2.1 Linguistic relevance

It has been observed in the literature that abstract number concepts can have different properties than pluralities of individuals (Rothstein 2017). Some examples include being prime, odd or a natural number, see (3). On the other hand, such properties cannot be predicated of pluralities. For instance, (4a) and (4b) have very different truth conditions than the corresponding assertions in (3a) and (3b), whereas (4c) is simply infelicitous.

Furthermore, mathematical statements such as those in (5) require abstract-counting numerals. As evidenced by the infelicity of (6), even numeral phrases including the virtually vacuous noun thing are incompatible with mathematical environments, which call for numeric values and not pluralities of objects.

In addition, certain grammatical constructions require arguments referring to number concepts rather than to pluralities. For instance, only abstract-counting numerals can be complements of count (up) to, as witnessed by the contrast between (7a) and (8a). Moreover, the dimension of comparison in (7b) and (8b) is different. While in the case of arithmetical entities, what is relevant is their relative ordering on the number scale, the truth conditions in (8b) concern the size of individual objects making up a plurality. For instance, the sentence would be false when comparing three onions and two pumpkins. Notice also that number agreement in (7b) and (8b) is different.

Yet another property of abstract-counting numerals is that they do not give rise to scalar implicatures (Bultinck 2005). While the most natural way to understand (9) is that the addressee needs to give the speaker at least three things to pay their debt, (10) lacks an analogous lower bounded reading and can only get an exact interpretation.

Finally, only object-counting numerals allow for modification by comparative and superlative modifiers as well as so-called prequantifiers, e.g., the English universal quantifier all (cf. Corbett 1978). While the sentences in (11) are regular felicitous expressions, the examples with abstract-counting numerals in (12) are funny.

So far, we have seen that there is robust evidence that the conceptual distinction between abstract and object counting is encoded in natural language semantics and grammar. Given that all the examples discussed heretofore include the very same morphological form, an obvious question regarding the relationship between abstract and object counting arises. ¹

2.2 Relating abstract and object counting

The contrasts discussed in the previous section call for an explanation of the dual life of cardinals such as English three. In order to account for the relationship between number concepts and pluralities, most mainstream approaches take the object-counting function of numerals as the basic one from which the abstract-counting meaning can be derived. For instance, Rothstein (2017) following Landman (2003) takes cardinal numerals to denote cardinal properties. As such they are intersective modifiers of type 〈e, t〉. ² The abstract-counting meaning is then derived by a special shifting operation which applies to a cardinal property and yields a number-denoting expression of type n.

Another account assuming essential object-counting semantics for numerals is proposed by Ionin & Matushansky (2018). They posit that all cardinals are predicate modifiers (type 〈〈e, t〉, 〈e, t〉〉) which provide the cardinality of a partition of a plural individual. In order to derive the abstract-counting meaning, they propose a null nominalizing suffix. Its function is to yield the number corresponding to the object-counting denotation of a given cardinal.

Finally, Kennedy (2015) also argues that the basic meaning of numerals is object-counting. On his account, cardinals denote quantifiers over degrees (type 〈〈d, t〉, t〉), e.g., three denotes a set of degree properties whose maximal value is 3. The abstract-counting meaning is then obtained as a result of the consequent application of the standard type-shifting operations be and iota to the degree-quantifier object-counting semantics.

Though in recent years treating abstract counting as derived seems to have become prevalent, there are also proposals deriving the object-counting meaning of numerals from a number-denoting core. Two examples of such accounts are Krifka (1995) and Hackl (2000). In Krifka's system the underlying meaning of cardinals amounts to the reference to a number concept. In addition, depending on a language this core meaning can be accompanied by an overt or covert object-counting element, e.g., a classifier. Likewise, Hackl starts with the abstract-counting meaning, which can be then shifted to a GQ-style object-counting determiner.

What both families of theories briefly described above have in common is that they presume a relationship between the abstract- and object-counting component of numerals, an assumption justified by the identical form of numerals like three in both types of uses. How they account for the relationship in question is represented schematically in (13a) and (13b), where abs and obj stand for ‘abstract-counting’ and ‘object-counting’, respectively, and α and β are operations responsible for the shifts.

On the assumption that morphological marking reflects meaning composition, the two approaches make different predictions regarding morphological make-up of cardinal numerals. In particular, (13a) predicts that across languages one should find numerals expressing α overtly. This means that cross-linguistically, abstract-counting numerals should exhibit a tendency to be morphologically more complex than object-counting numerals. On the other hand, (13b) predicts the opposite, i.e., object-counting numerals should be morphologically more complex than abstract-counting numerals.

In the next two sections, we will explore cross-linguistic morphological marking patterns focusing on the marking of the two types of numerals. In Section 3, we present data that call into question the idea that the object-counting meaning of cardinals is basic, as in (13a), while supporting the view offered in (13b). In Section 4, we discuss examples indicating that even though the view in (13b) is essentially correct, it is still too simplistic in that even abstract-counting numerals are not grammatically atomic, and break down into at least two smaller meaning components.

3.1 Syncretism

The first pattern we will investigate makes no overt distinction between abstract- and object-counting numerals. In other words, both functions are expressed by the same formal exponent. For instance, in English the prenominal modifier three is used to quantify over individuals in (14a). In this use, it has the same shape as the name of a number concept, see (14b). We call such numerals syncretic.

From a typological perspective simplex syncretic numerals seem to be the most widespread type of numerals across languages. In (15)–(17), we illustrate this type by examples from Czech, Turkish and Basque, respectively. ⁴

The syncretic pattern is not very revealing with respect to the relationship in question. In the next section, we examine cases where the abstract- and object-counting numerals differ formally.

3.2 Stacking

The second pattern involves a formal asymmetry between the two types of numerals in question. In particular, object-counting numerals are morphologically more complex, i.e., include an additional morpheme compared to their abstract-counting counterparts. We shall refer to the formal relationship as stacking and numerals that arise as a result of stacking will be called augmented.

Typically, the asymmetry is characteristic of classifier languages. In these languages, the additional morpheme used with object-counting numerals is often referred to as a classifier. For instance, in Mandarin, the numeral cannot appear bare when modifying an NP, see (18).

On the other hand, the numeral has to be bare in an abstract-counting environment, see (19). ⁵

It should be noted that there is usually more to the semantics of classifiers than just allowing for numerical quantification. Classifiers often have also other functions, namely they introduce certain requirements regarding the kind, shape and size etc. of referents of the modified noun (e.g., Aikhenvald 2000). However, gè belongs in the class of the so-called general or default classifiers, which can be used to count virtually any type of object. Thus, rather than conveying some specific characteristic of the counted object, we understand Mandarin gè to be at its core a pure object-counting marker. ⁶

The same pattern as in (18)–(19) is also attested in Vietnamese. ⁷ As witnessed by the contrast in (20), the numeral hai ‘two’ must combine with an additional morpheme in an object-counting construction, where cái is a general classifier for inanimate objects. ⁸

On the other hand, in an abstract-counting environment like (21), the cardinal has to appear bare.

Yet another example comes from Adang (Alor-Pantar), a Papuan language with a developed classifier system. Similar to Mandarin and Vietnamese, Adang object-counting numerals are bimorphemic, as demonstrated in (22), where paʔ is a general classifier (Robinson & Haan 2014, 250). On the other hand, abstract-counting cardinals are monomorphemic, see (23) (Klamer et al. 2014, 354).

Augmented numerals occur also in Indo-European. For instance, in Bhojpuri (Indo-Aryan) the morphemes go and t_˙ho function as general classifiers that can be used to quantify over an object of any type, size or shape. An example of an object-counting use is given in (24), where the bimorphemic form consisting of the numeral root and go is used as a prenominal modifier (Barz & Diller 1985, 166; adapted). However, in an abstract-counting context neither go nor t_˙ho is employed, see (25) (Barz & Diller 1985, 165).

The final example to be discussed here comes from Yoruba (Niger-Kongo), which is a non-classifier language. At least in some varieties of Yoruba, the distinction between abstract-counting and object-counting numerals is marked by the prefix m-. ⁹ As demonstrated in (26) and (27), only prefixed forms can be used as nominal modifiers whereas only bare numerals can express abstract counting. ¹⁰

Cross-linguistically, the asymmetry discussed in this section is a relatively frequent phenomenon. Other languages with simplex numerals exhibiting the stacking pattern reported in the literature include, e.g., Thai (Hundius & Kölver 1983), Japanese (Sudo 2016), Assamese (Indo-Aryan) (Kalita 2011, 184) and Teiwa (Alor-Pantar) (Klamer 2010, 144–145; Klamer et al. 2014, 351). We interpret this phenomenon as evidence for the view that the abstract-counting meaning is basic and the object-counting meaning is derived from it, recall (13b) in Section 2.2.

3.3 Suppletion

Finally, the third marking pattern to be discussed here concerns suppletive numerals. In such a case, for a given number there are two simplex morphologically unrelated forms one of which expresses abstract counting whereas the other is used in the object-counting function. Admittedly, this pattern is very rare and it is challenging to provide an uncontroversial instance of a suppletive abstract/object-counting pair. Nonetheless, Hurford (1998) gives the example of numeral suppletivism in Maltese in which there are two morphologically unrelated forms corresponding to the number 2, namely tnejn and żewġ (both ‘two’). As evidenced by the contrast in (28), only żewġ can convey the object-counting meaning as a nominal modifier (Borg 1974, 293). On the other hand, only tnejn is possible in an abstract-counting environment such as (29) (Borg 1987, 62). ¹¹

As already mentioned above, the suppletive pattern is very rare and if identified, it typically applies to only one numeral in a language. However, according to the literature it is not restricted to Maltese ‘two’. Other examples of simplex suppletive numerals reported so far include, e.g., the cardinals ū and hlɛh (both ‘one’) in Palaung (Austroasiatic) (Greenberg 1978) as well as the Ibani (Eastern Ijaw) numerals gbẹrẹ and ng $\underset{\dot{}}{i}$ ẹ (both ‘one’) and oyi and atie (both ‘ten’) (Obikudo 2016). In Section 7, we will discuss in detail an additional example from colloquial Mandarin (He 2015).

The final note on suppletion we need to make here is that stacking and suppletion may in fact combine. As a case in point, consider the Mandarin numeral ‘two’. The numeral has the form èr for abstract counting, see (30a) (from Po-Ching & Rimmington 2015, 26). In (30b), we see that the object-counting use features a suppletive root liǎng ‘two’, crucially accompanied by the general classifier gè (He 2015, 198). It is ungrammatical to use the abstract-counting èr in such phrases, see (30c) (He 2015, 198). In sum, the conclusion is that in the example (30b), stacking and suppletion combine.

3.4 A side note on compound numerals

In our investigation, we often found a relation (suggestive, even though possibly an imperfect one) between the shape of abstract-counting numerals and the shape of numerals within compound numerals. We use the term compound numerals to refer to numerals like two hundred five that arise by combining two or more basic numeral roots.

In order to see the parallel between complex numerals and abstract counting, recall the Mandarin example (19) repeated in (31). The example shows that classifiers are absent in mathematical statements.

As pointed out by He (2015, 195), the classifier is also (unsurprisingly) missing in arithmetical statements like (32), where the property of being a prime number (which, recall, is a property of abstract numbers) is predicated of a compound numeral composed of three other numerals (‘two’, ‘ten’ and ‘three’).

These three numerals jointly specify (by multiplication and addition) the number 23, which is, indeed, a prime number. What is interesting for us is that in order to arrive at the meaning of the compound numeral, the arithmetical operations such as multiplication and addition must be postulated among the numeral roots, namely [23 = 2 × 10 + 3]. Such operations are, in turn, defined over abstract-counting numerals, and we are therefore not surprised that there is no classifier in between any two numerals in (32).

Interestingly, the absence of classifiers in between numerals is also observed in object-counting contexts, see (33) from He (2015, 190).

He (2015) gives multiple arguments in favor of the proposal that the four basic numerals in (33) (i.e., èr, shí, sān and wàn) form a constituent to the exclusion of the classifier. According to He, the numerals are first joined together compositionally by a series of arithmetical operations [(2 × 10 + 3) × 10,000] and they jointly form a complex cardinal ‘two hundred thirty thousand.’ This internally complex compound cardinal is then used to quantify over the denotation of the noun, at which point the classifier becomes indispensable, turning the abstract-counting numeral into an object-counting numeral (cf. Cinque 2020). In sum, the absence of classifiers inside complex numerals suggests that individual numerals inside compounds are similar (both semantically and morphologically) to abstract-counting numerals.

This view is supported by the fact that the form of the initial numeral ‘two’ in (33) is èr. Recall from (30), repeated in (34), that èr ‘two’ is a dedicated abstract-counting form of ‘two’; see (34a). Therefore, the appearance of èr in the object-counting compound numeral in (33) indicates that inside such numerals we find the abstract-counting structure.

He (2015, 198–199) points out that the abstract-counting form èr is also found in additive numerals. This is interesting, because the sequence èr+clf is generally ungrammatical, recall (34b). However, in an additive numeral like ‘twelve’, the form èr must be used in object-counting contexts, see (35a). The form liǎng ‘two’, which is found before the classifier in (34c), is impossible in compound numerals, see (35b) (He 2015, 198). Once again, the fact that èr is used in compound numerals suggests that the internal structure of such numerals is similar to abstract-counting structures.

The reason why these facts are relevant to note is because abstract-counting contexts (arithmetical examples) are often missing in grammars. This limitation may be overcome by investigating compound numerals, which (in a number of cases) seem to mirror the structure of the Mandarin example in (35a).

Consider, for instance, the examples found in Upper Necaxa Totonac, as discussed in Beck (2004, 26–27). In Upper Necaxa Totonac, numerals in object-counting functions precede the noun and they are themselves preceded by a classifier, see (36a) for the numeral ‘one’ with a classifier for bunches. In (36b), we see the numeral ‘ten’ with a general classifier preceding it. This numeral is given here in order to see that the numeral ‘eleven’ in (36c) is composed by juxtaposing the roots of the two numerals in (36a–b).

The classifier is found only once in the compound numeral, suggesting that (on analogy with Mandarin) the numeral ‘eleven’ is formed first as an abstract-counting numeral [10+1]. Only after the compound numeral is formed, the classifier attaches to it, yielding an object-counting numeral. If correct, this suggests that Upper Necaxa Totonac provides us with yet another example of the stacking pattern.

The final example of this type we shall mention here comes from the Nilotic language Luwo (Storch 2014, 271). In this language, the numerals ‘one’ to ‘five’ are basic (not compound) and we list them in the first column in Table 2. It is interesting to note that they all start with an initial á, which is placed in italics and separated as an independent morpheme. The residue of the numeral is bold. The same strategy is applied to the numeral ‘ten’ at the bottom of the second column, which is also basic (not compound).

Let us now turn to the numerals ‘six’ to ‘nine’. These are contained in the second column. We can see that they are always composed of the numeral ‘five’, followed by bí (glossed as ‘plus’ in Storch 2014) and an appropriate numeral from the basic set. What is most relevant for our purpose is that the second numeral (after bí) lacks the initial á-. This suggests a structure where the á is a separate morpheme that attaches to the compound numeral as in: á-[5+1], á-[5+2], etc. This again suggests that the compound numeral is formed first (through addition), and only after the compound numeral is created, we attach the ‘general classifier’ á- in order to turn the abstract numeral into an object-counting numeral. As in Upper Necaxa Totonac, this suggests that Luwo numerals belong to the stacking type, since the general classifier is missing in compound numerals, which resemble abstract-counting structures.

The same conclusion is suggested by looking at the multiplicative numerals ‘forty’, ‘sixty’, ‘eighty’ and ‘hundred’ in the last column. These numerals contain the base jé- glossed as ‘person’ in Storch (2014), but obviously standing in for ‘twenty’. This invariant base is never preceded by á- and it is therefore classified as a syncretic numeral in our typology. What is relevant is that when jé- is multiplied, it is followed by the á-less numeral roots of the numerals ‘two’ to ‘five’. The absence of the general classifier á- once again suggests that the numerals 1–5 represent a stacking pattern that obtains between á-less abstract-counting numerals and object-counting numerals (the latter include the á-). ¹²

3.5 Summary

The data explored in this section (the stacking pattern discussed in subsection 3.2 in particular) suggest that across languages, the abstract-counting component of numerals is more primitive than the object-counting one. This is because of the fact that in the stacking cases, the object-counting numeral morphologically contains the abstract-counting numeral. In the next section, we explore cross-linguistic evidence suggesting that even abstract-counting cardinals involve more structure than typically assumed.

4.1 Syncretism

Let us begin the empirical discussion from complex syncretic numerals. A good example is Hawaiian, where the cardinals 1–9 always consist of a root and the prefix ‘e-. ¹³ As witnessed in Table 4 (based on Elbert & Pukui 1979, 158–160), the interpretation of ‘e- as an affix is corroborated by the fact that it is replaced by another morpheme in distributive numerals. ¹⁴

An example of an object-counting use of the numeral ‘e-lua ‘two’ in Hawaiian is provided in (37) where the prefixed form precedes the noun (Elbert & Pukui 1979, 159).

Interestingly, the very same shape is also used in mathematical discourse to express the abstract-counting function, as demonstrated in (38) (Pukui & Elbert 1986, 501; adapted). These data show that unlike, e.g., in Mandarin, Hawaiian cardinals are morphologically complex in both functions under consideration.

Another example of complex syncretic numerals comes from Sanzhi Dargwa (Northeast Caucasian). In this language, all basic cardinals except for ca ‘one’ are morphologically complex. They consist of a root and a special derivational suffix -(j)al, see Table 5 (based on Forker 2020, 130–135). ¹⁵ The affixal status of -(j)al is proved by the structure of ordinal and multiplicative numerals in which -(j)al is replaced by another element.

The bimorphemic forms in the left compartment of Table 5 are used as both object- and abstract-counting numerals. For instance, in (39), the cardinal xu-jal ‘five’ functions as a prenominal modifier (Forker 2020, 131; adapted).

The very same form is also used in abstract-counting environments such as (40) (Forker 2020, 137; adapted).

Having introduced the pattern of complex syncretic cardinals, we now turn to a language with complex numerals that exhibit the stacking pattern. ¹⁶ This language behaves similarly to, e.g., Mandarin or Yoruba, but features abstract-counting numerals that are morphologically complex.

4.2 Stacking

The relevant pattern is found in Vera’a (Vanuatu). In this language, the cardinals 1–5 are morphologically complex, containing a root preceded by the prefix vō-/ve-, see Table 6 (based on Schnell 2011, 73–74). ¹⁷

The status of vō-/ve- as a separate morpheme is evidenced by the fact that it is replaced by the prefix vag- in multiplicative numerals. Analogous prefixes on basic cardinals are attested in other Vanuatu languages as well, e.g., in Vurës (Malau 2016, 126).

In abstract-counting contexts like the one exemplified in (41) (from Schnell 2011, 83), we find the numeral in the same form as given in the table. This pattern is similar to the ones seen in the previous section, where abstract-counting numerals were morphologically complex.

What is different about Vera’a is that in NPs, i.e., in the object-counting function, the numerals appear with an additional morpheme ne referred to as a ‘ligature’ by Schnell (2011, 74), see (42).

It is interesting to note that according to Schnell’s description, the ligature is specific to numerals, i.e., it is not a general linking morpheme. In sum, Vera’a instantiates a pattern similar to Mandarin, because its object-counting numerals stack an additional marker on top of the abstract-counting numeral. The new thing about Vera’a is that the abstract numeral is already complex, so we have a sequence of three morphemes in Vera’a. ¹⁸

The Vera’a data suggests that the abstract- and object-counting functions consist of two and three semantics components, respectively, each of which can be exponed by a separate morpheme. Although instances of such a strongly agglutinative marking are scarce, the strength of the evidence from Vera’a indicates two things. First, morphologically complex abstract-counting cardinals suggest that we need a complex structure for them. And further, the data shows that even morphologically complex abstract numerals may be further augmented in their object-counting function.

4.3 Suppletion

The final pattern that we would like to discuss is attested in a subset of languages that would be classified as obligatory-classifier languages. In such languages, the numeral in the object-counting function is always accompanied by one of a potentially large set of classifiers. We depict this schematically in (43a). This is similar to Mandarin and other languages discussed in Section 3.2.

However, unlike in Mandarin, the languages that we shall discuss in this section retain the classifier in the abstract-counting function, as depicted in (43b). The particular classifier tends to be one of the set of object-counting classifiers. Following the tradition, we will call it the general classifier.

If we analyze the pattern in (43), we realize that such languages represent a mixture of two ‘pure’ systems. On the one hand, such numerals have something in common with the complex syncretic numerals discussed in Section 4.1. The similarity consists in that the particular form consisting of a numeral + clf 1 (the general classifier) has both an object-counting use and an abstract-counting use.

However, the language type in (43) is different from the pure complex syncretic pattern (as exemplified by the languages discussed in Section 4.1) in that the abstract-counting classifier is not always found in the object-counting function and may be replaced by other (more specific) classifiers. If we only focused on these other classifiers, we would get a pattern where the abstract-counting classifier is replaced by an unrelated (i.e., suppletive) object-counting classifier, as in (45).

Based on these considerations, we shall treat here the mixed type separately from the syncretic type, and, in fact, as closely related to the suppletive type. However, we admit that these languages do not represent the pure suppletive type depicted in (45). Such a pure suppletive type of language was not found among the languages we looked at. ¹⁹

As the first example of the mixed/suppletive type, consider Shuhi (Qiangic, Sino-Tibetan). This language has a relatively rich inventory of classifiers. They are obligatory in numeral phrases and their choice depends on the modified noun. One of them is ko ³⁵, which serves as the general classifier, see (46a) (Qi and He 2019, 65). ²⁰ The example (46b) contains a different classifier, namely ƫshu ⁵⁵.

Importantly, unlike languages such as Mandarin or Yoruba, Shuhi numerals require the general classifier ko ³⁵ also in abstract-counting contexts such as (47) (Qi & He 2019, 69).

A similar set of facts is found in Mokilese (Austronesian). In (48), we can see the examples of two classifiers (Harrison 2019, 91). The general classifier -w is seen in (48b). The pair nicely illustrates the general nature of -w.

The example in (49) again contrasts the general classifier -w in (49a) and the classifier -pas for long thin objects, see (49b). In (49c), we see the animate classifier men.

The abstract-counting context is shown in (50). We can see that the numeral walu- ‘eight’ is followed by the general classifier -w (Harrison 2019, 94).

A similar system is found in Abkhaz (Northwest Caucasian). In the morphological system of this language, nouns are grammatically classified as human or non-human, with human nouns including the classes of masculine and feminine nouns (Chirikba 2003, 24–25). Furthermore, the cardinals 2–10 can take two forms, as demonstrated in Table 7 (based on Chirikba 2003, 34–35 and Hewitt 2010, 33), each of which is clearly morphologically complex. ²¹

Forms with the suffix -ba are object-counting cardinals used to quantify over referents of non-human nouns, see (51a). On the other hand, numerals with the suffix -j°ə(k’) are dedicated for counting human individuals, see (51b) (Hewitt 2010, 35; transliterated and adapted).

Both types are morphologically marked, i.e., there is no structural asymmetry between the two forms. However, despite this fact, only non-human numerals can express the abstract-counting meaning, as evidenced by the contrast in arithmetical statements such as (52). ²²

Admittedly, none of the patterns discussed here is an unequivocal instance of abstract/object-counting suppletivism. Nevertheless, based on the reasoning presented at the start of this section, we treat this mixed type as a relevant case of suppletivism, because this is what one part of the system amounts to. The reason why it is important to capture the fact that the general classifier is replaced by other classifiers is because it is not necessary for it to be so. For example, in the Mayan language Chuj, the more specific classifier (called nominal classifier) may (optionally) appear in addition to the numeral classifier. For example, in (53), the animate numeral classifier wanh is followed by the nominal classifier nok’ ‘animal’ (Royer 2017, 33).

While we do not wish to suggest that Chuj represents a stacking pattern (there is much to be understood about the second – so-called nominal – classifier), we do think that the possibility of a system like the one in Chuj highlights the need to explain why the general classifier does not combine with other classifiers. And this is an essence of what we label as the complex suppletive pattern.

Before we propose how to account for the morphosemantic variation in cardinal numerals discussed so far, let us summarize our findings.

6.1 Key intuition

The core intuition behind our proposal is that numerals are at their core scalar entities. Specifically, we assume that each numeral is associated with an interval on the number scale. Assuming that the number scale has a non-arbitrary starting point corresponding to 0 and that each numeral designates a different upper bound, such intervals can be viewed as ordered in terms of their length.

It seems to us that there are at least three other strands of research that share a similar intuition. The first one concerns an attempt to explain the meaning of spatial and directional numeral modifiers as in above three and up to three (Nouwen 2016). Furthermore, in a number of approaches to degree semantics, degrees are taken to refer to intervals rather than to points on a scale (e.g., Seuren 1984; Kennedy 2001; Schwarzschild & Wilkinson 2002). Finally, the recent analysis of morphological marking patterns in comparative formation suggests a similar direction (Vanden Wyngaerd et al. 2020). ²³

6.2 The role of classifiers

The standard assumption regarding the role of numeral classifiers in languages such as Mandarin concerns the semantics of nouns (e.g., Chierchia 1998; Borer 2005; Rothstein 2010; Scontras 2013). That is because according to the received view the denotations of nouns in classifier languages are mass-like which makes them incompatible with the meaning of numerals. Thus, classifiers are required in order to compensate this semantic ‘deficit’ and turn the denotations of nouns into countable ones.

In this paper, however, we embrace an alternative view on the role of classifiers in numeral phrases (Krifka 1995; Bale & Coon 2014; Sudo 2016). One of the problems the received view encounters concerns languages such as Mi’gmaq (Algonquian) and Ch’ol (Mayan) in which certain numerals require classifiers whereas others appear only bare (Bale & Coon 2014). Furthermore, even in well-studied obligatory classifier languages such as Japanese certain quantifiers do not take classifiers (Sudo 2016). If the occurrence of the classifier were due to the semantics of nouns, those facts would be unaccounted for since there is no evidence that nouns change their meaning depending on a quantifier that modifies them. Consequently, according to the alternative explanation of the role of classifiers, they are required to compensate semantic ‘deficits’ of numerals rather than nouns. The idea is that in classifier languages numerals lack a semantic component that would allow them to quantify over entities, recall (13b) in Section 2.2. This role is, thus, assigned to classifiers.

This semantic view is also confirmed by a comparative study of the ordering of classifiers, nouns and numerals by Cinque (2020). What Cinque observes (building on previous work) is that classifiers never occur on the opposite side of the noun than the numeral; orders such as *numeral > N > clf or *clf > N > numeral are not found. This receives a natural explanation under the hypothesis that [ numeral + clf] form a unit/constituent, which is what we are proposing here as well. Furthermore, as long as the classifier is adjacent to the numeral, it may in fact be separated from the noun by the numeral, yielding the order clf > numeral > N. Such examples are attested, see, e.g., the data from Upper Necaxa Totonac in (36) or Hawaiian in (37). Both of these facts suggest that the syntactic structure contains a constituent [clf numeral], which does not split and is (as a unit) positioned with respect to the noun.

With the crucial assumptions introduced, let us now move to the details of the proposal.

6.3 Semantic features

In order to capture the morphological complexity of basic cardinals discussed in Section 3 and 4 and summarized in Table 8 in Section 5, we propose three semantic primitives: Scale, Num (for ‘number’) and Cl (for ‘classifier’), which we take to be cross-linguistically stable ingredients of the meaning of numerals.

The first feature, namely Scale, denotes a closed interval, i.e., a set of natural numbers (type 〈n, t〉), see (54a). As indicated by the subscript, the semantic content is lexically encoded, and thus specific for each distinct cardinal numeral. In other words, there is Scale ₁ for ‘one’, Scale ₂ for ‘two’ etc. What all Scale elements have in common is that their lower bound is always 0. They differ in what the upper bound is. For instance, in the case of the cardinal three, Scale ₃ denotes the set of integers between 0 and 3, as in (54b).

The second primitive is the syntactic head Num. As such it is an invariant functional element shared by all cardinal numerals. The meaning of Num is a function from intervals to numbers. In (55a), the maximization operator max takes a set of integers denoted by Scale and returns the greatest value from that set. As a result, it creates a name of a number concept (type n) compatible with abstract-counting contexts calling for numeric arguments. For instance, when applied to the interval [0, 3] denoted by Scale ₃, it will yield the number 3, see (55b).

The final component we propose is the Cl head, which introduces object-counting semantics. Its role is to turn a number into a counting expression corresponding to that number. In particular, we postulate that the meaning of Cl is a function from an integer to a predicate modifier supplied with the pluralization operation * (Link 1983) and the measure function #(P), which maps a plurality of individuals to a numeric value corresponding to the number of individuals making up that plurality, see (56a) (Krifka 1989). In other words, Cl is an expression of type 〈n, 〈〈e, t〉, 〈e, t〉〉〉 that creates an object-counting numeral. ^{24 , 25}

Combining the components in (54) and (55) in a compositional manner gives us the abstract-counting structure in (57). As one can see in the tree, the entire NumP refers to a number concept. Such a denotation is thus compatible with expressions calling for numerical arguments such as be a prime number or plus which we take to be of type 〈n, t〉 and 〈n, n〉, respectively.

(57)

Adding the Cl head as defined in (56) on top of (57) leads in turn to (58) which represents structure of the object-counting numeral. The semantic contribution of Cl derives us a predicate modifier which when combined with a noun returns a set of pluralities in the denotation of that noun whose cardinality is the number referred to by the NumP, e.g., 3 in the case of (58).

(58)

Our model is broadly based on the idea that the meaning components postulated above are uniformly structured across languages and they must all be pronounced, though languages differ in how they pronounce them. So, for instance, the English phrase three apples and the Mandarin classifier construction sān gè qiú ‘three balls’ get the corresponding denotations in (59a) and (59b), respectively (cf. Krifka 1995).

In the next section, we will show that it is possible to use the cross-linguistically uniform structures given in (57) and (58) and still account for language variation in the domain at hand, summarized in Section 5.

7.1 Simplex syncretic numerals

Nanosyntax is a realizational post-syntactic model of morphology. This means that syntactic structures are built first from abstract meaningful components, which are then mapped onto their pronunciation during the so-called lexicalization procedure. A crucial element in the lexicalization process is the lexicon, language-particular list of lexical items. In Nanosyntax, each lexical item is understood as a link between representations belonging to different modules. Concretely, they link a particular syntactic structure to a particular phonology and/or concept.

Due to the linking, lexical items serve as ‘translation’ instructions of sorts. We can read them as follows: if syntax generates the stored representation S, then this representation is linked to a phonological representation P, which is associated to the lexical entry. Under this view, lexical entries have the restricted format < S,P >, where each of S and P is a well-formed representation in the respective module. ²⁶

In order to have a concrete lexical item as an example, consider the English numeral three. This numeral can act as an object-counting numeral. The structure of object-counting numerals according to our proposal is as given in (60). In order to express the fact that this structure can be pronounced by the numeral three, we link the relevant structure to the phonology /θɹiː/. This linking is shown in (61). This numbered item corresponds to the lexical item for the numeral three.

As already said, the entry for three can be read as a ‘translation’ instruction: when syntax builds ClP, this structure can be spelled out as /θɹiː/.

In Nanosyntax, the decision whether a lexical item can (or cannot) spell out a particular structure is dependent on the notion of matching. The rule is that whenever matching between a lexical entry and the structure obtains, the lexical entry can spell out the structure. Matching is based on identity: whenever the syntactic structure is identical to something stored in the lexicon, the item that contains the relevant tree matches the syntax tree. This is encoded by the so-called Superset Principle, see (62).

The Superset Principle leads to the consequence that a lexical entry matches any structure that is contained inside the lexical entry. This means that when syntax only builds the abstract-counting numeral – with structure as in (63) – the lexical item for the numeral three can be inserted, because it matches the structure. Matching (recall) is defined by The Superset Principle: since the lexical entry for the numeral, repeated in (64), contains the structure in (63), matching obtains and spellout can take place. Spellout is encoded by the circle around NumP in (63). The circle in (64) just highlights the fact that the lexical entry literally contains a part that is identical to the structure spelled out.

As a result, when a language has a numeral with an entry such as (64), we get a syncretism between the object- and abstract-counting function.

7.2 Simplex suppletive pattern

Let us now address the question how the simplex suppletive pattern arises. This pattern is found in languages where a single numeral (e.g., ‘two’) has two lexical entries, each for a different function. As an example, recall the Maltese numeral ‘2’, which has the shape tnejn for abstract counting and żewġ for object counting, recall (28)–(29).

The lexical entries are given below. The entry for the object-counting numeral is shown in (65), the abstract-counting numeral is in (66).

With these entries in place, let us consider the spellout of the object- and abstract-counting forms. These are depicted in (67) and (68) respectively. In the case of the object-counting numeral, we only have one matching lexical item, namely żewġ. The reason is that the other numeral (tnejn) does not match the relevant constituent (it does not contain it). Therefore, the object-counting numeral is spelled out as żewġ.

Let us now turn to the abstract-counting structure in (68). This structure is contained in both lexical entries. Therefore, both entries match, and they can both be used to spell out the structure. In such a situation, competition among the entries arises. The competition is resolved according to the so-called Elsewhere Condition (Kiparsky 1973), which demands that ‘the more specific’ entry wins. In a system where insertion is governed by the Superset Principle, the most specific entry is the one that has fewer superfluous features. In the case of the abstract-counting structure depicted in (68), this is tnejn, whose entry has no superfluous features. On the other hand, żewġ does have one superfluous feature and it therefore loses in the competition. In sum, the pair of lexical entries in (65)–(66) gives rise to the simplex suppletive pattern.

7.3 The simplex stacking pattern

In the simplex stacking pattern (which is found, for instance, in Mandarin), we encounter (for the first time) a situation where two morphemes are needed to spell out a particular structure. We shall begin by providing the lexical entries for the numeral ‘three’ (sān) and for the default classifier gè; see (70) and (71) respectively.

We can see that the lexical entry for the numeral in (70) corresponds to the abstract-counting structure. Therefore, when syntax builds such a structure – as in (72) – the numeral can be used to spell it out. As a result, we correctly model the fact that we only see the bare numeral in the abstract-counting function in Mandarin.

Consider now the object-counting structure in (73). Here, the numeral cannot spell out the full structure because it does not contain it in its entry. (Hence, no match and no spellout.)

The maximal constituent matched by the numeral is NumP, circled in (73). This still leaves the head Cl and its projection ClP without spellout. Leaving projections without spellout is not allowed in Nanosyntax: in order to externalize meaning, all features must be spelled out (cf. Fábregas 2007). The intuition which we shall follow is that the two unlexicalized nodes in (73) are spelled out by the classifier gè. These are also precisely the two nodes that are contained inside its entry in (71).

In order to see how this intuition is technically implemented in Nanosyntax, we must introduce the idea of cyclic spellout and spellout-driven movement. Cyclic spellout in Nanosyntax means that every time a new feature is introduced in the syntax, it must be lexicalized before the derivation is allowed to continue. We state this in (74).

The consequence of adopting cyclic spellout is that structures are built in small steps, where each Merge F operation is immediately followed by spellout. For example, in order to build the structure of the object-counting numeral (corresponding to ClP), we must first build – and spell out – the abstract-counting numeral. Therefore, the tree in (72) can be considered the first step in the construction of the object-counting numeral in (73). The tree in (73) (with a circle around the NumP) then represents a stage in the derivation where we have added a new feature to the structure in (72), and we now need to spell out the new structure.

In order to see how exactly spellout operates, let us present in (75) the so-called spellout algorithm. This algorithm implements the idea of Cyclic Spellout, and it also introduces three different spellout options for an FP built by external merge.

The first option is to find a match for that FP and spell it out. By hypothesis, the Mandarin numeral ‘three’ is stored as a NumP, and it therefore does not match the whole tree in (76). What this means is that other derivational options in the algorithm in (75) must be explored.

The first rescue option that must be tried (since spellout failed in (76)) is to move the Spec of the complement (and then retry spelling out FP). However, since the complement of Cl has no Spec in (76), the option (75b) is undefined. As a consequence, the final derivational option is tried, see (75c). This derivational option requires us to move the complement of F, which is a step we show in (77).

The root node in (77) has the same label as the one on the right branch. The point here is not so much to suggest that this is an adjunction structure; the point is to say that it is the right branch that provides the label to the root node.

After complement movement, the structure in (77) can be simplified as in (78) with the trace of the movement ignored. The assumption that spellout movements do not leave a trace is not crucial, even though it is a standard part of the Nanosyntax toolbox. The reason why we follow this idea here is that ignoring the trace makes it easy to see when lexical entries match and when they don't. See Starke (2018) and Caha (2019) for more discussion.

The spellout algorithm now requires that after we move the complement, we try to spell out FP again. This time, matching is successful, because the general classifier gè, recall (71), can be inserted at the lower ClP, as in (79). As a result, the simplex stacking pattern is derived using the lexical entries in (70)–(71), and we also derive the correct ordering. The structure in (79) also replicates the intuition depicted in Table 9 to the effect that the classifier in Mandarin spells out Cl, while the numeral root spells out Scale and Num.

7.4 On the interaction between suppletion and stacking in colloquial Mandarin

At this point, we have shown how all the three simplex patterns arise. The current section turns to a particularly interesting set of facts surrounding the numeral ‘three’ in spoken Mandarin, relying on the description in He (2015, 199) and Qi & He (2019, 76–78). We argue that the facts provide a particularly compelling case for analyzing the patterns at hand by means of phrasal spellout.

As a background, recall that Mandarin generally exhibits the stacking pattern illustrated in (80a-b).

Interestingly, the spoken language has developed a special form for the object-counting numeral ‘three’, namely sā. This form is in (80c), and He (2015, 199) describes it as a “reduced form” corresponding grammatically to the Standard Mandarin combination sān gè ‘three clf’. The analysis of sā as an opaque combination of a numeral and a classifier is motivated by the fact that sā must combine with the noun directly, i.e., without a classifier; see (80c). In our system, sā is therefore an instance of a suppletive object-counting numeral with sān in (80a) as its abstract-counting counterpart.

The important point is that this description nicely illustrates the gist of our idea as to how morphologically simplex object-counting numerals arise, namely as an opaque realization of a complex structure containing the meaning components corresponding to the combination of an abstract-counting numeral and a classifier.

We will see shortly that the reduction of sān gè ‘three clf’ to sā is not phonological, but morphological in nature. In our proposal, the colloquial example in (80c) therefore requires us to posit a special lexical entry for sā, which only exists in the colloquial variety. The entry is as given in (81).

This lexical entry co-exists in the colloquial variety with two lexical entries. One of them is the lexical entry for the abstract-counting sān ‘three’, which the colloquial variety uses in arithmetical statements just like the standard variety. The second lexical entry is the entry for the general classifier gè, which is still used in the colloquial variety with other numerals. We give the two entries in (82) and (83), repeated from (70) and (71), respectively.

The coexistence of the three lexical entries in the colloquial variety allows us to illustrate the explanatory power of the theoretical tools that we have posited up to now. First of all, note that sā with the lexical entry as in (81) could be in principle used in mathematical statements, because it contains the abstract-counting structure. However, sān with the entry (82) is a perfect match for the abstract-counting structure, and we therefore predict that sā cannot occur in arithmetical environments. This prediction is borne out, see (84) (Qi & He 2019, 77).

A more interesting prediction relates to the relative sequence of steps in the spellout algorithm. To see it, suppose that the cyclic derivation has produced the abstract-counting numeral sān ‘three’. Suppose now that the head Cl is added, as in (85), repeated from (76).

The prediction – related to the ordering of steps in the spellout algorithm – is that when confronted with (85), the system first tries to spell out the whole constituent without movement. In the colloquial variety, this attempt leads to a successful match by sā, see (86). As a result, we predict that in colloquial Mandarin, sā (rather than sān gè) is going to be the spellout of (85), which is the case.

On the other hand, if complement movement were ordered before direct spellout, we would derive the form with the classifier (i.e., sān gè), despite the presence of the suppletive portmanteau sā ‘three.clf.’ Therefore, if the ordering of the steps were different, we would fail to derive the colloquial form even if it was contained in the lexicon, clearly the wrong result. In sum, we found support for the proposed ordering of the two relevant steps of the spellout algorithm.

Most curiously, some additional facts show that even though colloquial Mandarin prefers the mono-morphemic spellout sā ‘three.clf’ over sān gè ‘three clf’, this is only so if sān and the Cl head form a constituent. Since this is something that is predicted by the phrasal-spellout account of suppletion, let us discuss the relevant facts in the remainder of this section.

The relevant configuration arises when the numeral ‘three’ is a part of a transparent compound numeral, such as the Mandarin equivalent of ‘fifty three’. The structure of such numerals argued for in He (2015, 200) is as in (87). In this tree, we only elaborate on the fine structure of sān ‘three’, while the rest of the structure is simplified. The head that contributes the additive semantics is represented as &, following He.

Against this background, consider what our theory predicts when the Cl head is merged on top of the compound-numeral structure, as in (88). The spellout algorithm requires that ClP must be spelled out. The first attempt is to spell out ClP directly. This will fail in (88): even though the numeral sā can spell out Cl along with the NumP, the matching principle allows this only if the Cl head and the NumP are contained in the lexical entry of the numeral; recall the Superset Principle in (62). This is obviously not the case in (88); therefore, we predict that sā ‘three.clf’ cannot appear in the compound numeral ‘fifty three’.

The prediction is borne out. As the data in (89a) show, it is indeed impossible to use sā ‘three.clf’ inside the compound numeral. Instead, the analytic combination sān gè is found, as in (89b).

The example in (89b) (where sān is directly followed by gè) shows that the numeral sā does not arise as a result of a simple phonological or morphological rule that operates on the basis of linear adjacency between sān ‘three’ and gè ‘clf’. Rather, it operates on the basis of syntactic constituency, which is predicted by our account where suppletive object-counting numerals like sā arise as a result of phrasal spellout.

Finally, let us turn to the issue of how we can generate the correct form in (89b). Note that when the spellout algorithm fails to directly spell out Cl in (89), it proceeds to movements. The first movement to be tried is Spec-movement, recall (75b). The phrase in (88) has a Spec, namely the phrase ‘fifty’. However, even if that phrase is removed, the & head (responsible for addition) is in the way of successful spellout of Cl and the NumP. Therefore, the option that ultimately succeeds is the one that moves the complement of the Cl head to the left, as schematically depicted in (90). This gives rise to the structure in (91) (with the trace removed), and the remnant ClP is spelled out by the classifier, see (92).

As a final point in this section, we note (following He 2015) that the data discussed above support the idea presented in Section 3.4 that, in at least some cases, compound numerals form a complex constituent composed of abstract-counting numerals. Only after the compound numeral is formed, Cl is merged in order to turn the abstract-counting compound numeral into the corresponding object-counting numeral. (This does not exclude the possibility that in other cases, the structure may be more complex, recall footnote 12.)

7.5 Complex syncretic numerals

In this section, we turn to complex numerals. Recall that complex numerals are bi-morphemic already at the abstract-counting level. In the current model, this entails that they spell out separately Scale and Num. Consider, for instance, the case of the Sanzhi Dargwa numeral xu-jal ‘five’. This complex numeral can be used both as an abstract-counting numeral and also as an object-counting numeral, recall (39)–(40). We have therefore classified it as a complex syncretic numeral.

The lexical entries needed to capture this pattern are given in (93)–(94). The first lexical entry (namely xu ‘five’) in (93) spells out just Scale (reflecting the intuition in Table 9). The remaining features are included in the lexical entry of jal, see (94). ²⁷

In order to see how these lexical entries lead to the result xu-jal in both object- and abstract-counting uses, consider the derivation below. We start by assembling the lowest Scale ₅ node and we spell it out as xu, using the lexical item (93). Once the spellout of Scale ₅ succeeds, the abstract-counting structure is derived by the addition of Num, see (95).

The NumP in (95) fails to spell out, because there is no match. (Neither the entry for xu (93) or jal (94) contain the structure.) Therefore, we must use the other options of the spellout algorithm, specifically complement movement. (96) shows the configuration after Scale ₅ undergoes complement movement from below the Num (the trace is ignored). In this configuration, the remnant NumP node may be spelled out by jal. This correctly yields the form xu-jal in (97) as the spell out of the abstract-counting structure.

Consider now how the object-counting numeral is derived. Following the algorithm, we first merge the feature Cl on top of the abstract-counting numeral. The abstract-counting numeral is in (97), and merging Cl yields (98). Spellout without movement fails here, since the top node of (98) (ClP) is not contained inside any single entry.

Therefore, as customary by now, other clauses of the spellout algorithm are invoked. The first option to be tried is Specifier movement, recall (75b). The movement is shown in (99).

The result of Spec movement (with the trace ignored) is given in (100). After Spec movement, we have to retry ‘spell out FP’, where FP corresponds to the remnant ClP. This ClP is contained inside the lexical entry of the suffix jal, and the suffix is therefore inserted, see (101). Note that as a part of the process, the ‘old’ spellout of NumP in (100) is replaced by the ‘new’ spellout of ClP in (101). This is called overriding in Nanosyntax; see Starke (2009) and other Nanosyntax references for more on overriding, here we only make this explicit.

Before we conclude, let us make a general remark about syncretism. In the current theory, syncretism is always the result of the fact that one and the same lexical item may spell out two different structures. In the simplex syncretic pattern (e.g., three), the numeral would either spell out all three meaning components of object-counting structures, or just Num and Scale in the abstract-counting use. In the complex pattern, this is very similar. The ambiguity boils down to the fact that the suffix -jal either spells out only the Num feature, or Num+Cl. This is also as coded in Table 9.

7.6 Complex suppletive numerals

Let us now consider complex suppletive numerals. The derivation of this type is very similar to the way complex syncretic numerals work, the only difference is that instead of having an ambiguous suffix (like jal), we will have two different suffixes: one for Num+Cl, and the other one for Num only.

The lexical entries needed for this are given below. They are modeled on the basis of Mokilese, recall (48)–(50). In Mokilese, we find the general classifier -w in abstract-counting contexts, and we therefore hypothesize that numerals spell out only Scale, see (102). The general classifier -w spells out Num, see (103). In (104), we give the lexical entry for the classifier men, which is suffixed to the numeral in the object-counting function when animates are counted. The crucial point is that -men replaces -w, and we therefore attribute to it both the feature Cl and Num.

The entries as given above will derive a pure suppletive system – which (as the reader may recall from the discussion in Section 4.3) we have not been able to find yet. For the sake of the discussion, we will nevertheless show how the derivation of such a pattern proceeds, using the entries as given above. Later, we shall also derive the actual attested patterns that mix suppletion and syncretism. (That will require us to say something about the difference between different classifiers.)

The derivation of a pure suppletive type proceeds in exactly the same steps as in the preceding Section 7.5. First, Scale is spelled out by the numeral. The feature Num is added. There is no single morpheme to spell out both, therefore Scale moves to the left and the remnant NumP is spelled out by -w, see (105). Note that -men can also be inserted here, but it loses in competition with -w. Note also that the structure has the same abstract shape as the one used for Sanzhi Dargwa, recall (97).

The tree in (106) represents the object-counting numeral. It is abstractly also the same structure as in Sanzhi Dagwa, recall (101). The only difference between Mokilese and Sanzhi Dargwa is that Mokilese has a dedicated suffix for each of the right branches in (105) and (106), while Sanzhi Dargwa had just a single marker for both. This is then how an idealized complex suppletive pattern arises. ²⁸

Now in reality, recall from (49a) that the general classifier -w can also appear (as an ‘unmarked’ classifier) in object-counting use. Therefore, we will have to attribute to it a lexical entry capable of spelling out Cl, which makes it – as things stand – identical to Sanzhi Dargwa -jal. This is empirically correct, since both morphemes can actually be used in both object- and abstract-counting uses.

However, by attributing Cl to -w, we also make it indistinguishable from -men, which is wrong; -men does not appear in abstract-counting contexts. The answer we suggest here emanates from the observation that classifiers themselves may be morphologically complex. To see that, consider the following dataset from Nepali. Nepali is a simplex stacking language. In the abstract-counting use, numerals are bare, see (107) (Turnbull 1982, 45).

When numerals are used in the object-counting function, they must be accompanied by a classifier. Example (108a) shows the general classifier wota, while (108b) exemplifies the human classifier jana.

Interestingly, as pointed out by Allassonnière-Tang & Kilarski (2020, 129–130), the general classifier (though not other classifiers) agrees in gender with the head noun and shows up in two different shapes, namely wot-a for masculines and wot-i for feminines.

What these data show is that the general classifier wot- (the realization of a pure Cl node), may be followed by additional (agglutinative) markers that further restrict the applicability of the general classifier to masculine/feminine nouns respectively.

Extrapolating this observation to Mokilese, we may think of the non-general classifiers (that are opposed to a general classifier) as lexicalizing (in a manner of phrasal spellout) additional features apart from Cl. For instance, the Mokilese situation could be modeled as a combination of the following two entries, see (110) and (111).

(110) shows the updated entry for the general classifier. Compared to the original version in (103), it can also spell out the Cl head. With this lexical entry, it can appear in both object- and abstract-counting environments, cf. the Sanzhi Dargwa jal in (94). The animate classifier men has now been updated to include an additional Anim feature, which (unlike the general classifier -w) imposes an additional restriction on the nature of the counted object. Note that since the classifier -men also includes the features Num and Cl, it does not stack on top of the general classifier, but replaces it.

7.7 Complex stacking

The final type to be addressed is the complex stacking type. In this type, exemplified by Vera’a, the abstract numeral is complex, e.g., vō-ru(ō) ‘prfx-two’, and an additional marker stacks on top of the numeral in the object-counting function, e.g., ne vō-ru(ō). This can be modeled by proposing that each marker spells out exactly one feature as specified below.

We shall now run the derivation using the markers as specified above. As a caveat, we point out that if we take the entries to be as in (112)–(114), we shall generate a mirror-image suffixal paradigm (i.e., instead of ne vō-ru(ō) we shall get ne ru(ō)-vō ne). This can be fixed by changing the shape of lexical entries into a different format that leads to prefixation (cf. Starke 2018, De Clercq & Vanden Wyngaerd 2018), but we do not take up this issue for reasons of space, since the derivation of prefixal paradigms in Nanosyntax would take us too far afield.

With this caveat in place, our starting point is the fact that the derivation of the abstract-counting numeral will proceed as in Mokilese and Sanzhi Dargwa, yielding the tree in (115). The reason for this is that the numeral can spell out only Scale; for num to be spelled out, Scale must move to the left, and the remnant NumP is spelled out by the marker vō.

Now the difference between Vera’a and the other languages is that Cl is spelled out by a separate marker still, see (114). (In Mokilese and Sanzhi Dargwa, Cl was spelled out jointly with Num.) This leads to the fact that the derivation is forced to another step of complement movement, producing the completely analytic structure in (116).