2.1 Entailment patterns

Let us first discuss sentences with NPs in which numerical frequency adjectives modify deverbal nouns describing a participant of an event denoted by the verb such nouns are related to. Consider the entailments in (7)–(9).

In each of the pairs above, the sentence with the NP modified by the numerical frequency adjective, i.e., (7a), (8a) and (9a), entails that Kim participated in n-many complete events described by the verb from which the relevant participant noun was derived, where n is specified by the numerical frequency adjective. So in (7b), (8b) and (9b), Kim took part in two winning, two surviving and two qualifying events, respectively. Therefore, at first sight it might seem plausible to deduce that expressions such as two-time simply quantify over eventualities encoded in the deverbal root of the modified noun. Crucially, however, numerical frequency adjectives can also combine with nominals that are not derived from verbs. For instance, let us reflect on the examples in (10)–(12).

The fact that (10a) entails (10b) suggests that there is a hidden event of acquiring the property of being a champion in play and that, in the absence of any overt deverbal morphology, that event is introduced by the adjective two-time. In a similar vein, the relationship between (11a) and (11b), on the one hand, and between (12a) and (12b), on the other, indicates the relevance of an act of gaining a socially relevant role or capacity by Kim, i.e., becoming a senator and a captain, respectively. Of course, this perspective also extends to the examples with deverbal participant nouns in (7)–(9) since, e.g., Kim’s winning Pulitzer Prize is equivalent to her becoming a Pulitzer Prize winner. Therefore, I conclude that numerical frequency adjectives operate on hidden events of acquiring a certain property or a role by specifying the number of such “becoming” events.

Let us now turn to the behavior of multipliers. Consider the entailment pattern demonstrated in (13)–(15).

The entailments illustrated above indicate a concealed complex inner part-whole structure of the events denoted by the entire NP modified by double. In particular, one infers from (13a) that in the murdering event there were two victims. But this means that there were two parts of that event each of which could be described as a murder in its own right. These parts could be defined temporally as subevents or spatially as regions of space occupied by the casualties. For instance, in a knife-stabbing scenario killing each of the two victims would be a part defined temporally, i.e., as a particular subevent. On the other hand, in a bombing scenario, where two persons die simultaneously as a result of an instantaneous explosion, the two relevant parts would be defined spatially, i.e., as regions of space occupied by the victims. In a similar vein, (14a) and (15a) entail that that strike and that play, respectively, involve two subevents that somehow seem essential for the whole event, see (14b) and (15b).

Notice that the same effect is also attested in sentences with nominals denoting individuals, see (16)–(18) (Wągiel 2019; 2020).

For instance, the sentence in (16a) entails that the Pschent is a crown consisting of two elements having a property comparable to that of the whole, see (16b), namely an ancient Egyptian crown that combines the White Hedjet Crown of Upper Egypt and the Red Deshret Crown of Lower Egypt. Similarly, (17a) and (18a) mean that the objects referred to by the subject DPs comprise two relevant parts, specifically two patties and two barrels, respectively.

Given the entailments discussed above, it follows that phrases including numerical frequency adjectives and multipliers such as (1a) and (1b) designate, respectively, a plurality of two events of acquiring a property denoted by the modified NP and a plurality of two essential comparable parts constituting an event (or an individual) denoted by the modified NP.

2.2 Scopal properties

The second set of phenomena to be discussed here concerns scope. Interestingly, the quantificational effects of both two-time and double are anchored to a particular entity or event designated by the modified nominal. In other words, the expressions in question cannot outscope the NP, which results in a collective-like behavior. For instance, consider the available interpretations of the sentences in (19).

It is well-known that a sentence such as (19b) can mean either that Kim and Ida met two champions in total or that they met two champions each. One might expect that the occurrence of the numerical frequency adjective inside the object DP would result in a similar ambiguity. However, while (19a) can mean that Kim and Ida either met one champion in total or one champion each, it is impossible to distribute championships over different individuals. Thus, Kim and Ida had to meet a champion with two titles and a reading on which they met two different one-time champions is unavailable. This shows that two-time displays no scopal ambiguities and its meaning is anchored to an entity denoted by the modified noun.

Since expressions such as two-time appear to be a special class of frequency adjectives like frequent and occasional (see, e.g., Stump 1981; Zimmermann 2003; Schäfer 2007; Gehrke & McNally 2015), it is reasonable to test whether they share some of their properties. Interestingly, unlike frequency adjectives such as occasional (but similarly to frequent), two-time does not give rise to adverbial interpretations (first observed by Bolinger 1967), compare (20) and (21).

While (20a) is synonymous to (20b), (21a) cannot be understood as equivalent to (21b). Similarly to frequency adjectives such as frequent, two-time triggers only the so-called internal reading, see (22)–(23).

Perhaps it is worth noting that this is somewhat reminiscent of the fact that multiplicatives like two times differ from frequency adverbs such as often in that they lack relational readings, compare (24) and (25) (Doetjes 2007).

Turning to multipliers, their quantificational effects are also anchored to an event described by the modified NP. For instance, consider (26).

While (26b) is true either if Kim and Ida witnessed a total of two or a total of four murders, (26a) does not show a comparable ambiguity that would result in distributing the victims over different eventualities. Again, the meaning of double is anchored to a singular event. Consequently, the sentence cannot mean, e.g., that Kim witnessed the murder of Jack, whereas Ida witnessed the murder of Steve.

Let us now compare the semantic behavior in (26), as discussed above, with adverbial numeric quantification in the VP domain. It is well known that, depending on a position in a sentence, multiplicatives such as twice can count either entire events or event-internal acts, i.e., subevents that are relevant parts of a whole (e.g., Cusic 1981; Andrews 1983; Cinque 1999; Zhang 2017). To illustrate, let us consider the ambiguity of the sentence in (27).

On the event-external reading, (27) means that on two separate occasions the salesman rang the bell once, whereas the event-internal interpretation states that on one occasion they rang the bell two times. Furthermore, in a configuration such as (28) the preverbal multiplicative, i.e., twice, unambiguously counts the number of independent knocking events, whereas the postverbal three times quantifies over their parts, i.e., particular knocks within each of those knocking events (Cinque 1999).

Importantly, though there are two pluralities in (28), the sentence does not have an interpretation that would resemble a cumulative reading. On such an interpretation the sentence would be true if there were (exactly) two separate knocking eventualities and the total number of knocks were three, i.e., on one occasion John knocked once and on another he knocked twice. Hence, similarly to double in (26a), the quantificational effect of the postverbal three times in (28) is event-internal.

2.3 Distribution

In order to determine the distributional properties of two-time and double, I conducted a corpus study based on the Corpus of Contemporary American English (COCA), which is a representative corpus of English containing more than one billion words of text from various genres (Davies 2009). The study was complemented with a series of Google queries.

Let us first discuss the distribution of two-time. Based on the list of most frequent nominal collocates in the COCA, I conclude that two-time combines with nouns denoting social roles or capacities that can be classified into four groups. The first class consists of nouns describing award recipients or generally persons whose achievements received some sort of formal recognition, e.g., champion, winner, laureate, honoree, medalist, titleholder and recipient, see (29) for some attested COCA examples.

The second class is closely related to the first one and comprises competition participants, e.g., nominee, finalist, Olympian, qualifier, runner-up, competitor, candidate and loser, as illustrated by the sentences in (30).

The third class describes positions with a term, e.g., president, governor, prime minister, senator and captain, see some attested examples in (31).

Finally, nouns in the fourth class designate other socially salient capacities that are typically related to some kind of conventionalized situation, e.g., felon, graduate, divorcee, husband, patient and survivor, see (32).

All of the expressions discussed above are human nouns that designate social functions, capacities or roles. Crucially, it must be possible to acquire a given role repetitively so that there can be multiple “becoming” events associated with that role. This property can be detected by the became again VP test and it gives rise to the distributional constraint illustrated in (33)–(34).

Since the roles of a champion, a senator and a felon are properties that can be acquired repetitively, (33a) is felicitous, and thus (33b) works fine. On the other hand, the reason why the sentences in (34b) are weird is that being a person, an adult and a German are not properties one can repetitively gain, as demonstrated in (34a).

Furthermore, nominals in each of the four classes discussed above, recall (29)–(32), describe conventionalized roles that are typically acquired during a codified ceremony, e.g., an awards ceremony, an inauguration or a wedding. This suggests the importance of social convention in defining which roles are sufficiently significant to keep track of when individuals assume them. Consequently, only nominals designating socially salient functions are predicted to be felicitous with numerical frequency adjectives, see (35).

Though being a designated driver, see (35a), a birthday girl, (35b), and a life of the party, (35c), are all social roles that one can perform multiple times, out of the blue all of the sentences above sound odd. The reason is that it simply seems that the properties in question are not socially salient enough to be associated with a conventionalized ceremony, which would justify counting how many times an individual acquired them. However, in a proper context nouns denoting “insignificant” stage properties such as passenger (Barker 1999) can also be modified by two-time. For instance, consider the fragment in (36).

Similarly, the roles of a visitor and a driver can count as salient enough if a proper context is set, see (37)–(38).

Finally, in the context of a global pandemic, the second COVID-19-caused hospital admission of a single person is worth reporting, as witnessed in (39).

Let us now investigate the distribution of multipliers when they combine with event-denoting expressions. Based on the COCA collocation list complemented with a number of Google searches, I conclude that double selects for nominals that can designate complex eventualities.⁵ In general, such expressions fall into three classes. The first class contains nominals denoting actions affecting multiple objects, e.g., murder, homicide, date and play, see (40) for sentences excerpted from the COCA.

Nominal expressions in the second class describe actions involving quick repetitions, e.g., jump, kick, somersault, flip and shift, and (41) gives some attested examples.

Finally, the third and final class consists of eventualities that can have multiple aspects or consequences, e.g., whammy, crisis, victory and defeat, see (42).

All of the expressions discussed above describe situations with a complex internal part-whole structure. How it is conceptualized depends on a particular class. For instance, in the case of actions involving quick repetitions such as repeated kicks, jumps or flips a sequence of movements is perceived as a single event consisting of two essential subevents. Therefore, such eventive part-whole structures are construed in terms of temporal partitions. However, other types of complex eventualities could be partitioned along the spatial dimension, e.g., if the participating objects were affected simultaneously but occupied different regions of space. The role of multipliers is, thus, to determine how complex the internal structure of a given eventuality is irrespective of how exactly that structure is defined.

2.4 Data Summary

Let us now summarize the findings presented in this section. Both numerical adjectives and multipliers are cross-linguistically widespread expressions that involve reference to the corresponding number, e.g., the number 2 in the case of two-time and double. Crucially, it turns out that they both operate on hidden components within the semantic structure of the entire NP. Numerical frequency adjectives such as two-time differ from frequency adjectives such as occasional in that they lack adverbial readings, and thus they pattern with frequent. They combine with nouns denoting socially salient roles that are typically officially acquired in the course of a conventionalized ceremony. Consequently, expressions such as two-time quantify over covert events of acquiring of a given role by an individual. On the other hand, multipliers combine with nominals denoting complex events and operate on their internal part-whole structure. In doing so, they quantify over essential parts of an event, i.e., the parts that are conceived of as inherent to a particular event or even having a property comparable to that of the whole. In both cases, the quantificational effects of two-time and double are anchored to a particular individual or event, i.e., the expressions in question do not enter any scopal relationships. Finally, both what counts as a socially salient role and what is an essential part might be context-dependent.

With the empirical landscape clarified, let us now turn to the issue of how the intriguing semantic behavior of numerical frequency adjectives and multipliers can be accounted for. Before I present my proposal, let me spell out the conceptual framework that I will employ in the analysis.

3.1 Events

In general, I adopt a standard neo-Davidsonian framework (e.g., Carlson 1984; Dowty 1989; Parsons 1990). For that reason, I assume an extended ontology with the domain of events D_v. Events belong to the independent ontological category of the primitive type v and are conceived of as concrete spatiotemporal particulars, i.e., objects with a location and time (the idea dating back to Davidson 1967). Furthermore, events are associated with individuals participating in them via thematic roles such as agent, theme, instr (for ‘instrument’) and ben (for ‘beneficiary’) etc., which are part of the subatomic semantics of event-denoting expressions.

Furthermore, following Grimshaw (2011) I take all deverbal nominals with the exception of participant and result nouns to denote properties of events. In addition, I assume that similarly to event-denoting verbs they encode thematic roles. For instance, consider (43).

While the noun murder is a function of type ⟨e,⟨e,⟨v,t⟩⟩⟩, see (43a), the PPs of Kim and by Ida saturate the theme, see (43b), and the agent argument, see (43c), respectively. In the absence of linguistic expressions saturating the individual arguments, I assume that the variables corresponding to the agent and the theme get existentially bound in the course of composition.

Finally, in the spirit of Bach (1986) I assume that, similarly to individuals, events display part-whole relations. This means that we can talk about pluralities of events in an analogous manner as we talk about pluralities of individuals. For instance, let us assume that the plural noun murders has the denotation as in (44), where * is the pluralization operator defined as a closure under sum formation (Link 1983).⁶ In other words, it takes a set of singular events and yields a set containing these singular events as well as all pluralities one can obtain by summing them up. As a result, the noun murders can be used to refer to a plurality of murdering events.

With the basics related to the domain of events in place, let us now move on to the assumptions concerning quantification.

3.2 Numeric quantification

Following the standard literature on numerals (e.g., Rothstein 2017), I assume the domain of numbers D_n. I take numbers to be independent ontological objects of type n which, unlike individuals or events, are ordered. Furthermore, I posit that at some level numerals relate to abstract number concepts (the idea dating back to Scha 1981), and thus can be used to refer to objects of type n. In fact, for the purpose of this paper I will simply presume that basic cardinal numerals are names of numbers, i.e., that their meaning is as in (45).

As for numeric quantification, I adopt the main idea behind the theory of Krifka (1989) that counting is performed via additive extensive measure functions, i.e., operations that associate individuals or events with natural numbers, see (46), where o and ⊔ stand for overlap and sum formation, respectively.

The general idea is that when a measure function μ is applied to a plurality of objects of any type, it maps it onto a natural number corresponding to the number of objects constituting the plurality in question. For instance, for a plurality of two murders, it yields the number 2.

The final set of assumptions concerns augmenting the ontology with an additional category of roles. Since such an extension is rather non-standard, I will discuss the arguments for adopting it in slightly more detail.

3.3 Roles

Recent research on restrictive as-phrases and German copular clauses (Zobel 2017) as well as on Slavic derived social collective nouns (Wągiel 2021a) provides arguments for distinguishing between two types of human nouns, specifically class nouns like man and role nouns such as judge. The division corresponds to the distinction between individuals and abstract social roles advocated in the literature on theoretical computer science, knowledge representation and conceptual modelling (e.g., Sowa 1984; Steimann 2000; Loebe 2007). Roles differ from individuals in that they are independent social constructs representing functions, potentials and obligations of individuals who bear them. As such they occupy social space, which is an intricate web of expectations, duties, entitlements, prohibitions and other social dependencies (Wągiel 2021a). Though such a perspective is not widely adopted in natural language semantics, a similar notion of capacity has been proposed by de Swart et al. (2007), whereas Anderson & Löbner (2018) postulate the relevance of roles as part of higher-level social ontology.

As demonstrated by Zobel (2017), there are a number of linguistic phenomena sensitive to the distinction between class and role nouns. For instance, in languages such as German, Dutch and French role nouns can appear without the indefinite article in copular clauses while class nouns can never occur bare in this position (de Swart et al. 2007; Geist 2014). In English, a use of a role nominal without the indefinite article is also possible, but only for nominals designating single roles, as in (47).

Furthermore, predicates such as earn 3,000 euros are infelicitous in constructions with as-phrases containing class nouns, as witnessed by the contrast in (48).

In addition, there is an interesting difference in logical arguments of the form in (49)–(50). In the first premise in (49a), the descriptive content of the definite description consists of a class noun. As a result, the conclusion in (49c) is valid. On the other hand, if the first premise contains a role noun, as in (50a), the parallel inference does not arise, see (50c).

Note also that there might be certain properties of roles that do not apply to the individuals that bear them. For instance, in (51) being outmoded and dangerous does not pertain to any person but rather to the abstract object denoted by the DP this role.

Finally, there is no simple one-to-one correspondence between roles and their bearers since a single role can be performed by several individuals or even by no individual at all, compare the sentences in (52).

Given the contrasts discussed above, it seems that the evidence for the significance of roles in natural language is robust. Let us now discuss how they can be modelled within a formal system. In this paper, I will follow Zobel (2017) in simply assuming an additional domain alongside the more traditional domains of individuals and eventualities. In particular, I assume the domain of roles modelled as primitive ontological objects of type r ϵ D_r.⁷ As social constructs independent of their bearers, roles are conceptualized as capacities or functions of individuals. Consequently, role nouns differ from class nouns in that they denote properties of roles rather than properties of individuals. While nouns such as man are predicates of type ⟨e,t⟩, see (53a), nouns like judge are expressions of type ⟨r,t⟩, see (53b).

Crucially, despite the fact that individuals and roles are ontologically distinct objects, they are tightly related since the former typically perform the latter. In Zobel’s system, this relationship is captured by the shifting operation called play, which for a set of roles returns a set of individuals associated with those roles via events in which they perform them.

With all the background assumptions in place, let us now proceed to the formal account of the semantics of two-time and double.

4.1 Accounting for two-time

I propose that numerical frequency adjectives such as two-time quantify over events of acquiring of a socially salient, conventionalized role by an individual. For this purpose, let us define the measure function #(P), see (54), where ind specifies eventualities that are conceptualized as countable individuated units in the extension of P.

In prose, for a property of events P, the operation # would return a measure function yielding a number of eventualities individuated as singular spatio-temporal particulars of which P is predicated. In atomicity-based accounts, such eventualities would of course be modeled as atomic events. However, there are also alternative approaches that postulate the notion of an integrated whole (defined one way or another) instead of that of an atom (see, e.g., Moltmann 1997; Grimm 2012; Wągiel 2018). Furthermore, given the arguments that atomicity-based approaches fail to provide a satisfactory account for numeric quantification over parts of singular objects (see Wągiel 2018: Ch. 7), I do not want to commit here to such a view.⁸ Therefore, I will simply assume that whatever the mechanism of individuating units in the extension of P for the purpose of counting is the proper one, ind(P) yields a set of such units.

Let us add four more components. First of all, I assume the thematic role ben (for “beneficiary”), which yields the individual for whose benefit the action was performed. Second, I propose a special type of eventualities, which I call “becoming” events. Formally, they are specified by the operator bc (for “become”), which relates eventualities and roles in such a way that the beneficiary of the becoming event acquires the relevant role.⁹ The third ingredient that I postulate is the measure function #(bc)(r), which allows for the quantification over becoming events. Specifically, it yields a number of acts of assigning a role to a beneficiary individual. Finally, I posit a special higher-order property that defines conventionalized roles by which I mean socially salient capacities whose acquisition is typically associated with some kind of ceremony or other type of formal recognition. I will represent this property with conv_c(P), where the subscript c corresponds to a free variable whose value is provided by the context.

I postulate that in English the four components introduced above are encoded in the morpheme -time, which is shared by all numerical frequency adjectives in this language. The proposed semantics is provided in (55).

According to (55), -time denotes a function that requires a numerical input and returns an expression that counts the number of events of acquiring of a conventionalized, socially salient role (specified by the modified NP) by an individual. Typically, the general world knowledge concerning social life determines whether a particular role noun meets the conv_c(P) presupposition. However, a special context can also provide sufficient background for satisfying this condition, recall the discussion of (36)–(39) in Section 2.3.

Let us now demonstrate step by step how the semantics in (55) allows us to derive the truth conditions of a simple sentence such as (10a), repeated here as (56). Consider the tree in (57) (for the sake of clarity, I ignore the intricacies related to tense and aspect etc.).

First, the numeral two, which is an expression of type n, recall (45), combines with the morpheme -time in order to saturate its number argument, recall (55). As a result, we obtain a modifier that takes a predicate true of conventionalized roles and returns a quantificational expression equipped with the measure function #(bc) that would yield the number 2 for a plurality of two events if during each of those events an individual defined as its beneficiary acquired a role described by the modified role predicate. Next, two-time combines with the noun champion, which denotes a property of roles. Since common world knowledge provides that the role of a champion is tightly associated with a conventionalized becoming event, the definedness condition of two-time, i.e., the presupposition conv_c(P), is met and the result of the composition is a function from individuals to a function from roles to truth values. At this point, I assume existential closure over the role variable. In particular, the phrase two-time champion denotes a set of individuals who participated as a beneficiary in two events of acquiring the role of a champion. Finally, after combining with the subject proper noun Kim we obtain the truth conditions of the entire sentence. Specifically, (56) is true if Kim became a champion twice, which are the correct truth conditions.

Having discussed the proposed semantics of two-time, let us now turn to the analysis of double.

4.2 Accounting for double

While two-time counts events that are merely not immediately apparent, multipliers are expressions of a very different kind. That is because modifiers such as double are subatomic quantifiers, i.e., they do not target wholes but rather quantify over parts of an event denoted by the modified noun. As argued by Wągiel (2018), singular count nouns denoting entities encode complex internal part-whole structures that can be targeted by certain kinds of partitive expressions. In this paper, I propose that such subatomic structures also exist in the domain of events. In particular, I argue that singular events involve internal part-whole relations that are structured (see Simons 1987) and that natural language is sensitive to this phenomenon. For instance, proportional partitives modified by ordinal numerals enforce an interpretation based on the notion of subatomic structured parthood, see (58).

Crucially, the phrase first half in (58) does not designate an arbitrary part of the trip, but rather it specifies a structured sequence of subevents defined in terms of temporal contiguity. Consequently, the sentence is true only if what was wonderful constitutes a convex, i.e., gapless, chunk of the event.

In order to account for the subatomic quantificational effects of expressions such as double, I propose that they introduce the measure function ⊞(P) defined in (59), where esntl_c(P) yields parts that are perceived as essential for an event relative to a property P and context c.

The ⊞(P) measure function counts only parts of individuated singular, i.e., countable, events. Numeric quantification is ensured by employing the # operation. As a result, ⊞(P) provides a cardinality of parts of an event that are essential for an ascription of P. Typically such parts seem to have a property comparable to that of the whole, e.g., a double murder consists of two subevents each of which can be characterized as a murder in its own right, as discussed in Section 2.1. However, it is not always the case. To see why, let us consider the examples in (60).¹⁰

The expressions axel and lutz refer to figure skating elements in which a skater jumps, spins in the air once and successfully lands. Crucially, the phrases double axel and triple lutz in (60a) would not be true of an event in which the agent propels themself, spins in the air, lands and repeats the entire sequence since that would count as performing a single axel twice and a single lutz three times, respectively. Instead, the expressions in question designate events of a single jump with two and three spins in the air, respectively. Though performing every part of the sequence is necessary for a skating element to be evaluated as a double axel/lutz, the spin seems to be the most difficult action, and thus is perceived as what I dub the essential part. In a similar vein, double burpee in (60b) does not refer to an exercise in which one repeats the entire sequence of moves (that would be two burpees), but rather to a burpee with two push-ups after the squat thrust.

For the reason discussed above, I keep the notion of “essential part” somewhat underspecified. Though what counts as such a part is often conventionalized, different conceptualizations are possible under different circumstances, recall the discussion of temporal and spatial partitioning in Section 2.1. Therefore, for the purpose of this paper, I believe that it is sufficient to define it as in (61), where c stands for “context”.¹³

Since English multipliers are morphologically opaque, I assume that there is no reason to decompose them and that the reference to the number 2 is already encoded in the semantics of double, as provided in (62).

Thus, double is a subatomic quantifier that is dedicated to counting essential parts of an event relative to the property denoted by the modified event-denoting nominal.

With all the ingredients in place let us now consider how the pieces fit in. The tree in (64) gives a derivation of the sentence in (13a), repeated here as (63) (again ignoring all the issues related to tense etc.).

I assume that since there is no lexical material specifying the agent and the theme of murdering events designated by the deverbal noun murder, recall (43a), the corresponding individual variables are existentially closed before murder combines with double. The modifier double, recall (62), takes a predicate of events and returns a function with the in-built measure function ⊞(P) which yields the number 2 for each event that consists of two essential parts, i.e., subevents that typically have a property comparable to the property of the whole event. Therefore, after double and murder combine, the resulting phrase is an expression denoting a set of double murders. Finally, the subject DP that crime refers to an object of type v and as such it fits the event argument slot of double murder. Consequently, we obtain the meaning of the whole sentence. Specifically, (63) is true if that crime was a murder consisting of two parts each of which can be conceived of as a murdering event in its own right, i.e., a murder with two victims.