In of-verbs, it is always the internal argument that provides the reference point for the event to be considered excessive. In other words, the event is evaluated as excessive for the internal argument in question. For instance, in (2), the baking was too long and/or intense for the cake in question; in (20) the rising was too steep for the airplane in question; in (9), the exercise was excessive for Jón’s body; and in (25a), the heating event was too long and/or intense for the room in question. A further example is given in (27). More cases of excessive of-verbs are listed in the Appendix (Sect. 7.1).

In (27), there is a feeling that the referent of the internal argument is negatively affected by the event. Of-predicates do not seem to come with a semantic implication of affectedness, however. For example, of-rísa ‘over-rise’ does not, in itself, imply affectedness of the airplane.

Of- predicates typically correspond to degree achievements such as widen, cool etc. (see for instance Dowty 1979; Kennedy & Levin 2008; Rappaport Hovav 2008; McNally 2017 on degree achievements). They express a mapping between the temporal trace of the event and some scalar property exhibited by the Theme (e.g. temperature). (28) is a relevant example.

In some cases the change in the Theme is based on an underlying incremental predicate. For instance, the verb ala ‘to feed’ – as well as its excessive counterpart, of-ala ‘overfeed’ – implies eating. As in the case of degree achievements, we can, however regard the event as describing change undergone by the Theme (e.g., an animal that is fed). We will therefore analyse such cases – which are invariably transitive – with the same semantics used for (intransitive) degree achievements.

The semantic analysis provided in this section is formulated with reference to the work of Kennedy & Levin (2008) dealing with verbs such as widen or cool (cf. also Rappaport Hovav 2008; McNally 2017).9 Consider (29) (from Kennedy & Levin 2008: 161).

Example (29) cannot be analyzed as a change-of-state predicate leading to the state of being cool – say, become(cool(the.soup)) – as the soup is not necessarily cool after cooling 17 degrees.10 Rather, (29) says that the difference in temperature that the soup underwent as a result of the cooling event is 17 degrees. Degree achievements thus imply a comparison: the difference between the temperature of the soup at the beginning and the end of the cooling event amounts to 17 degrees. This is implemented in the analysis proposed by Kennedy & Levin (2008). They regard gradable adjectives as measure functions which map pairs of entities and points in time to degrees. Adjectives measuring temperature are thus interpreted as two-place functions of type ⟨e,⟨i, d⟩⟩, cf. (30a) (i and d are the types of intervals and degrees, respectively, as in Champollion 2017; points in time are regarded as special cases of intervals). If a measure function of this type is applied to the soup at the time of speaking, in context C, it returns a degree, cf. (30b). Note that we assume a ‘rich’ notion of context, which covers indexical parameters of time, place and speech act participants as well as the Common Ground (Krifka 2007; Clark 2012; Gast 2022).

As degree achievements denote change along some dimension, Kennedy & Levin (2008) introduce ‘difference functions’, which return the difference between the degree associated with some pair ⟨x,i⟩ and some other degree δ.11 In order to capture the difference between two degrees measured at the beginning and the end of an event, Kennedy & Levin (2008) furthermore define ‘measure of change functions’. A measure of change function “takes an object x and an event e and returns the degree that represents the amount that x changes in the property measured by m as a result of participating in e” (Kennedy & Levin 2008: 173, boldface original). Measure of change functions thus represent the lexical meaning of degree achievements like cool.

In our treatment of measure of change functions we assume temporal traces of events, represented as λe[τ(e)] (cf. Krifka 1989; 1992; 1998; Champollion 2017). These functions map events to intervals and are thus of type ⟨v,i⟩ (v being the type of events). Measure of change functions corresponding to a measure function m will be represented by a subscript Δ and an arrow identifying the orientation of the scale (↑, ↓) for dimensions that allow change in either direction. They are defined in (32) (in a slightly different format than the one used by Kennedy & Levin 2008). init(e) and fin(e) represent the beginning and end of event e, respectively. Measure of change functions are of type ⟨e,⟨v,d⟩⟩.12

The meaning of (29) can be represented as shown in (33a): the difference between the temperature of the soup at the beginning and the end of the event e amounts to 17 degrees. The denotation of the verb cool is shown in (33b).

Excess-marking with of- indicates that the change undergone by some Theme y in some event e surpasses a degree of change that y would have undergone in a ‘normal’ course of events. We use the term maximum norm value for this degree. Consider, again, (34) (=(28)).

Example (34) says that the degree of change in temperature in the room and, hence, the resulting temperature, was higher than the given normalcy standard. We assume that the normalcy standard is a function of the conversational background (Kratzer 2012). In (37) we define a maximum norm function that returns the maximum norm value for a given event type (extensionally, a set of events) with a specific Theme. The function fCnorm \documentclass[10pt]{article} \usepackage{wasysym} \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \[ f^{{\thinspace}norm}_{C} \] \end{document} maps the world of evaluation (w₀) to a “stereotypical conversational background” (Kratzer 2012: 37; cf. (35)). The function max selects the maximum value of an ordered set (cf. (36)). The maximum norm function takes as its argument a set of events in which the Theme y undergoes change along some dimension m and returns a degree δ.

The denotation of ofhita herbergið ‘overheat the room’ can accordingly be represented as shown in (38).

The analysis of of- when combining with degree achievements may be more obvious for verbs such as overheat (‘become too hot’) or overfill (‘become too full’) than it is for verbs like overbake, which do not appear to lexicalize an underlying adjective. We assume that predicates of this type generally encode some measure function. Measure functions are regarded as lexical atoms that constitute the backbone of degree expressions (cf. Champollion’s 2017 ontology). The natural language correlates of these dimensions can be recovered from passive participial forms of the relevant verbs. For example, the measure function of bake can be expressed as baked. Degrees of ‘bakedness’ can be linguistically expressed with proportional degree modifiers13 as shown in (39).

Using the measure of change function bake_Δ the VP of (40) (=(2)) can be analyzed as shown in (41).

Predicates with yfir + anph comprise transitive and unergative verbs, e.g., predicates of consumption (borða ‘eat’, drekka ‘drink’) and predictes of activity (baka ‘bake’, lesa ‘read’, spila ‘play’, skreyta ‘decorate’). Relevant examples were given in (10), (14b), (15a) and (18) above, and more verbs are listed in the Appendix (Sect. 7.2). For further illustration, consider (42), which suggests that Jón is tired after too much dancing.

From a semantic point of view we can distinguish two major types of predicates combining with yfir + anph. The first class comprises incremental Theme predicates.14 These predicates are ‘gradual’ in terms of the classification provided by Krifka (1989; 1992). The property of ‘graduality’ “comprises uniqueness of objects, mapping to objects, and mapping to events” (Krifka 1992: 42). For example, the VP drink a glass of wine denotes a gradual event.15 The VP drakk yfir sig (cf. (15a)) indicates excessive change in the incremental Theme, i.e., the beverage.

The second type of predicate that combines with yfir + anph is commonly taken to be compatible with cognate or hyponymous objects (cf. Hale & Keyser 1993; Kuno & Takami 2004; Sailer 2010 and, in particular, Kallulli & Oltra-Massuet 2022, which deals with scalar such modification). A typical example of a cognate object is provided by a dance, see (43a). (43b) is an example of a hyponymous object.

Note that in the literature on cognate objects, the denotation of this class is often extended beyond objects without a clear etymological relationship to the verb. For example, Kallulli & Oltra-Massuet (2022) regard the Albanian noun gjumë ‘sleep’ in (44) as a cognate object of the verb fle ‘sleep’:

The term ‘cognate object’ thus stands for objects that are implied by the verb and denote either an effected object or the event described by the predicate, according to Sailer (2010).

The availability of cognate objects is often regarded as a diagnostic for unergativity (see for instance Levin & Rappaport Hovav 1995), though there are also cognate object verbs (in English) that are not normally regarded as unergatives (most notably, die). Unergative verbs are accordingly analyzed as underlyingly transitive (e.g., Hale & Keyser 1993). We will assume that unergatives comprise two semantic components, an event description in the verb itself, and a description of the degree of change in the internal argument (which subsumes cognate objects).

What cognate objects have in common with incremental Themes is that they can measure out the event in question. For example, the cognate object dance can be quantified as in He danced two/three/four dances. In fact, some kind of specification or modification is often regarded as being obligatory for the use of a cognate object. Cognate object predicates thus also exhibit a mapping between the temporal trace of the event and the denotation of the internal argument.

In most examples given so far, there is an external argument (subject of a transitive or unergative predicate) with a human referent who carries out an action more than normally. The implication is mostly that s/he is tired, exhausted or fed up. However, affectedness of the external argument seems to be a pragmatic inference, rather than being semantically encoded. There are cases that allow alternative interpretations. For example, the verb phrase skreyta yfir sig (lit. ‘decorate over anph’, see (14b)) seems to have two potential types of implications. Either the referent of the external subject is fed up or exhausted, or the room, or whatever was decorated, has been spoiled. The latter reading is akin to the interpretation of the corresponding of-predicate (of-skreyta) and would in fact more naturally be rendered in this way. (45) is a similar case. Either the child is tired after too much whining, or the speaker is annoyed.16

The main point of reference for our semantic analysis of yfir + anph-predicates is the work by Kennedy (2012). We will start with incremental Theme predicates and then turn to cognate object predicates. Kennedy (2012: 117) (adopting a proposal made by Krifka 1989; 1992) analyses incremental Themes of the type illustrated in (46) as shown in (46b) (only the denotation of the VP is shown; ‘NU’ stands for ‘natural units’ and is interpreted as a classifier; NU_Δ returns a difference value).

According to this analysis, the VP of (46) denotes a set of events such that the change in the amount of dumplings, measured in ‘natural units’ (NUs), in the course of some event e, amounts to ten.

The amount of change associated with an eating event is, to some extent, analogous to the degrees of change measured in degree achievements. However, there are important differences (cf. also Rappaport Hovav 2008; Kennedy 2012 on differences between degree achievements and incremental Theme predicates). Most importantly, it is hard to conceive of a measure function returning absolute degrees for Themes of incremental Theme predicates, at some point in time t in the course of an event e. While the measure function temp can deliver a value at any given point in time (for some object), there is no obvious analogue for the amount of food consumed at a given point in time t. Rather, the amount of food eaten is incremental at t_i; i.e., it is the sum of the food consumed at all points in time t_1…h, h ≤ i.

In incremental Theme predicates, there is a (one-to-n) mapping from the stages of an event to the parts of the Theme. We capture this relationship by assuming that incremental Theme predicates imply what we call an ‘incremental function’. Incremental functions associated with a predicate will be represented by the relevant predicate carrying a subscript Incr, e.g., eat_Incr for the verb eat. Remember that we adopt Krifka’s (1989, 1992) temporal trace function (Sect. 4.1). In addition, we make the following assumptions:

For example, if at point t_i of the temporal trace τ(e) of an event e ten dumplings have been eaten, it is true that eat_Incr(t_i) = 10NU(dumpling) (note that we treat NU as a classifier that maps non-quantized predicates to quantized predicates; the value returned by a classifier can thus be multiplied by a number – here, 10).

In analogy with our treatment of degree achievements, we can define, for any incremental function P_Incr, an incremental change function P_Δ, which returns the difference in the Theme between the beginning and the end of an event e (init(e), fin(e)). In the definition in (48) we use the difference operator Δ to represent the difference between two Themes that stand in a part-whole relation to each other (θ_jΔθ_i is that part of θ_j that is not part of θ_i).

We can now represent the meaning of the VP eat the ten dumplings as shown in (49). Note that there is a separate eating predicate, which encodes manner. Alternatively, we could, for instance, use the event description devour. Note furthermore that the predicate eat does not take a separate internal argument according to this analysis. The Theme, which is incremental, is represented as the value returned by the incremental change function eat_Δ(e).

Our treatment of incremental Theme predicates can also be applied to cognate object predicates. Cognate objects like (He danced a) dance are either analysed as nominal instantiations of the events denoted by the corresponding verb, or as ‘effected’ or ‘resultant’ objects (Kuno & Takami 2004; Sailer 2010). In either case, their denotation is homomorphic with the temporal trace of the event. If we analyse cognate objects as nominal copies of the event, the homomorphism is trivial, as it is a mapping from the temporal trace τ(e) of an event e to itself. If we regard them as separate (effected, resultant) objects, the homomorphism holds between the temporal trace and those (meronymically structured) objects. We therefore assume that the semantics for incremental Theme predicates – with a target domain ordered by a meronymic relationship – carries over to cognate objects.

yfir + anph (with incremental predicates) can be interpreted in analogy to of-. While of- expresses that the difference between two degrees δ_init and δ_fin, measured at the beginning and the end of an event e, is greater than the maximum norm value for the event and Theme in question, yfir + anph expresses that the incremental change in the Theme resulting from e exceeds the maximum norm value relative to the Agent. As this function references the Agent, it carries the superscript ^Ag. It is defined in (50).

We can use the maximum norm function to represent the meaning of (51) (=(10)) as shown in (52). The maximum norm function delivers the maximum norm value, i.e., the amount of food that would normally be eaten in any event of the type of e (here, an eating event), in context C, if Kim is the Agent of e. Our analysis of (53) (= (42)) – with a cognate object – is shown in (54).

In our analysis of of-predicates we adopt Kennedy’s (1999) treatment of degrees, with minor modifications.17 We assume that an analysis á la Wellwood (2014; 2015) would work just as well though (see also Note 9). Consider the comparative in (55), analysed by Kennedy (1999: 220) as shown in (56).

The AP distant represents a measure function, and Deg establishes a comparison between the degrees to which Mars and Pluto are distant (from the deictic centre, i.e., the earth).18 More generally (and abstracting away from linear order), this structure can be represented as shown in (57). A positive (non-comparative) form with a degree specification takes the form in (58).

Degree achievements have a similar underlying structure as gradable properties. Consider example (29) again (The soup cooled 17 degrees). Such examples can be regarded as verbal analogues of adjectival comparatives. We assume the structure in (59) for the vP cool 17 degrees. The v_deg-head merges with the root representing the (downward-oriented) temperature scale, yielding another node of category v, which corresponds to the denotation of the verb cool as represented in (33b) above. Note that we follow Folli & Harley (2005) in assuming that there are different flavors of categorizing heads such as v, with different denotations.

The vP in (59) denotes a relationship between an event and an object, such that the object cools 17 degrees in its course. The denotations of v_deg, v_cool and vP in (59) are shown in (60). The constituent will be spelled out with an appropriate lexical item, most likely cool.

Of-predicates can be analyzed in parallel fashion to (29) (The soup cooled 17 degrees), differing from this example in the denotation of the v-head, and the number of arguments within v P. For illustration consider (61), which is analogous to (29) insofar as it relates to a scale of temperature. Its syntactic representation is provided in (62). We focus on the vP here, assuming that the Agent is introduced by a Voice head, projecting a VoiceP (Kratzer 1996; Alexiadou & Anagnostopoulou & Schäfer 2015).

The v-head in (62) has the denotation shown in (63a). It takes three arguments, two of which are saturated within vP: a measure function, and an argument specifying the standard (the room) for the norm function. The vP denotes a set of events such that the degree of change undergone by the room in question exceeds the maximum norm value for that room, cf. (63c).

The structure of incremental Theme predicates and unergatives is more complex than the one with degree achievements. Recall from Sect. 4.2 that we analyze vPs headed by an incremental Theme of the type of (64a) as shown in (64b).

Our syntactic representation of (64a) is shown in (65). We assume a completely parallel structure for cognate object predicates like dance. The vP denotes an event of eating whose extent is specified in a separate degree Phrase (degP). This phrase consists of a deg-head, an incremental function in the complement position , and a specification of the amount consumed in spec,degP. We assume that the standard of comparison in spec,degP moves to spec,vP, where it saturates the internal argument position of v.

The denotations of the deg-head, degP, v and vP in (65) are shown in (66).

Sentences with yfir sig have the same vP-structure as shown in (65). The deg-head takes an incremental function as its complement. It imposes a selectional restriction on spec,degP, to the effect that it only accepts anaphors (mig, dig, sig) as arguments. The anaphors are bound to the agent of the event (spec,VoiceP), from which they inherit a referential index (Büring 2005) and are thus interpretable in situ. Consider again example (67) (=(10)), which can be analyzed according to the structure provided in (68).

The syntax-semantics mapping for (67) is shown in (69). The deg-head takes an incremental function (in the complement position) and a standard of comparison (in the specifier position) as its arguments, returning a property of an event. The rest of the derivation is analogous to (66).