez	godzinę
	J.	prf-read.ipfv-pst	book.acc	in	for	hour
	(i	potem	po-szed-ł	spa-ć).
	and	then	prf-go-pst	sleep-inf
	‘Jan read a book #in / for one hour and went to bed afterwards.’

In (45), a so-called delimitative verb occurs, which is derived by merging the prefix po- with a morphologically imperfective verb stem; see also the Russian examples in fn. 14.

Tatevosov (2020) argues extensively that despite involving more overt morphology, the delimitative in Slavic resembles Turkic NCAs both derivationally and interpretationally. First, like Turkic NCAs, (45) is perfective and atelic. Second, the delimitative shows parallelism to Turkic NCAs in featuring both FA and PS interpretations. Example (45), based on an incremental theme verb, is an example of the latter. When the delimitative comes on top of an MMFP description like ‘open the door’, an FA reading parallel to Turkic (1)–(3) obtains. (Admittedly, this latter option is more readily available for the delimitative in Russian than in Polish). Third, the restrictions on the distribution of NCAs discussed in more detail in subsequent sections are the same in both Turkic and Russian.

To conclude, in what follows, I will try to come up with a theory of why the space of possibilities is divided in a particular way in Turkic. I will not try to explicitly extend my observations and generalizations to English and similar languages. A few speculations on this topic can be found in Tatevosov and Ivanov (2009). Nor will I attempt to come up with generalizations of how the Mandarin/Thai system is different from the Turkic one. I believe that a full-fledged typological study is needed for deeper cross-linguistic generalizations that has to determine the entire set of parameters of variation in the NCA domain as well as regular restrictions on this variation17.

3. Constraints on Non-Culmination

3.1. A Few Observations

So far we have seen that if an accomplishment predicate allows for a non-culminating interpretation, then a plausible analysis may involve something like the PART operator extracting proper non-final parts of an event from the extension of a predicate. However, (14)–(16) show that the culmination is entailed for accomplishments like ‘set up a tent’, ‘turn the light off,’ or ‘bury a person’. Other predicates of the same type that occur across different Turkic languages are, for example, ‘put a shirt on’, ‘take a pill’, ‘give out a book’ (e.g., in a library), and many others. Example (12b) is repeated as (46):

(46)	Karachay–Balkar
	??marat	beš	minut	šatır-nı	quru-du.
	M.	two	minute	tent-acc	set.up-pfv.pst3sg
	‘Marat tried to set up a tent for five minutes.’

Intuitively, ‘set up a tent’ in (46) resembles MMFP accomplishments discussed so far (e.g., ‘tear a thread’ and ‘wake up a person’ in (1b)-(2b)) in that the change of state only occurs at a final part of the activity. But rather than yielding an FA-interpretation where non-final parts of a setting-up-a-tent event are described, such predicates produce no non-culminating reading at all. Why?

Before we start approaching possible answers, two more observations reported in Tatevosov (2020) are worth taking into account.

One observation is that not only may MMPF accomplishments fail to derive a non-culminating reading, but the same can happen to INCR accomplishments, even though in a different way.

In (46), the unavailability of a non-culminating reading does not seem to depend on any linguistic material other than the verb itself. In (47), in contrast, the acceptability of a non-culminating reading of the INCR accomplishment ‘read’ varies with the choice of an incremental internal argument:

(47)	Karachay–Balkar
	alim	ike	minut	/	sekunt	OKroman-nı /	OKstatja-nı /
	A.	two	minute		second	novel-acc	article-acc
	OK/?qaʁıt-nı	/	? qaʁıtcıq-nı	/	?/??abzac-nı /	??ajtım-nı /
	letter-acc		note-acc		paragraph-acc	sentence-acc
	???söz-nü	/	*xarf-nı		oqu-du.
	word-acc		symbol-acc		read-pfv.pst.3sg
	‘Alim spent two minutes/seconds reading a novel/article/letter/note/paragraph/sentence/word/symbol.’

As (47) suggests, the larger the spatial extent of the internal argument is, the easier it is for an activity to stop before it has been read. The limiting case is ‘read a symbol’, which strictly entails culmination. The pattern is easily replicated with other incremental theme verbs like ‘dig up’ in (9b), ‘write’, ‘draw’, ‘sing’, ‘plow’, and so on.

The other observation is: in (46), the unavailability of a non-culminating interpretation is determined by the lexical meaning of the verb and not dependent on a context. For many other accomplishment predicates, their non-culminating potential is restricted contextually. Consider ‘open the door’ from (3b) repeated as (48), where it is supplied with two specific contexts:

(48)	Tuba Altai
	Scenario 1. The lock in the door is broken. The agent tries to open the door with the key, then applies a picklock, then uses a crowbar, then tries to disassemble the lock, etc. At some point, he gives up. (In fact, the door can be opened by hitting it with a sledgehammer, but the agent does not have a sledgehammer.)
	*Scenario 2. The door is opened by typing a digital code that consists of a sequence of numbers, e.g., 2-5-9-6. After typing “2” and “5”, the agent stops.
	amɨr	eki	minut	eʒik-ti	ač-tɨ.
	A.	two	minute	door-acc	open-pfv.pst3sg
	‘Amyr spent two minutes opening the door.’

What defines a culmination in (48) in both scenarios is the door attaining a state of having any non-zero degree of openness. In the first scenario, the activity can stop without the door entering this state. In the second scenario, this cannot be the case. Therefore, the availability of a non-culminating reading seems to be dependent on what kind of opening activity is entailed or presupposed by the context. Whereas opening the door by applying different tools to a broken lock can stop before the change of state occurs (Scenario 1), typing a digital code has to culminate (Scenario 2)18.

In the next section I will review an account for similar restrictions in Russian proposed in Tatevosov (2020) and point out that there is a class of examples where this account does not seem to make correct predictions19.

3.2. An Attempt at an Explanation

The existing analyses of NCAs (Koenig and Muansuwan (2001); Bar-el et al. (2005); Tatevosov (2008); Koenig and Chief (2008), Tatevosov and Ivanov (2009); Martin and Schäfer (2012, 2017); Altshuler (2013), Altshuler (2014); Demirdache and Martin (2015); Martin (2015)) are preoccupied with what happens when a non-culminating interpretation is available. The question of what is going on in examples like (14)–(16) and (46)–(48), where such an interpretation is inappropriate, and how they are different from examples like (1)–(3) and (9)–(11) does not seem to have been much discussed.

Apart from Koenig and Chief (2008), cited in Section 2.4, one specific proposal is offered in Tatevosov (2020). Tatevosov builds on the intuition that the difference between NCAs and accomplishments that entail culmination is to be found in the characteristics of the activity component of a complex eventuality description embedded under PART. The predicate in (49) is a partial semantic representation of ‘open the door’ parallel to (33)–(34):

(49)	λe.∃e′[ openA(e) ∧ … ∧ openCS(e′) ∧ cause(e′)(e) ∧ mmfp(e′)(e)]

In (50), the relevant activity component e falls under open_A and brings about an opening e′ of the theme at its minimal final part.

Intuitively, in case of accomplishments that do entail culmination, (14)–(16) and (46)–(48), this component must be structured in a specific way, as described in (50):

(50)	Activity component of accomplishments: the temporal arrangement hypothesis
	For NCA-negative predicates, contextually salient subevents that are parts of the activity component of an eventuality description show unique temporal arrangement (UTA).

Consider (48) again in Scenario 2, where the door is opened by typing a digital code and a non-culminating reading is not available. ‘Open the door’ is an MMFP accomplishment: the door changes its state from ‘closed’ to ‘open’ at the very final part of the activity, right after the final number in the sequence is typed. In Scenario 2, the activity component, e_P, consists of four subevents, (51a–b), arranged in a specific order in (51c):

(51)	a.	e2: typing of 2	b.	eP= e2 ⊕ e5 ⊕ e6 ⊕ e9
		e5: typing of 5
		e6: typing of 6	c.	e2 «T e5 «T e6 «T e9
		e9: typing of 9

If the subevents occur in a different order or some of them are skipped over or occur more than once, their sum is no longer an activity that opens the door. This is shown in (52), where the green item represents an activity that falls under || open the door || in this scenario, and red items are examples of activities that do not.

(52)

In other words, in a context where e_P opens the door (and nothing else opens the door), any activity composed of typing numbers can only be in the extension of || open the door || if it is identical to e_P. This is what Tatevosov (2020) calls unique temporal arrangement: for an eventuality to be a process component of predicate it has to start in a specific way, to finish in a specific way, and for any of its non-final parts there must a specific follow-up.

Now consider the same sentence under Scenario 1, where the relevant subevents of the activity look like (53):

(53)	ekey = using a key
	epicklock = using a picklock
	ecrowbar = using a crowbar
	esledgehammer = using a sledgehammer

In this scenario, there is no unique arrangement of subevents into an activity. As long as a sledgehammer opens the door, all that matters is that e_sledgehammer is the final subevent in a process. Other subevents can be absent or occur in whatever order, since they make no causal contribution to opening of the door at all. The lack of causal dependence of the change of state on the non-final subevents tells Scenarios 1 and 2 apart: as an anonymous reviewer insightfully points out, in (51) all subevents are necessary for an activity to open the door (see also Tatevosov and Ivanov (2009)). In (54), e′′, e′′′, e′′′′′, e′′′′′′ all are activities that open the door, so all of them can form a process component of the denotation of ‘open the door’.

(54)

The activity component of (54) thus fails to have the property of unique temporal arrangement.

I will not go into further technical detail here; the informal outline of the idea will suffice for our purposes20.

Tatevosov (2020) suggests that the unique temporal arrangement of the activity component is a sufficient condition for the inability of a predicate to derive a non-culminating interpretation. All UTA predicates are NCA-negative. The material discussed so far seems to conform to this generalization. If a predicate is characterized by UTA, neither FA nor PS readings are available; the culmination is entailed. Moreover, this material indicates that UTA may also be a necessary condition: all the above accomplishments that entail culmination (‘set up a tent’, ‘turn the light off’, etc.) show UTA.

In (48), in Scenario 2, the UTA character of the activity is contextually entailed; the set of eventualities in the extension of ‘open’ is restricted to those that match the scenario. Events in this restricted set are arranged in a unique way, as we have just seen. Under a different context, e.g., Context 1, no unique arrangement of subevents into an opening activity is entailed: in this scenario, one can arrange opening the door activities in various orders, repeat or skip their non-final parts, etc.

The UTA character of the predicates like ‘set up a tent’ and ‘turn the light off’ in (14)–(16) seems to be fully determined by the lexical material they are composed of. Events described by these predicates consist of very specific subevents that have to occur in a very specific order. For example, setting up a tent requires laying down a tarp on the ground, laying the tent onto the tarp, connecting the tent poles, inserting the poles into the flaps, raising the tent, and so on. The same is true of activities referred to by ‘turn the light off’ and ‘bury a person’. They are sequences of actions such that if they are performed in a wrong temporal order or some of them are skipped over, the overall sequence does not count as a turning-the-light-off or burying-a-person activity anymore.

The pattern in (47) can be analyzed in the UTA perspective as follows. If one reads a novel, nothing imposes any specific temporal ordering on the reading process except for the structure of the novel itself. Even though any novel is organized into a linear order, it allows the reader substantial flexibility: one can go back and forth, skip some parts and re-read others. This leaves space for the reading subevents to be arranged in a non-unique way, which ensures that the predicate will be UTA-negative. But the smaller the size of an argument is, the more difficult it is to partition an event e in different ways (see Rothstein (2004, pp. 111–12) for related observations). For example, one does not normally conceive of a sentence of an average length as consisting of parts that can be read in various orders (even though some sentences in À la recherche du temps perdu possibly can). As we reach the limiting case, reading a symbol, the activity cannot be properly partitioned into subevents in any admissible context at all. If this happens, the predicate comes out as trivially having the UTA property, which predicts, correctly, the lack of a non-culminating reading.

If this reasoning is correct, it may be taken to support UTA: non-culminating accomplishments are constrained by whether the activity element of an accomplishment eventuality description is characterized by unique temporal arrangement.

In the next section, however, we will see two cases problematic for a UTA account. One class of examples reveals that the lack of UTA is not sufficient for NCAhood, hence an analysis solely based on UTA is incomplete. The other class casts doubt on the generalization that being UTA-negative is a necessary condition. While each of the problems can be fixed with an additional stipulation, together they point towards the conclusion that UTA is only a way to go as long as there is no better alternative.

3.3. Problems for Unique Temporal Arrangement

In this section we will see that upon closer scrutiny, UTA and NCA predicates are not co-extensive. One can find both

◾

UTA-negative predicates that do not allow for non-culminating readings, and

◾

UTA-positive predicates that do.

Let us start with the former, a class of non-UTA predicates that fail to accept a non-culminating interpretation. Consider (55):

(55)	Tuba Altai
	*Scenario 3. The door is opened by typing a sequence of any five numbers pressed in whatever order. The agent slowly types “2”, “9” and “7” and stops.
	amɨr	eki	minut	eʒik-ti	ač-tɨ.
	a.	two	minute	door-acc	open-pfv.pst3sg
	‘Amyr spent two minutes opening the door.’

In this scenario, no unique temporal arrangement is entailed or presupposed, but a non-culminating reading is nevertheless unavailable.

Similar scenarios can be replicated for many other MMFP accomplishments, especially those where the activity subevent does not come with much descriptive content. Such accomplishments can easily admit contexts where a change of state (‘waking up’, ‘tearing a thread’, etc.) is brought about by a machine after pressing random buttons several times in a random order. The speakers’ judgments seem to be very clear: in such scenarios, a non-culminating reading is not an option.

Therefore, (55) is a non-UTA predicate, which is nevertheless obligatorily culminating; the lack of UTA does not suffice for licensing a non-culminating interpretation.

No less problematic are cases of the other type, where UTA is contextually entailed, but an NCA reading is nevertheless available. Consider (56):

(56)	Chuvash
	Context. The agent is assembling an IKEA wardrobe using the assembly instruction.
	?/OKvaɕa	ɕirɘm	minut	ʃkap-a	puɕtar-tɕ-ɘ.
	V.	twenty	minute	wardrobe-acc	assemble-pfv-pst
	‘Vasja spent twenty minutes assembling a wardrobe.’

Above, in (10), we have seen that ‘assemble a model’ is an NCA. Moreover, ‘assemble a model’ does not show UTA: under normal circumstances, there typically is more than one arrangement of the subevents that add up to an assembling activity.

Example (56) is minimally different in that a process of assembling IKEA furniture is linearly ordered by the assembly instruction. Moreover, it is common ground that if the linear order is not followed, an activity does not normally culminate into a piece of furniture getting successfully assembled. In the scenario in (56), therefore, UTA seems to be contextually entailed in much the same way as in (48) with ‘open the door’ on the digital code scenario. One would expect that an NCA interpretation should be degraded for the same reason as in (48), but this does not happen.

This is even more surprising in light of the observation that ‘assemble a wardrobe’ shows substantial parallelism to ‘set up a tent’ in (46). As an anonymous reviewer points out, “both predicates refer to a gradual creation of an object”, and “in both cases, there are preparatory stages that do not involve the object directly but are necessary for the process to reach its end”. Yet, (56) is much easier to get than (46).

Again, the same effect can be replicated with a lot of other accomplishments by forcing them into similar scenarios where an activity is temporally ordered by an instruction.

One more relevant fact about the examples like (55) and (56) is that the deviance of (55) as well as the acceptability of (56) can be manipulated by changing the length of a chain of subevents that form the activity component of a description.

If we keep the overall setting of the Scenario 3 the same, but increase the number of subevents the activity consists of, the sentence improves. According to the native speakers’ judgments, the non-culminating reading of ‘open the door’ is acceptable under the Scenario 4.

(57)	OK/?Scenario 4. The door is opened by typing a sequence of any 50 numbers in whatever order. After typing the first 15 numbers, the agent stops.

In case of (56), the other way around, replacing a wardrobe with a stool, which only needs driving a few screws to be assembled, leads to a substantial decrease in acceptability:

(58)	Chuvash
	Context: One has to drive three screws in order to assemble a three-legged stool. After driving the second screw, the agent stops.
	??vaɕa	viɕɘ	minut	pukan-a	puɕtar-tɕ-ɘ.
	V.	three	minute	stool-acc	assemble-pfv-pst
	‘Vasja spent three minutes assembling a stool.’

Therefore, it looks like what matters in both cases is a relative complexity of the activity: the more subevents it is composed of, the easier it is to access a non-culminating reading. This seems to suggest that one can fix a UTA analysis by introducing an additional requirement:

(59)	Threshold condition (TC) on NCAs
	The number of contextually salient subevents in the activity component of an accomplishment eventuality description has to be above a certain contextually determined threshold.

The condition (59) sets up the lower limit on the number of subevents in the activity component. Example (55) comes out highly degraded because this number falls below the limit. Increasing the amount of subevents in (57) makes the sentence improve.

The difference in acceptability between (56) and (58) can be attributed to the same effect. Assembling a wardrobe is a considerably more complex activity with many more subevents than assembling a stool. As such, the former satisfies the threshold condition, but the latter does not.

However, assuming that TC is operational alongside with UTA does not give us a lot of empirical advantages. One still needs a theory of why the amount of subevents representing the relative complexity of the activity makes a difference. In addition, a question is bound to arise about how UTA and TC conditions interact. For the vast majority of accomplishments, UTA alone predicts the availability of non-culminating reading, as discussed in Section 3.1 and Section 3.2, so the role of TC is such semantic derivations is not clear. The other way around, in cases like (55)–(58), the workings of TC do not seem to be sensitive to whether a predicate is UTA-positive. For both UTA and non-UTA predicates alike, (55) and (56), the increasing complexity of an activity subevent facilitates a non-culminating reading. Needless to say, unless the relationship between UTA and TC is made entirely clear, empirical predictions of a theory of NCAs that relies on both conditions are very difficult to assess.

For this reason, I will explore an alternative in the next section. I will argue that the availability of non-culminating readings can be connected to what I will call distance to the culmination.

4. Distance to the Culmination

4.1. A Few Intuitions

The temporal theory of NCAs outlined above posits that the lack of unique temporal arrangement of the activity component, UTA, is a characteristic property of NCAs. In addition, the discussion in Section 3.3, with all the reservations mentioned there, suggests that the amount of subevents in the activity component is of relevance, as well, which gives rise to the threshold condition, TC, in (59). There is no obvious implicational relation between these two characteristics: both UTA-positive and UTA-negative predicates show TC-effects. However, even if UTA and TC cannot be reduced to each other, I believe it is possible to view both of them as manifestations of a deeper semantic generalization.

Informally, this generalization, equidistance to the culmination (ED), is stated in (60):

(60)	Equidistance to the culmination (informal, first version)
	NCAs describe a proper non-final part e of an activity component of an accomplishment event description such that the initial and final bounds of e are equidistant to the culmination.

Let me introduce the intuition behind ED through the broken lock example again. It is repeated as (61).

(61)	Tuba Altai
	Scenario 1. The lock in the door is broken. The agent tries to open the door with the key, then applies a picklock, then uses a crowbar, then tries to disassemble the lock, etc. At some point, he gives up.
	amɨr	eki	minut	eʒik-ti	ač-tɨ.
	A.	two	minute	door-acc	open-pfv.pst3sg
	‘Amyr spent two minutes opening the door.’

Example (61) describes an eventuality e that occurs in the evaluation world. This eventuality is a non-final part of a complete activity that opens the door. In the given scenario, it consists of a number of contextually salient subevents listed in (53) above. The crucial fact about e is that at the end of it the agent has not made any progress as to having the door opened. At the point where her activity stops, the change of state is as far from being realized in the evaluation world as at the point where the failed attempt starts. By the end of e, the evaluation world does not start looking more like the worlds, whatever they are, where e culminates.

The digital code scenario is different. Intuitively, every next number typed correctly on the digital keypad lock makes the culmination closer to realization. In (62), the more subevents from (51) occur in the actual world, the closer the moment is where the door opens.

(62)	*Scenario 2. The door is opened by typing a digital code that consists of a sequence of numbers, e.g., 2-5-9-6. After typing “2” and “5”, the agent stops.

This is the intuition behind the idea of relative distance of the initial and final bounds of a preculminating part of an eventuality to the culmination. (60) says that for NCAs this distance is the same. This condition is satisfied in (61), but not in (62), hence the contrast in acceptability.

For (60) to be a theory of NCAs, one has to make the notion of (equi)distance to the culmination formally precise. But before attempting that, let me explore empirical expectations derivable from (60) a bit further.

The obvious question to ask is how the idea of the culmination not coming closer to realization in the evaluation world is related to UTA. Generalizing over the digital code scenario, it seems uncontroversial to say that if an activity consists of subevents temporally arranged in a unique way, then with every subevent unfolding, the culmination approaches; the evaluation world looks more and more like one of the culminating worlds. With UTA, the initial and final bounds of an eventuality cannot be equidistant to the culmination. UTA is thus sufficient for ED.

However, it is not necessary. It is not the case that the culmination only comes closer to realization if the subevents show a unique temporal arrangement. It is exactly the scenario in (55), repeated as (63), where the culmination is coming closer with every subevent without there being a unique temporal arrangement.

(63)	*Scenario 3. The door is opened by typing a sequence of any five numbers in whatever order. The agent types “2” and “7” and stops.

If one has to push random five buttons to open the door, then, metaphorically, at the very beginning the culmination is five pushes away from the evaluation world. After two pushes, the distance goes down to three pushes.

UTA, therefore, asymmetrically entails ED.

The next question is: does ED not look very wrong when applied to the predicates that show the partial success non-culminating reading? One class of such predicates are incremental accomplishments like ‘dig up a vegetable patch’, ‘assemble a model’, ‘read a book’, or ‘write a letter’, discussed in Section 1 and Section 2 (see (9)–(11) and (47)). For incremental predicates like ‘read’, the problem with ED in (60) is that they show mapping to subobjects: the more an eventuality extends in time, the larger part of the incremental theme is involved. The culmination is defined by the spatial extent of the theme: as soon as the entire book is involved in reading, a reading eventuality culminates. Therefore, under normal circumstances, any part of reading a book occurring in the evaluation world makes the culmination closer to realization, just by virtue of the fact that at the end of any reading eventuality more is read than at the beginning. The same holds for any incremental theme predicate, of course, as well as for other types of incremental predicates including incremental path predicates, degree achievements and the like. On the partial success reading, such non-culminating accomplishments entail that some change has occurred.

ED thus needs to be amended. The amendment I propose appears in (64):

(64)	(Equi)distance to the culmination (informal, amended version)
	NCAs describe a proper non-final part e of the activity component of an accomplishment event description such that the difference in the distance to the culmination between the initial and final bounds of e is insignificant in the current context.

The amendment in (64) is based on the following intuition: If the subject reads a novel, then in the course of a reading event the volume of what has been read is, of course, increasing. But when described by an NCA, this increase makes very little difference for the culmination, reading the entire novel, to become part of the actual world. Despite the fact that the initial and final bounds of the non-culminating reading are not literally equidistant to the culmination, the change in the distance is small enough to be disregarded.

Similarly, non-culminating sentences based on other incremental accomplishments (‘dig up a vegetable patch’, ‘assemble a model’, and so on) convey that while some digging up or assembling activities occur, by all contextually relevant criteria their contribution to making complete digging up and assembling events part of the actual world is negligible.

This view allows to make sense of the restriction illustrated in (47), where the acceptability of an incremental NCA co-varies with the size of the incremental theme, in the following way. The less extended an object being read is, the more difficult it is to come up with a context where a portion of reading activity makes no substantial difference for the culmination to come closer to realization. The contribution of reading 30 pages of À la recherche du temps perdu into reading the entire text is not substantial. With 30 pages out of more than 3000, it is relatively easy to set up a context where the initial and final bounds of reading count as equidistant to the culmination for all relevant purposes. But reading two sentences in a five-sentence note designates a significant advance on the way to the culmination, and a context compatible with equidistance is much more difficult to accommodate. As before, reading a symbol is a limiting case, since it does not have identifiable proper parts, let alone proper parts insignificant from the point of view of the culmination.

From the perspective of the amended version of the equidistance condition in (64), the TC effects discussed in Section 3.3 can be treated in a parallel way. In (57), repeated as (65), the actual agent’s activity consists of a few pushes out of very many:

(65)	OK/?Scenario 4. The door is opened by typing a sequence of any 50 numbers in whatever order. After typing the first 15 numbers, the agent stops.

Again, this activity does not make the distance to the culmination significantly smaller. In this respect, (65) contrasts with (63), where it is much harder to accommodate that two pushes out of four occur without the culmination coming closer to realization. The same reasoning extends to the assembling scenarios in (56) and (58). In (56), 20 min of assembling the wardrobe are compatible with there being no substantial progress in having the wardrobe fully assembled. In (58), upon driving two screws out of three, it would be very difficult to come up with a context that supports a parallel inference.

In a similar way, the contrast between (56), ‘assemble a wardrobe’, and (46), ‘set up a tent’, can be understood. As we have just seen, under appropriate circumstances, assembling a wardrobe can be construed as a complex activity consisting of a lot of subevents that occupy an extended interval. It would not be difficult to find a context in which an NCA sentence describes an interval where no easily identifiable progress is made as to achieving the culmination. Temporal bounds of this activity will then count as contextually equidistant to the culmination.

In contrast, setting up a tent, under normal circumstances, involves just a few conventional operations. As a result, it is extremely difficult, in any context, to pick up a part of the activity where the culmination is assessed as not being closer to realization at the final bound compared to the initial bound. (I am grateful to an anonymous reviewer who encouraged me to reflect on the difference between ‘assemble a wardrobe’ and ‘set up a tent’.)

Generalizing somewhat, one can say that what matters here is a relative “size” of a complete, culminating activity and its actualized temporal part occurring in the evaluation world under an NCA description. The smaller this ratio is, the easier it is to find a context where the change occurring in the course of an activity is disregardable in terms of the distance to the culmination.

It should be emphasized that whereas UTA effects and TC effects discussed above can hardly be reduced to each other, both fall out of the equidistance condition. Under normal circumstances a unique temporal arrangement of subevents of the activity component is incompatible with the equidistance. But longer chains of subevents make it easier to accommodate an equidistance context.

In (64), dependence of the equidistance on the context predicts that what counts as an (in)significant advance towards the culmination can be determined not only by the relative size of an activity and its actualized temporal part, but by other, sometimes very non-trivial factors as well. For instance, NCAs like ‘read a paragraph for an hour’ or ‘read a sentence for 10 minutes’ that sound odd out of the blue, improve in a context entailing that reading requires more effort than normally. If the text is written in a language not very familiar to the agent, esoteric, or merely not legible, it is entirely possible that one can make little or no visible progress after an hour of reading. This is what happens in (66), for example, given that it is common ground that reading the Phaistos disk has never culminated in our world.

(66)	Karachay–Balkar
	alim	ike	saʁat	fest-ni	disk-ni	jüs-ü-ne
	k.	two	hour	phaistos-gen	disc-gen	surface-3-dat
	zaz-ıl-ʁan-ı-n			oqu-du.
	write-pass-pfct-3-acc			read-pfv.pst3sg
	‘Alim spent two hours reading the inscription on the Phaistos disk.’

Another type of contextual variability can have to do with the agent’s teleological perspective. For instance, unlike reading an entire novel, reading an entire newspaper is not typically the agent’s goal. Imagine a reader who normally reads “Politics”, “Opinions”, “Science”, and “Arts” and always skips “Health” and “Sports”. This morning, he reads “Politics” and “Opinions” and stops. This makes a non-culminating sentence like ‘He read the newspaper for 15 minutes’ true, with the chance for a reading event to culminate being exactly the same at the beginning and at the end: around zero. In general, an agent whose intention is not to read the text to the end can interrupt reading at any point, and at any point the likelihood of a culmination would be more or less the same—very small21.

The above discussion emphasizes the significance of contextual inferences in determining if an NCA interpretation is available. An anonymous reviewer points out that there is a more general question: can lexical constrains based on compositionality be changed with an appropriated context, or, in other words, can we pragmatically alter the semantic properties of the predicates and, if so, when?

I believe that nothing that has been said so far forces us to assume that this should be the case.

On the current view, the context can interact with NCAs in two ways. First, it can restrict the extension of a predicate to a set of relevant eventualities. Second, as has been discussed above, it is relevant for deciding whether distance to the culmination of the initial and final bounds of an eventuality is significant.

I will say more about the second point in the next section. At the moment, let me briefly address the first point.

Recall that non-culminating predicates like (38)–(39), repeated as (67)–(68), describe non-final parts of activities that bring about a change of state. In (67)–(68), these activities fall under the denotations of λe. wake_A(e) and λe.assemble_A(e).

(67)	λe.PART(λe′. ∃e″ [ wakeA(e′) ∧ … ∧ wakeCS(e′′) ∧ cause(e′′)(e′) ∧ incr(e′′)(e′)])(e)

(68)	λe.PART(λe′. ∃e″ [assembleA(e′) ∧ … ∧ assembleCS(e′′) ∧ cause(e′′)(e′) ∧ mmfp(e′′)(e′)])(e)

A natural assumption about the predicates like λe.wake_A(e), λe.assemble_A(e), λe.open_A(e), λe.read_A(e), and the like is that their extensions in a world are very rich: they contain all waking up, assembling, opening, and writing activities admissible in that world.

Furthermore, some of these predicates are radically underspecified for descriptive content. This is the case with the so-called result verbs (discussed first by Rappaport Hovav and Levin (1998); see Beavers and Koontz Garboden (2020) for recent reflections). The meaning of ‘open the door’ only tells us that any e from the extension of λe.open_A(e) has to be an eventuality that brings about a change of state of the door, but does not specify further descriptive characteristics of this eventuality.

The role of the context here is to narrow down a very extensive set of activities (more extensive for result verbs like ‘open’, arguably less extensive for manner verbs like ‘write’) to a more restricted contextually relevant set, as described, e.g., by the Scenarios 1–4 in (61)–(63) and (65). If this more restricted set contains eventualities that can satisfy conditions on the non-culminating interpretation, as, e.g., under the Scenarios 1 and 4, an NCA is licit.

Therefore, contextual information can strengthen the content of an activity predicate, but it is not supposed to weaken or override it. The effects we observe with ‘open’ and similar verbs are essentially strengthening effects: some contexts (e.g., Scenarios 1 and 4) facilitate an NCA interpretation while others block it. On the current view, this happens because the initial very weak denotation of open_A is restricted to entirely different contextually relevant sets of eventualities.

We expect, correctly, that if a predicate comes with more specific inferences associated with its lexical content, strengthening effects will be weak or non-existent. We have already seen that, e.g., ‘set up a tent’ cannot derive an NCA more or less independently of the context. On the view advocated in this paper, this happens because the extension of λe.set.up_A in an accomplishment structure like ‘set up a tent’ in (47) consists of eventualities incompatible with the derivation of an NCA to begin with. Contextual narrowing down of this extension is correctly predicted to have little consequences for the availability of an NCA.

The opposite view—that in certain contexts an NCA becomes available because in these contexts the denotation of a predicate is forced to change/coerce—is very difficult to maintain.

If the above reasoning is on the right track, the next step would be to make the notion of (equi)distance to the culmination more explicit and to address culmination-related contextual inferences. This is attempted in the next section.

4.2. Modeling Equidistance

The empirical success of the generalization in (64) depends on whether it is possible to make the concept of (equi)distance to the culmination sufficiently explicit. What does it mean exactly that the distance to the culmination between the initial and final bounds of an eventuality is the same or contextually insignificant? The purpose of this section is to offer an outline of an answer to this question. While many technical details of what is laid out below require further elaboration, I hope that a quasi-formal outline of the idea will suffice for the purposes of the current study22.

I argue that the idea of the culmination being close or coming closer to realization can be given the following content. When an eventuality from the extension of an accomplishment predicate unfolds, there are options for the evaluation world to develop into a world where it culminates as well as into a world where it does not reach the culmination. We can say that the distance to the culmination decreases if in the course of an eventuality non-culminating options are eliminated in favor of culminating ones.

Let us look at the digital code scenario, which entails UTA, from this point of view. In this scenario, the door is opened by an eventuality e which is the sum of e₂, e₅, e₆, and e₉ events of pushing the corresponding buttons on the keypad lock, (62). At the left bound of e, there is one way for it to culminate: all of e₂, e₅, e₆, and e₉ have to occur in a world. Let us call all worlds in which this happens culminating worlds for e. At the beginning of e, all of them are an open possibility. On the other hand, the space of possibilities available at the left bound of e contains quite a lot of non-culminating worlds for e. These are the epy worlds in which e is incomplete or does not occur at all, i.e., the worlds where only e_2, e₅, and e₆, but not e₉ occur, the worlds where only e₂ and e₅ occur, the worlds where only e₂ occurs, and the worlds where even e₂ is not realized. Once e₂ has happened, we find ourselves in a slightly different position. After e₂, the number of available culminating worlds stays the same: the worlds where all of e₂, e₅, e₆, and e₉ occur. But the number of available non-culminating worlds is reduced, since the option that even the initial part of e has not occurred is now cut off, and the world of evaluation has more chances to develop into a culminating world than before “2” was pushed. The same happens at the next step and so on. In this type of scenario, the proportion of non-culminating options in the total population goes down with every subevent occurring.

To make this idea more explicit one can take Thomason’s 1984/2002 (Thomason 2002, pp. 216–17) forward branching world-time model as a point of departure, which is employed (directly or in a modified way) in a number of studies of modality, tense, and aspect, e.g., Condoravdi (2002), Deo (2009) and other related work.

Thomason defines his T × W frames as a quadruple <W, T, <, ≈>, where W and T are sets of worlds and times, respectively, < is the temporal precedence relation with the usual properties, and ≈ is a three-place equivalence relation between times and two worlds. For a given world w and time t, ≈ defines historical alternatives to w, worlds that share the past with w up to and including t and can only differ as to what is future with respect to t. We write “w ≈ _t w′” for “w′ is a historical alternative to w through t”. For all w, w′ ∈ W and t, t′ ∈ T the following holds: if w′ is a historical alternative to w through t, and t′ precedes t, then w′ is historical alternative to w through t′. Historical alternatives can thus be thought of as possible variants of development of a world, starting from a certain time.

The set historical alternatives through the times t and t′ can be depicted as a branching structure in (69):

(69)

In (69), all the worlds are historical alternatives to each other through t. The worlds w, w₃, and w₄ are historical alternatives to each other through t′. The diagram in (69) makes it clear that as time advances, the set of historical alternatives to a particular world decreases: what is an available alternative at some time can cease to be such at a later time, but no new alternatives can become available.

Let e be an activity component of an eventuality that falls under an accomplishment description P_w0 in the evaluation world w₀. Let e_a be its temporal part that occurs in the evaluation world and is described by a non-culminating sentence. Let INI(e_a, w₀) be the time where e_a starts. Then, the entire set of historical alternatives to w₀ through INI(e_a, w₀) falls into two parts: culminating worlds, in which e_a develops into an eventuality from the extension of P_w0, and non-culminating worlds, where e_a stops before the culmination23. On a non-culminating construal, the evaluation world w₀ itself is a non-culminating world.

Let R_NC(e)(t)(w) be the ratio of non-culminating worlds to all worlds for an eventuality e at time t in a world w:

(70)	RNC(e)(t)(w) = |{w′: w ≈t w′ ∧ ¬culm(e)(w′)}| / |{w′: w ≈t w′}|

Now we can use R_NC to measure for the distance to the culmination. At INI(e, w), the initial bound of an eventuality in a world, there can be quite a lot of non-culminating worlds within the set of historical alternatives. R_NC can be close to 1. As time advances, some of the historical alternatives are closed off, among them potentially being both culminating and non-culminating ones. We can say that the culmination is coming closer to realization if the proportion of non-culminating worlds in the entire population of historical alternatives decreases, i.e., if R_NC goes down.

Consider an illustration in (71):

(71)

In (71), the line in bold represents an eventuality from the extension of an accomplishment predicate P_w0, which has a blue part e_a occurring in the evaluation world w₀. The eventuality starts at t (=INI(e, w₀)) in all of the historical alternatives to w₀ through t except in w₅. At t′ it stops in w₃ and w₄, but continues in w₀, w₁, and w₂. At t′′, it stops in w₀, but continues in w₁ and w₂. Finally, at t′′′ it stops in w₂, but continues and culminates in w₁. The culmination here and in the diagrams below is symbolized by a circle.

Now we can say that the distance to the culmination decreases from the initial and final bounds of e_a, i.e., from t to t′′ in w₀. This is so because by t′′ the non-culminating world w₅, w₃, and w₄ cease to be historical alternatives to w₀. If {w₀, w₁, w₂, w₃, w₄, w₅} are all worlds that there are, then R_NC(e)(t)(w₀) is 5/6, since the culmination is not reached in five out of six historical alternatives to w₀, namely, in w₀, w₂, w₃, w₄, and w₅. R_NC(e)(t′′)(w₀) is 2/3, since through t′′ three historical alternatives are available, and the culmination is not attained in two of them. A positive difference between R_NC(e)(t)(w) and R_NC(e)(t′′)(w) indicates that t and t′′ are not equidistant to the culmination. For there being equidistance, this difference has to be negligibly small, according to (64).

It should be pointed out that R_NC(e)(t′′′′)(w₁) is 0, since the only historical alternative available in w₁ at t′′′′ is w₁ itself, a culminating world. This seems to be an intuitively plausible result.

Illustration (71) is a possible representation of the structure of historical alternatives for the digital code scenario, where the door is opened by pushing the buttons 2, 5, 9, and 6 in that specific order, (72a-b). This is shown in (73).

(72)	a.	e2: pushing “2”	b.	e= e2 ⊕ e5 ⊕ e9 ⊕ e6
		e5: pushing “5”
		e9: pushing “9”
		e6: pushing “6”

(73)

In (73), historical alternatives are individual worlds. Note that while w₃ and w₄ are different historical alternatives, this difference has nothing to do with the eventuality in question: in both worlds the same part of e, e₂, occurs, and only this part. Whatever makes w₃ and w₄ different is irrelevant for the development of an eventuality and its culmination. We can therefore consider w₃ and w₄ one option for an eventuality that culminates in w₁. Options for an eventuality will thus be classes of historical alternatives in which the same part of it occurs:

(74)	o1(e) = {w: e2 occurs in w and no other part of e occurs in w } = {w3, w4}
	o2(e) = {w: e2 ⊕ e5 occurs in w and no other part of e occurs in w } = {w0 }
	…

Considering w₃ and w₄ one option, (74), and letting R_NC count options rather than worlds, we arrive at (75):

(75)

The R_NC values for e with respect to the option {w₁} at different times are shown in (76):

(76)	Time	t	t′	t′′	t′′′	t′′′′
	RNC w.r.t. w1	4/5	3/4	2/3	1/2	0

As (76) makes explicit, the value or R_NC monotonically decreases with time, representing the fact that in every world where the eventuality goes on, the culmination is coming closer to realization.

As seen in (75), on a UTA scenario, at any time an activity can either stop or have one specific continuation. With any other continuation, it is no longer an activity that can develop into an eventuality that falls under the description in question.

The broken lock scenario is crucially different in that at any time an eventuality can go whichever way without losing its potential of culminating into opening of the door. It is only required that the final subevent is e_sledgehammer; other subevents from (77) can be arranged in any possible way.

(77)	ekey = applying a key (orange line)
	epicklock = applying a picklock (blue line)
	ecrowbar = applying a crowbar (red line)
	esledgehammer = applying a sledgehammer (green line)

The structure of historical alternatives is shown in (78):

(78)

In (78), in any world at any time the same options are available. After any of the subevents in (77) except e_sledgehammer, an activity can stop without reaching the culmination (black lines in (78)), or be followed by any of the subevents. If followed by e_sledgehammer, an activity culminates, other subevents can be further extended. The reader can check for herself that on this set up, any choice of times and options will yield the same R_NC. This is the case of equidistance.

The structure behind the Scenario 3 in (63) involving a fixed number of pushes of arbitrary buttons is shown in (80). Let us assume for the sake of exposition a simplified version of the Scenario 3 where there are exactly two different buttons, “1” and “2” in (79), and to open the door one has to make three random pushes. As in (78), options where the activity stops are shown by black lines.

(79)	a. e1 = pushing “1” (blue line)
	b. e2 = pushing “2” (red line)

(80)

This setup involves eight (2³) culminating options and 7 (1 + 2 + 4) non-culminating ones. At the initial bound of the activity, R_NC is thus 7/15 for any world. After two pushes, there are three options available in any world where an activity continues: an activity stops or a push of any of the two buttons makes it culminate. R_NC at this time is thus 1/3, which is by 2/15 smaller than at the initial bound. Note that the same difference obtains between R_NC at t and at t′′ in (75), as (76) shows. In the scenario in question, therefore, despite the lack of UTA, as time advances, the number of non-culminating options decreases faster than that of culminating ones. The lack of equidistance in (75) and (80) correctly predicts that on these scenarios the speakers’ judgments pattern together.

TC-effects whereby a sentence improves with longer chains of subevents is what one can expect, as well. Imagine that in (63) instead of four pushes ten are required. Then, the number of culminating worlds is 2¹⁰, while that of non-culminating worlds is 2⁰ + 2¹ + … + 2⁹. After two pushes, the former will be 2⁸, while the latter 2⁰ + 2¹ + … 2⁷. The reader can calculate R_NCs before and after two pushes and see that the difference between them is about 0.002 instead of 2/15 in the previous case. This is what is meant by (64) as a (contextually) insignificant distance to the culmination between the initial and final bounds of a relevant eventuality.

TC-effects for UTA scenarios like (58) are not surprising either. UTA scenarios involve non-culminating options being eliminated one at a time, as (75) makes evident. Therefore, the difference in R_NC between the k-th subevent and m-th subevent in a chain of n subevents will be n-k-1/n-k – n-m-1/n-m. With n increasing, the difference becomes negligible. I believe this provides a natural basis for characterizing the intuitions behind the TC-effects described in the Section 3.3.

Finally, consider incremental accomplishments like ‘read a novel’ or ‘dig up a vegetable patch’ where the acceptability of a non-culminating reading depends on the size of the incremental argument. The discussion in the previous section suggests a parallelism between this effect and TC effects characterized above. Indeed, the structure of options for incremental predicates resemble the one in (80) except for being an instance of permutation without repetition. (As an idealization, let us assume that these predicates adhere to mapping to subevents, i.e., the same part of an object is not read/dug up more than once.)

Let e be an activity component of an incremental accomplishment description that consists of three subevents, corresponding to three parts of its incremental theme. Such an activity can, for example, be digging up three parts of a small vegetable patch or reading a note consisting of three sentences, as in (81). Such a setup yields a considerably degraded sentence under normal circumstances, as we saw in (47) with ‘read a note’.

(81)	a. e1 = reading / digging up the first part of the object (blue line)
	b. e2 = reading / digging up the second part of the object (red line)
	c. e3 = reading / digging up the third part of the object (green line)

The structure of historical alternatives is shown in (82):

(82)

At the initial bound of an eventuality, there are six (3!) culminating and 10 (1 + 3 + 6) non-culminating options, with R_NC being 5/8. After two subevents occur, R_NC comes down to 1/2. The difference between the two comes close to other non-equidistant cases in (75) and (80). Again, with the number of subevents increasing, which happens if an incremental object is more structured, more spatially extended or more fine-grained, the difference decreases radically. If a chain consists of ten subevents rather than three, then after two subevents occur in the evaluation world, the R_NC-difference between the initial and final temporal bounds of their sum will be less than 1/10⁻⁴. Exactly as the discussion in the previous section suggests, it should not be difficult to come up with a context where this difference is negligible.

Now I am in the position to formulate a more refined version of the ED:

(83)	(Equi)distance to the culmination (refined version)
	NCAs describe a proper non-final part e of the activity component of an eventuality that falls under an accomplishment event description in a world w such that the difference value RNC(e)(INI(e,w))(w)—RNC(e)(FIN(e,w))(w) falls below μC, the negligibility threshold in the context C.

According to (83), an eventuality e that falls under a NCA is a proper non-final part of an eventuality from the extension of an accomplishment predicate. For any world w, e should satisfy the following condition: the difference in the proportion of non-culminating options (i.e., ways in which w can develop into a world where e does not culminate) at the initial and final bounds of e in w should be negligibly small relative to the current context.

The difference value in (83) seems to be the simplest possible metric for the equidistance to the culmination, but it is certainly not the only admissible one. I believe that (83) can be formulated in terms of probability distributions, for example. I will leave an exploration of the alternatives for a future study, however.

If equidistance to the culmination is operational in constraining the distribution of non-culminating accomplishments, the last question to address is: why is it operational?

4.3. Culmination and the Meaning of Activities

Equidistance to the culmination suggests that a non-culminating accomplishment describes an eventuality that does not reach the culmination in the evaluation world, and moreover, that the culmination does not come closer to being realized in that world. The literature on eventuality types (aspectual classes, actionality) starting from Vendler (1957) is not unaware of predicates that can be characterized in a similar way.

Predicates of activities and states describe eventualities that do not and cannot culminate hence trivially (i.e., at any interval in any world) meet the equidistance condition. The reader can check for herself that (83) would always yield 0 if the number of culminating worlds in the population of historical alternatives is 0, which is the case with activities and states. Specifically, classical non-derived activities like walk would remain activities if defined in terms of the notion of equidistance. I believe this is a desirable result24.

Therefore, a natural hypothesis relating the equidistance to the culmination and the distribution of NCAs would consist of the two parts in (83)–(84):

(84)	NCAs as activities
	NCAs are perfective activities.

(85)	Activity as equidistance
	Equidistance to the culmination is a defining characteristic of activities.

If being reinterpreted as an activity is necessary for the derivation of NCAs, (83), and if equidistance to the culmination is what makes activities actually activities, (84), then the question why equidistance is part of the semantics of NCAs can receive a satisfactory answer.

According to (83), ‘Kerim walked for two hours’ and ‘Kerim read a novel for two hours’ (or ‘Kerim opened the door for two hours’, impossible in English, but possible in Turkic) are both activity descriptions in the perfective past. The only difference has to do with the fact that walk is a lexical predicate of activities, whereas NCAs are created out of predicates of accomplishments by combining PART with an accomplishment vP.

Impossible NCAs of the form [PFV [ PART [_vP …. ] ] ] discussed throughout this paper should thus be impossible because the combination of PART with a vP-level eventuality description fails to denote a well-defined activity. The infelicity/ungrammaticality observed in (14)–(16), (47)–(48) and similar examples should result from the failure of ‘setup a tent’, ‘open the door’, ‘read a word/a symbol’, etc., to present themselves as an activity after combining with PART.

Hypothesis (84) is not the received view of what activities are. Garden variety activities like ‘walk’, ‘cry’, ‘eat apples’ are typically separated from accomplishments (‘walk to the station’, ‘eat an apple’, ‘open the door’) by well-known characteristics like the subinterval property (Taylor 1977) and cumulativity/quantization (Krifka 1989, 1992, 1998; Rothstein 2004).

However, these characteristics are not helpful in isolating the class of predicates that fail to derive NCAs. It is not difficult to see that after applying PART to an accomplishment description one would always get a non-quantized event predicate. This is so because any part of a non-final part of an eventuality is also a non-final part of this eventuality. If analyzed in temporal terms, [ PART [_vP …. ] ] will have the subinterval property, for similar reasons. In this respect, [PART [set up a tent]], [PART [read a novel]], [PART [open the door]] are all alike: they pattern together with true activities, no matter if they are capable of deriving an NCA. If activities are defined by mereological characteristics like quantization or by the subinterval property, activityhood cannot be what isolates the class of impossible NCAs.

But if part of the definition of activities is equidistance to the culmination, we arrive at a much clearer picture of what NCAs are. The peculiarity of accomplishments that allow for a non-culminating construal is that eventualities in their extensions can have activity-like temporal parts, which do not make any detectable contribution to the culmination coming closer to realization. For instance, the extensions of ‘tear a thread’ or ‘open the door’ admit eventualities where non-final parts of the activity component bring about no change at all. After PART applies to such an accomplishment description, these temporal parts characterized by equidistance to the culmination will be in the extension of the resulting predicate [ PART [_vP …. ] ]. The latter will therefore be a predicate of activities. If the extension of the original predicate provides no such eventualities, the PART-predicate will fail to be an activity, and the derivation of an NCA will not succeed.

Note that equidistance to the culmination as a property of activity predicates is easy to overlook if non-derived lexical activities are only taken into account, since the culmination of ‘walk’, ‘cry’, etc. is never at issue.

It is only in derived environments, i.e., with predicates that are created by applying PART to an accomplishment description, where the significance of (84) fully reveals itself. It allows us to discriminate between different types of PART-predicates, because not all of them would come out activities under (84). In particular, || [PART [set up a tent ]] || and similar descriptions will be told apart from || [PART [read a novel] ] ||. The latter will be an activity but the former will not.

Therefore, the view that a [ PART [_vP … ]] is a predicate of activities allows to reduce the derivation of NCAs to the derivation of perfective lexical activities like ‘walk’. Moreover, assuming that equidistance to the culmination is what all activities have in common predicts the restrictions on NCAs discussed above. These seem to be welcome consequences.

A separate question is what forces [ PART [_vP … ]] to be a predicate of activities, i.e., what goes wrong if PART comes on top of predicates like ‘set up a tent’ or ‘open the door’ on the relevant scenarios.

To the best of my knowledge, this question has never been addressed in the literature. But if the above reasoning is correct, one hypothesis suggests itself. There is one property that || [PART [set up a tent ]] || and other predicates that cannot output an NCA have in common: they do not fall under any of the established eventuality types. They are not accomplishment predicates. The culmination which is part of the denotation of || set up a tent || is no longer in the denotation of the corresponding PART-predicate. But they are not activity predicates either, as we have just seen, since eventualities in their extensions fail to show equidistance to the culmination. If being neither an activity nor an accomplishment is offensive for the derivation, that would explain why NCAs in (12b)–(14b) and the like cannot be derived.

One can imagine two sources for this restriction. One source can be output constrains on operations on eventuality descriptions. Such constraints would define what is a well-formed output of the application of PART. The other would attribute the same effect to the input requirements that aspectual operators (in particular, PFV) impose on their arguments.

The first option can involve a principle that constrains operations on eventuality descriptions in the following way:

(86)	Closedness restriction on eventuality types
	The set of eventuality types in a given language is closed under any operation on eventuality descriptions.

If (86) is a principle of the grammar, no operation can create an eventuality type that does not exist independently in a given language. But predicates like || [PART [set up a tent ]] ||, which are neither activities not accomplishments, will represent such a new eventuality type that (86) rules out.

The other option would be to suggest that predicates that belong to no eventuality type are not offensive by themselves, but what is problematic is their interaction with other elements of the derivation. In the case at hand, that would imply that PFV, which takes [ PART [_vP … ]] as its argument (see (21)), can accept predicates of accomplishments and predicates of activities, but not predicates that fall under neither of these categories. A possible implementation of this option would be to have PFV only defined for well-established eventuality types, as in (87).

(87)	Domain restriction for PFV
	|| PFV || = λP: P falls under an eventuality type.λt.∃e [P(e) ∧ τ(e) ⊆ t]

At the moment I cannot offer a full assessment of relative benefits of (86)–(87). However, it is not difficult to imagine what kind of empirical and theoretical considerations can be taken into account to decide between these two options.

One observation that seems to support (86) is: attested types of operations on event structure seem to conform with (86). For example, causatives and applicatives across languages can change an eventuality type of an input predicate, but do not seem to be able to create new eventuality types. For example, when a causative comes on top of a stative like ‘know’, the result is typically ‘let know’, ‘inform’, and the like, a well-defined accomplishment.

An argument against (86) would be that PART does not only appear in the derivation of NCAs. Presumably, it is also present in the structure of the imperfective clauses, which do not show restrictions parallel to what is observed with NCAs. As was discussed above, while an NCA construal is unavailable for ‘set up a tent’ and similar predicates, a corresponding progressive sentence is readily available in (iii), fn. 18. If both NCAs and the progressive involve PART, (86) cannot be maintained. But to what extent this argument is irrefutable depends on whether one can completely exclude a theory in which NCAs and the progressive/ imperfective do not share PART and are derived by two distinct mechanisms. An independent reason to doubt that both NCAs and the progressive/imperfective are based on the same PART is that many languages with the progressive do not have NCAs of the type discussed above. This is not expected if PART is operational in both configurations.

On the other hand, assuming (87) comes with implications which invite us for a lot of further reflection. Condition (87) can hardly be viewed as a universal characteristic of all aspectual operators. The progressive can make reference to any non-final parts of the activity components of accomplishment descriptions without this component being a well-defined activity. If one wants to say that (87) is a universal property of the perfective, not shared by the progressive, one has to explain why the two semantic aspects are different. If (87) is viewed as a domain restriction for a specific grammatical morpheme in a specific language or group of languages, the expectation would be that it should not be impossible to find languages where the same morpheme appears without such a restriction. Therefore, to find out if (87) is tenable, one should study domain restrictions associated with semantic aspects in more detail and identify existing patterns of cross-linguistic variation in this area of the grammar.

A lot of further reflection is needed to decide between the two options in (86)–(87). Assessing possible pros and contras would be a subject for a separate study, however. In the current one, I cannot have these questions settled, but I hope to have outlined a minimal prospect for a further inquiry. This having been said, I am ready to proceed to the concluding section of this article.

5. Summary

I distinguished between three subclasses of accomplishments that differ as to whether they allow for the failed attempt reading, partial success readings or none of them. Relying on the observations independently made in the literature, I assumed that the essential part of the semantic structure of non-culminating predicates is the partitive operator, PART. I argued that we need a decompositional analysis of accomplishments whereby the activity and change of state components are represented independently. The difference between failed attempt and partial success accomplishments is determined by the relation between these two components. The failed attempt interpretation obtains it this relation is mapping to the minimal final part, whereas the partial success interpretation is due to the incremental relation originally proposed by Rothstein (2004). Then, I explored possible factors that isolate a class of accomplishments which fail to derive non-culminating sentences. The discussion of these factors resulted in the hypothesis that attributes the inability to produce an NCA to the property I called (equi)distance to the culmination. NCAs can describe a proper non-final part e of the activity component of an accomplishment event description only if the distance to the culmination between its initial and final bounds is contextually insignificant. This hypothesis naturally leads to the conclusion that the derivation of non-culminating accomplishments is essentially a process of creating an activity eventuality type out of an accomplishment eventuality type, equidistance to the culmination being the property that defines the type of activities.