Toward a uniform account of the degree modifier cut'

This section examines the distribution and semantic contribution of cut' in the adjectival domain.

Adjectives in the Positive Form

Cut' is perfectly compatible with minimum standard adjectives, which lexicalize lower-closed scales (scales that involve a minimal value, cf. [Kennedy, McNally, 2005]). A search in National Corpus of Russian renders such phrases as the following: cut' izognutyj `slightly bent', cut' zametnyj `slightly noticeable', cut' vypuklyj `a bit protruding', cut' slysnyj `slightly audible', cut' sonnyj `a bit sleepy', cut' sersavyj `a bit rough', cut' vlaznyj `a bit wet'. The resulting phrases denote a set of individuals that possess the property lexicalized by the adjective to a very low degree. The degree is entailed to exceed the lower boundary of the scale but, at the same time, to be very close to this boundary. It is also worth noting that x is cut' P entails that x is P. This is illustrated below:

(2) a. Линия чуть изогнута.

Linija cut' izognuta.

line cut' bent

`The line is slightly curved.'

ENTAILS:

b. Линия изогнута.

Linija izognuta.

line bent

`The line is curved.'

We can tentatively represent the semantics of (2a) as in (3a). The meaning of (2b) is provided in (3b) for the sake of comparison. It can be seen from the formulae that the entailment relation between the two sentences indeed holds as specified above.

(3) a. 3d [bent(d)(the line) & d >c min(Sbent)]

where Sbent is the scale lexicalized by the predicate bent and >c stands for the `higher than and very close to' relation.

b. 3d [bent(d)(the line) & d > min(Sbent)]

J

Note that for every d and d' it holds that d >C d' entails d > d'. In prose, if d is higher than and close to d', then d is higher than d'. It therefore follows that (3 a) entails (3b).

Let us now turn to additional types of scales. Cut' is incompatible with adjectives that lexicalize upper-closed scales (the ones that involve a maximal value):

Analogously, cut' is generally bad with relative adjectives which lexicalize totally open scales in their positive form:

Cut' with Modified Adjectives

The facts illustrated above do not mean that he distribution of cut' is limited to minimum standard adjectives. This item can also apply to adjectives of other types if the latter do not appear in their simple, positive form, but rather combine with certain degree modifiers. In other words, cut' can appear on top of certain degree modifying expressions. For instance, it can combine with comparative adjectives, as is illustrated in (7):

(6) Миша чуть выше Лены.

Misa cut' vyse Leny.

Misha cut' taller Lena...

GEN

`Misha is slightly taller than Lena.'

In (6), cut' applies to an adjective that has already combined with comparative morphology. The standard of comparison corresponds to Lena's (maximal) height. Due to the comparative modification, the subject is entailed to be mapped to a degree on the scale that exceeds the standard. The presence of cut' further specifies that the distance between the two degrees is very small. Thus, while Misa is entailed to be taller than Lena, his height exceeds Lena's very slightly. The truth conditions of (6) can be represented as in (7):

(7) 3d 3d' [tall(d)(misha) & tall(d')(lena) & d >c d']

Once again, x is cut' P entails that x is P. For instance, (6) entails that Misha is taller than Lena. Cut' is also compatible with adjectives modified by the suffix -ovat, whose semantics is investigated by Kagan and Alexeyenko (2011a, b) and which, under its relevant use, contributes the meaning of excess (`slightly too P'). For reasons of space such adjectives will not be discussed here; however, the analysis proposed in this paper applies to them as well.

A Uniform Analysis of Adjectival cut'

On the basis of the data discussed in the previous subsections, we can conclude that cut' is systematically associated with the following meaning components:

(i) cut' specifies that the argument possesses the property denoted by the AP to a degree that exceeds the standard of comparison (the latter is independently provided by the environment). cut' further specifies that the distance between these two degrees is very small;

(ii) for every P, it holds that: x is cut' P ^ x is P.

The entailment relation represented in (ii) has been shown to hold in all the environments mentioned above in which cut' is acceptable. The meaning component in (i) also systematically accompanies cut'. Whether the standard is the lower boundary of a scale or a linguistic standard supplied e.g. by the comparative bolee...cem `more than', the degree to which the argument is mapped is consistently entailed to be located slightly above this standard.

Further, the proposed analysis captures the incompatibility of cut' with maximum standard and relative adjectives in their positive form. This incompatibility essentially means that cut' cannot apply to the maximal degree on a scale and to a distributional, comparison-class-based, standard, associated with relative adjectives. The first observation is particularly easy to explain. cut' relates the argument to a degree that is higher than the standard. However, if the standard constitutes the maximal value on the scale, it is impossible to possess the property to a higher degree, not even to a slightly higher one. Kagan and Alexeyenko (2011a, b) argue that -ovat cannot apply to this kind of standard for exactly the same reason.

The case with distributional standards is a bit more complex, but here too, a reason for incompatibility is available. In particular, a distributional standard is known to be characterized by a particularly high degree of vagueness. Where exactly does the boundary between tallness and lack of tallness lie? Even once the context is fixed and the comparison class is determined, there remains the famous problem of borderline cases: for some individuals whose tallness is close to the standard, it is still difficult to determine whether they fall under the denotation of tall. (Cf. e.g. [Kennedy, 2007] for discussion.) The precise boundary is difficult or plausibly even impossible to determine, and speakers will disagree with one another as to where exactly it should be located. Given this kind of inherent vagueness, how can we guarantee that a degree to which an individual is mapped is higher than and at the same time very close to the standard? Since the location of the standard is unknown, it is impossible to measure a distance between this and another value with a high degree of precision, nor to guarantee that such a distance is indeed extremely small, in some sense minimal. In the linguistic literature, additional degree modifiers have been claimed to be incompatible with distributional standards for the same reason (cf. e.g. [Winter, Rotstein, 2004] on almost, [Kagan, Alexeyenko, 2011a] on -ovat). We assume that cut' is no different in this respect.

Formally, the semantics of adjectival cut' is represented in (8), where the standard of comparison (ds) is to be provided by the environment which includes pragmatic context:

(8) [[cut']] = ZPZx.3d [P(d)(x) & d >c ds]

The compositional semantics of (9) below is provided in (10).

`This line is slightly curved.'

(10) [[cut']] = ZPZx.3d [P(d)(x) & d >c ds]

[[izognuta]] = Zd'Zy.bent(d')(y)

[[cut']]([[izognuta]]) = Zx.3d [bent(d)(x) & d >c ds]

The function applies to the standard of comparison associated with the stem, and we get:

Zx.3d [bent(d)(x) & d >c min(Sbent)]

[[cut' izognuta]]([[eta linija]]) = 3d [bent(d)(this-line) & d >c min(Sbent)]

3. Verbal cut'

The modifier cut' can also appear within verbal projections and apply to VP

`By that time the water had already cooled a bit.'

The clauses in which cut' appears entail that one of the arguments, in all these examples, the subject, undergoes a certain change (in temperature, in openness and in size, respectively). cut' specifies that the change is very small. In other words, it measures the change that takes place in an argument.

The most natural way to capture the semantics of verbal cut' is by relating to the notion of a degree of change, introduced by [Kennedy, Levin, 2002]. They argue for the existence of a type of verbs, verbs of gradual change, that contribute scales to sentential semantics. Such verbs denote a change that takes place along these scales, i.e. a change in the degree to which an argument is characterized by the gradable property. More precisely, the argument is entailed to possess the property at the end of the event to a higher degree than at its beginning (cf. also [Hay et al., 1999; Rothstein, 2008]). For instance, the verb widen denotes a change in the degree to which its argument is wide. At the beginning of the event the subject is entailed to be less wide than at the endpoint of the event. The semantics of verbs of gradual change involves a `degree of change argument', the degree to which a participant undergoes an increase in the relevant property between the beginning point of the event and its endpoint. One way to represent the semantics of such verbs is illustrated in (12) for the verb widen. (13b) provides the truth conditions of (13a). The formalism is based on [Kennedy, Levin, 2008] and [Kennedy, 2010], slightly adapted in order to fit the framework assumed in this ф paper; the examples are based on [Kennedy, 2010, p. 8-10]. |

(12) [[widen]] = kdkxke.wideA(d)(x)(e) о

where wide^ is a function that for a degree d, individual x and event e renders as the output the truth value of the proposition x has widened to degree d in the course of e.

(13) a. The canyon widened 30 kilometers.

b. 3e [wideA(30km)(the-canyon)(e)]

[Kennedy, Levin, 2008] further propose that the degree of change can be treated as a degree on a derived scale, which is identical to the scale originally introduced by the predicate except for the fact that its lower boundary is reset to the degree associated with the beginning point of the event. This is the degree to which the argument possesses the gradable property at the time when the event begins. This degree corresponds to a zero change. Any degree that is higher than this minimal point corresponds to an increase that takes place in the course of the event.

We are now in the position to account for the contribution of cut' in sentences like (11). This item measures the change that an argument undergoes along a scale by imposing a restriction on the degree of change argument of the verb. In particular, it specifies that the degree of change is very low. The change does take place, and so the degree of change is higher than zero, but at the same time it is very close to zero. Naturally, this contribution is very close to the contribution of cut' within the adjectival domain. cut' applies to a predicate that lexicalizes a lower-closed scale and provides the same information about the degree argument of the predicate as it does with gradable adjectives. The degree to which the argument is mapped is entailed to be slightly higher than the standard of comparison (in this case, the lower boundary of the scale).

Formally, we propose that verbal cut' has essentially the same semantics as its adjectival counterpart, except for the fact that this time it combines with properties of events; therefore, an event argument is added to the degree and individual type arguments that were present in the adjectival domain. Further, the standard of comparison for verbal cut' is fixed as the lower boundary on the derived scale, the one along which the degree of change is measured. The compositional semantics of (11b), repeated below as (14), is provided in (14'):

(14) Интенсивность чуть уменьшилась.

Intensivnost' cut' umen'silas'.

intensity cut' lowered

`The intensity lowered a little.'

(14') [[cut']] = ZPZxZe.3d [P(d)(x)(e) & d >C min(Sp)]

[[umen'silas']] = Zd'ZyZe'.lowA(d')(y)(e')

[[cut']]([[umen'silas']]) = /.x/.e.3d [lowA(d)(x)(e) & d >c min(SlowJ]

[[cut' izmen'silas']]([[intensivnost']]) =

= /.e.3d [lowA(d)(the-intensity)(e) & d >c min(SlowJ]

The event variable gets bound by existential closure:

[[Intensivnost' cut' umen'silas']] =

= 3e3d [lowA(d)(the-intensity)(e) & d >c min(SlowA)

Note also that in both the adjectival and the verbal domain, cut' p entails p. For instance, (14) entails that the intensity lowered.

Temporal cut'

Data and Intuitions

Let us now turn to the temporal-aspectual use of cut'. It should be emphasized that this use is not productive in modern spoken Russian; it is found primarily in literary texts and poetry. In spoken language, temporal cut' is likely to be substituted by the word tol'ko, literally `only'. Still, this use is present in the language and native speakers understand this type of cut' and have intuitions regarding its appropriateness. It is therefore worth asking whether and how temporal cut' is related to the phonologically identical item that is found in the adjectival and the verbal domains.

Temporal cut' is illustrated in such examples from National Corpus of Russian as (15) and (slightly modified) (16). Under the temporal use, cut' appears to the left of the subject. The interpretation of cut'p intuitively seems to be close to that of `as soon as p'.

For the sake of illustration, let us concentrate on (16).

`As soon as the borders were opened, former Soviet people forgot about the fraternal Golden Sands.'

`As soon as Alex left, a letter from Irina Nikolaevna arrived.'

The role of cut' is to contribute a relation of temporal ordering between the two events: the arrival of the letter and Alex leaving. The former is entailed to take place immediately or almost immediately after the latter. The contribution of cut' can be divided into two parts: (i) Alex' departure temporally precedes the arrival of the letter, and (ii) the temporal traces of the two events are very close. None of these meaning components is entailed by the corresponding sentence without cut'.

Crucially, it is easy to see that these components are very close to the meaning contributed by the other uses of cut'. A certain degree (this time one on a time scale) is asserted to be higher than and close to another degree (presumably the standard of evaluation). This suggests that we deal with the same item applied to a different syntactic and semantic domain. But in order to compare temporal cut' to the other uses, we need to consider the compositional contribution of the former in some detail.

Temporal cut' and Reference Time

How do we formalize the semantic contribution of temporal cut'? Let us concentrate on the meaning component of temporal precedence. cut' specifies that the event that falls under the denotation of the VP in its clause is followed by another event. (As a rule, temporal cut' appears in a past tense clause, and both events are understood to precede the time of speech). Thus, cut' makes sure that the temporal trace of the event is ordered not only relative to the time of speech but also relative to the time of an additional eventuality.

In this respect, the contribution of cut' is in part similar to the function of past perfect in English. The past perfect form indicates that the event in question does not only precede speech time but also precedes the time of another past eventuality (or at least another temporal interval in the past that is somehow made salient). For instance, in (17), past perfect makes sure that John left before Mary's arrival:

(17) Mary arrived when John had left.

Since [Reichenbach's, 1947] important work, this kind of temporal ordering is captured using the notion of reference time (or topic time). This is the temporal interval relative to which the event time is viewed or evaluated. Thus, in (17), the time of John's departure is evaluated relative to another salient temporal interval, in particular, the time of Mary's arrival. The relative configuration of event time (E), reference time (R) and speech time (S) depends on the tense and aspect of the clause. Most relevantly for our purposes, past perfect is characterized by the following configuration: E_R_S.

Event time precedes reference time, and both precede speech time. A detailed review of linguistic analyses of tense and aspect that are based on these notions can be found in [Borik, 2002].

As noted above, cut' is partly similar to past perfect in that it, too, establishes a relation between event time and another salient temporal interval, relative to which event time is evaluated. We therefore propose that cut' establishes a relation between event time and reference time. (The latter overlaps with the event time of the second clause.) It specifies that reference time follows event time (E_R), and also that the two intervals are close on the time scale.

Formalizing the Relation between cut' and Reference Time

Let us begin with the intuition behind the formal analysis that is developed in this section. The attachment of cut' to a clause signals that the time of the event reported in the clause (e.g. Alex leaving) is evaluated relative to the time of another event, which takes place immediately after the first one. Assuming that the latter temporal interval constitutes the reference time, cut' specifies that this reference time is located - on a time scale - higher than, but at the same time close to, the time of the reported event. The standard of comparison in this case is constituted by event time (the time of Alex leaving). This is a salient degree on a time scale, which is independently contributed by the sentence, since events always take place in time and, thus, come together with their temporal traces. Reference time, which constitutes a degree on the time scale as well, is entailed to be located above this standard but close to it.

We now turn to the formal, compositional semantics of cut'-clauses. We will largely follow the approach to the temporal-aspectual structure at the syntax-semantics interface developed by [Ramchand, 2004], although we will not follow her analysis of perfectivity. Ramchand takes the Asp(ect) Phrase area as the location where reference time is introduced to the semantics of a clause and a relation is imposed between reference time and event time. In turn, the TP domain is responsible for determining the relation between reference time and the time of speech. It is AspP that is of interest for our current purposes.

(18) [[Asp]] = ZPZt.3e[P(e) & t є T(e)]

The Aspect head introduces a temporal argument (which corresponds to reference time) to the semantics of the clause and further specifies that reference time is included in event time.

As discussed above, we assume that alternative relations between R and E are possible. Specifically in cut'-clauses, E precedes R. This leaves open two options. We can assume that perfective Asp leaves the relation between

E and R undetermined (since perfectivity is compatible with more than one such relation). Alternatively, assuming that the relation between R and E is systematically determined at the level of Asp, this could be the place where the configuration E_R is established. Perfectivity per se does not require this particular configuration but is compatible with Asp that specifies this relation. For the purposes of presentation we will choose the second alternative; however, nothing crucial hinges on this choice, as will become clear below.

Yet another point where the semantics we assume for Asp differs from (18) is has to do with quantification over events. We assume that Asp head does not contribute an existential operator that binds the event argument; rather, the latter gets bound by an independent mechanism like existential closure.

Let us now consider the compositional semantics of the clause cut' Alex usel.

(19) a. [[vP]] = Ze. [left(alex)(e)]

b. [[Asp]] = ZPZtZe. [P(e) & t > r(e)]

c. [[AspP]] = [[Asp]]([[vP]]) = ZtZe. [left(alex)(e) & t > r(e)]

d. [[cut']] = ZPZe. 3d [P(d)(e) & d >c ds]

e. [[AspP]] = ZdZe. [left(alex)(e) & d > r(e)]

f. [[cut']]([[AspP]]) = Ze.3d [left(Alex)(e) & d > r(e) & d >C ds]

g. [[cut' AspP]] = Ze.3d [left(Alex)(e) & d > r(e) & d >c T(e)]

h. [[cut' AspP]] = Ze.3d [left(Alex)(e) & d >c r(e)]

We propose the semantics of temporal cut' in (19b). The item keeps functioning as a degree expression: it applies to a property that has a degree argument and imposes the `>c' relation between this degree and a standard of comparison. The latter is provided either by the linguistic environment or by the discourse.

It may seem that cut' cannot successfully apply to AspP due to type mismatch: the AspP does not involve a degree argument. However, we should remember that temporal intervals constitute degrees on time scales; thus, the denotation of AspP does contain a certain type of degree argument. For the sake of uniformity, and since temporal cut' applies specifically to scales with the temporal dimension, we can substitute the degree variable d in the semantics of cut' by the variable over temporal intervals t. But in order to keep the semantics of temporal cut' as close as possible to the semantics of its other uses, we will instead substitute t by d in the denotation of AspP . (with the implicit assumption that the degrees involved in its semantics are specifically on a time scale). We get (19e). Temporal cut' (unlike the other uses of this item) applies to AspP. The result is (19f). The standard of comparison is a degree on a time scale, and such a standard is linguistically contributed by AspP. This is the temporal interval of the event, represented above as r(e). We thus get (19g). Finally, since d >C r(e) entails that d > r(e), the representation of the denotation of cut' AspP is reducible to (19h). The resulting meaning after cut' applies to AspP is that the reference time appears slightly higher on the time scale (i.e. comes slightly later) than the time of the event of Alex leaving.

Negative cut'