2.1 A short history of numeral classifier analyses

For both Krifka (1995) and Chierchia (1998), who, to our knowledge, are the first to propose the classifier-for-numeral and classifier-for-noun theories (respectively), nouns in a language with numeral classifiers denote kinds rather than sets of entities. For them, part of the function of the classifier is to semantically access the atoms and plural entities that are members of the kind.

One feature that is shared among all semantic analyses of classifiers is the necessity to identify atomic entities in the set denoted by the noun. The definitions of atom and atomic are given in (3). An atom cannot have any subparts, and an atomic set is made up of entities that are themselves composed of atoms. For example, the set {a,b} contains only atoms and is an atomic set; whereas, the set {a,b,ab} contains both atoms (a,b) and a sum (ab) and is also an atomic set.

Previous analyses of numeral classifiers vary on a number of important semantic points that are connected to the extended structure of noun phrases. These points of variation include (i) the semantic type of the noun, (ii) the semantic function of the classifier, and (iii) the denotation of the numeral. We discuss these points of variation in turn below.

Among the semantic analyses for classifiers, there are three semantic types that have been proposed for nouns: (i) kinds (e^k), (ii) sets of atoms and pluralities, or (iii) sets of atoms. Having kinds as the starting denotation is proposed by Krifka (1995), Chierchia (1998), Jenks (2011), and Dayal (2012), among others. Bale & Coon (2014) give the starting denotation of the noun as a set of atoms. For Wilhelm (2008), Nomoto (2013), and Bale et al. (2019), the noun denotes an atomic join-semilattice, containing both atoms and pluralities.

There are roughly four ways that the function of the classifier has been characterized: (i) as a proper semantic mediator between the numeral and classifier (Krifka 1995; Wilhelm 2008; Jiang 2012), (ii) as changing the denotation of the noun (Chierchia 1998; Jenks 2011; Dayal 2012; Nomoto 2013), (iii) as a measure function argument required by the numeral (Bale & Coon 2014; Bale et al. 2019), and (iv) as semantically vacuous (Cheng & Sybesma 1999).¹ Numeral classifier semantics is intricately connected to the semantics of numerals, since together they make up the strategy for numeral modification in a language.

For the semantics of numerals, there have been three main proposals: (i) numerals of type n; (ii) entity-measuring numerals (e.g., the numeral two combines with a nominal predicate that denotes a join-semilattice and yields only the entities that consist of two atoms); (iii) plurality-forming numerals (e.g., the numeral two combines with a nominal predicate that denotes only a set of atoms, retrieves sums of two members of the set).

Krifka (1995) and Wilhelm (2008) propose that the numeral is type n and that it does not have any measure semantics of its own. In such theories, the classifier mediates between the numeral and the noun and provides a measure function. Chierchia (1998) does not specify what sort of numeral to use, but since the noun denotes a set of atoms and pluralities before it combines with a numeral, it would follow that the numeral measures entities. Bale et al. (2019) also use a plurality measuring numeral for their analysis. For analyses where the noun denotes a set of atoms when it combines with the numeral, it is necessary to have a plurality forming numeral. Jenks (2011),² Jiang (2012), Dayal (2012), Nomoto (2013), and Bale & Coon (2014) use a plurality forming numeral, meaning that the numeral forms pluralities where each plurality has a set number of sub-entities.

Summarized in Table 1 are the analyses of classifiers discussed in this section. The second column shows the starting type of the noun assumed by the analysis of the reference given in the first column. The third column shows the semantic type of the classifier. The final column shows the function of the numeral in each theory—“n” represents numerals that form a basic semantic type of their own, “|Entity|” stands for numerals that are entity measuring, and “*” stands for numerals that are plurality forming.

The four proposed classifier functions reflect the degree to which the classifier function is tied to numeral modification versus other semantic functions in the nominal domain. If the classifier is functioning as a semantic mediator between the noun and numeral or as a measure function, we would expect it to appear only in contexts that include numerals. If the classifier functions to affect the noun denotation, we might see it in the absence of a numeral. Therefore, we claim that two broad categorizations into classifier-for-noun and classifier-for-numeral theories can cover the primary distinction between accounts in terms of their predictions. We categorize Krifka (1995), Wilhelm (2008), Jiang (2012), Bale & Coon (2014), and Bale et al. (2019), located in the upper part of Table 1, as classifier-for-numeral theories. Chierchia (1998), Jenks (2011), Dayal (2012), and Nomoto (2013), located in the lower part of Table 1, make up the classifier-for-noun theories. If the classifier is semantically vacuous, as proposed by Cheng & Sybesma (1999), we would not expect to see it optionally added to influence semantic interpretation.

We will demonstrate that these two types of numeral modification strategies are present cross-linguistically, exemplified by Ch’ol and Shan in subsequent sections. Scontras (2022), similarly, argues for cross-linguistic variation in numeral modification strategies, but this comes about through variation in the selection of the measure function, which is intended to capture the cross-linguistic differences in number marking.

While we believe that the core difference between these theories lies in the the semantic role of the classifier in numeral modification, we will make use of specific analyses in order to demonstrate the predictions that follow from both families of theories. For classifiers-for-numerals, we will assume an account closely related to that of Bale et al. (2019). For classifiers-for-nouns, we will use a theory that builds on Jenks (2011), Dayal (2012), and Nomoto (2013) (as opposed to Chierchia 1998), since, as will be made clear in the next section, the atomizing function of the numeral classifier will turn out to be an important semantic contribution of classifiers-for-nouns.

2.2 Noun type and denotation

One crucial point of variation important in comparing these two families of theories is the starting denotation of the noun. In classifier-for-noun theories, numeral classifiers change the denotation of the noun in some way, whereas in classifier-for-numeral theories, they do not. The previous section included discussion of the noun type proposed in a variety of theories. We assume that all nouns are type ⟨ e, t ⟩, adopting the noun typology proposed by Deal (2017).³ According to Deal, all languages have a mass/count distinction, and there are two possible countability distinctions made in languages: sums-based and parts-based. A sums-based distinction is a distinction based on whether a noun is cumulative or not. The definition of cumulativity, as defined by Deal (2017), is in (4). The noun ‘water’ is an example of a cumulative noun: any sum of the parts of water is also water.

A language with a sums-based distinction would have both nouns that are cumulative and the opposite—quantized, as defined in (5). Bare singular count nouns such as ‘dog’ denote quantized predicates. In terms of the previous distinction between Atom and Atomic, quantized nouns only include Atoms, so they are by default Atomic.

A parts-based distinction, by contrast, depends on whether a language has a distinction between nouns whose denotations are Atomic or not. Thus, a language may have only cumulative nouns, but those cumulative nouns may be atomic or non-atomic.

For Deal, there are three possible noun denotations available cross-linguistically: quantized nouns, atomic join-semilattices, and non-atomic join-semilattices. Quantized nouns are sets of atoms, and thus are atomic but not cumulative; a noun denoting an atomic join-semilattice is both atomic and cumulative; a noun denoting a non-atomic join-semilattice is cumulative but not atomic. As an example, Mandarin Chinese is a language that has been proposed to have a parts-based distinction but not a sums-based distinction (Doetjes 1997). This would mean that noun denotations in Mandarin Chinese are either atomic join-semilattices or non-atomic join-semilattices. Thus, the denotation of Mandarin gou ‘dog’ would be atomic and cumulative. The denotation of English dog would be quantized (a set of atoms), and the denotation of dogs would be atomic and cumulative.

In this paper, we focus primarily on sortal classifiers, in contrast to measure numeral classifiers. A sortal classifier is one used with an atomic noun (often called ‘count nouns’), whereas a measure classifier is one that can be used with a non-atomic noun, which we will call ‘mass nouns’. Atomic nouns are further divided into quantized nouns (consisting of atoms) and cumulative nouns (i.e., atomic join-semilattices—consisting of atoms and sums). We will assume that Shan and Ch’ol nouns are all count (atomic) or mass (non-atomic), and we focus on count nouns which are associated with sortal numeral classifiers. The base denotation of a noun in examples for the two languages will be an atomic join-semilattice. There is more to say about plural marking, but, aside from a brief discussion in Section 3.2, this topic is beyond the scope of this paper. While we do not go into detail whether atomic nouns in the two languages discussed are quantized or cumulative, we will discuss how this distinction factors into our proposal.

2.3 Two strategies for numeral modification

As discussed in the previous subsection, we take as a point of departure that the principle difference between classifier-for-numeral and classifier-for-noun theories comes from the role of the classifier and numeral in numeral modification. We represent this distinction using differences in the denotation of the numeral and the classifier. The particular numeral denotations we use to illustrate this distinction pattern with the intersective (for the classifier-for-numeral type) and subsective (for the classifier-for-noun type) numeral distinction (Bale et al. 2010; Nomoto 2013). However, for Nomoto (2013), the distinction is less connected to numeral-classifier constituency, which we emphasize here.

Using the noun denotation in (6), we show how each theory derives the meaning of two dogs in a numeral-classifier language. (6) denotes a set containing atoms and pluralities of dogs—an atomic join-semilattice.⁴ We also assume for purposes of illustration that the denotation of the noun in (6) only contains three atomic entities and their sums.

Below, we demonstrate how each theory derives ‘two clf dogs’, to mean the set of pluralities of two dogs.

In classifier-for-numeral theories, classifiers are needed for numerals because the numeral in (7) requires an extra semantic argument in order to compose with the noun (Krifka 1995; Wilhelm 2008; Bale & Coon 2014; Bale et al. 2019). The classifier in (8) saturates the first argument of the numeral in (7), where μ_# is a variable over measure functions.⁵ The measure function is specified for the particular noun it ultimately combines with, e.g., as proposed by Krifka (1995), and it counts the number of atoms/units that make up a sum in the noun denotation.⁶

The numeral in (7) combines with a measure function m and predicate P and denotes the set of individuals x such that x has the property of P and the measure of x according to m is 2. (8) is a measure function which gives the number of atoms in a given plurality (Wilhelm 2008: 55). The numeral in (7) takes the classifier in (8) as its first argument. The noun combines with the numeral classifier constituent to yield the set of pluralities of two dogs, as shown in (9).

In classifier-for-noun theories, the classifiers are needed to mediate between a noun, as in (4), and a numeral, as in (10) (Chierchia 1998; Jenks 2011). We assume a classifier-for-noun theory following Bale et al. (2019), who use a theory for numerals proposed by Ionin & Matushansky (2006).⁷ The numeral needs to combine with an atomic predicate and returns the set containing all sums with the predicate property that have a cardinality of 2:

(10) is a function that takes a predicate P and yields the set of entities x such that x can be partitioned into two parts and x has the property P. As such, the numeral in (10) generates a set of pluralities from a set of atoms. This is different from (7), where the numeral creates a subset with the right cardinality. The type of numeral in (10) must combine with a set of atoms. (11) defines the partition relation Π between a set S of individuals that forms a cover, defined in (12), of some plurality x.

Since the noun in (6) is not composed solely of atoms, classifiers, as in (13), are needed to quantize the set denoted by the NP.

The function in (13) yields the set of x, such that x has the property P and there is no y with the property P that is a sub-part of x. When it combines with a noun predicate, this classifier generates a set containing atoms with that property. The classifier in (13) first combines with a noun allowing for the numeral in (10) to then combine with the classifier-noun constituent, as shown in (14).

2.4 Predictions of each theory

Though derivationally distinct, each theory produces the same meaning for two dogs: a set containing pluralities made up of two individual dogs. For classifier-for-numeral theories, the numeral takes the classifier as a measure function, and then combines with the noun. For classifier-for-noun theories, the numeral cannot directly combine with the noun, and so a classifier is needed to individuate the members of the nominal predicate to create a set of atoms.

Despite producing similar meanings, the two theories make different predictions regarding the distribution of numeral classifiers. These predictions are summarized in Table 2.

In Section 4, we compare Ch’ol and Shan, which have been described as having numeral classifiers. We show that while Ch’ol shows evidence for predictions 1 and 4, Shan shows evidence for predictions 2 and 3, supporting our proposal in (15).

Before going through the predictions in Table 2, however, Section 3 provides an introduction to classifiers in Ch’ol and Shan and an analysis of their syntactic structures.

3.1 Ch’ol classifiers

Ch’ol is a Mayan language of the Ch’olan-Tseltalan branch, spoken in southern Mexico by approximately 254,000 speakers (INEGI 2020). There are three main dialects of Ch’ol—Tumbalá, Tila and Sabanilla. The data in this paper come from the Tumbalá and Tila dialects. The patterns were consistent across both dialects with respect to the behavior of classifiers investigated in this paper.

Numerals in Ch’ol obligatorily appear with classifiers; numerals 1–20 are given in Table 3. Numerals 1–19 are given with the generic classifier -p’ej. The Ch’ol numerical system is vigesimal and has a base-twenty classifier, -k’al, which appears in the classifier slot. Other classifiers for multiples of twenty include -bajk ‘four hundred’ and -pijk ‘eight thousand’.

Today, many speakers, including monolinguals, only use Ch’ol numerals up to six and numerals borrowed from Spanish for higher numerals (Vázquez Álvarez 2011: 160). Classifiers are always used with Ch’ol-based numerals, as in (16). However, as noted in Bale & Coon (2014), the Spanish-based numerals are ungrammatical with classifiers, as seen in (17).⁹

Arcos López (2009) identifies at least 180 classifiers, though he notes that this is not an exhaustive list. Examples of common classifiers are found in Table 4.

While the generic classifier -p’ej and the classifier for humans -tyikil are of unknown origin (Arcos López 2009), many other classifiers are derived from positional and transitive verb roots (see Arcos López 2009 and Bale et al. 2019 and also Haviland 1981 for Tsotsil). Positional roots are a large and distinct class of roots that convey information about the position or configuration of an object (see e.g., England 1983; Haviland 1994; Henderson 2019). The numeral classifier -bujch, used to count objects which are propped up, seated, or leaning against something, is derived from the positional root buch ‘seated’. The animal classifier -kojty is derived from the positional root koty ‘standing on four legs’.

The position or shape of a noun is relevant for the choice of classifier, meaning that the same noun could be counted with more than one classifier, as in (18). Depending on the shape or position of a tree, a different classifier is used; See also Bale et al. (2019: (39)).

Numeral classifiers in Ch’ol only occur with numerals and the interrogative quantifier jay- ‘how many’, which we take to be the interrogative version of a numeral. Classifiers in Ch’ol are ungrammatical with quantifiers (19a)–(19b), demonstratives (19c), and modifiers (19d).

To summarize, Ch’ol classifiers are used with Ch’ol-based numerals but not Spanish based ones. They are largely derived from positional roots and vary based on the noun and other semantic information related to the object. Ch’ol classifiers only appear with numerals and the interrogative version of the numeral.

3.2 Shan classifiers

Shan is a Kra-Dai language of the Southwestern Tai branch, spoken in Myanmar and surrounding countries by approximately 4.6 million speakers (Ethnologue 2021). While classifiers in Thai, a related Tai language, have been investigated in detail (e.g., Iwasaki & Ingkaphirom 2005; Piriyawiboon 2010; Jenks 2011), there have been very few descriptions or analyses of Shan numeral classifiers. Cushing (1887) first describes classifiers in Shan, calling them ‘numeral auxiliaries’, which denote “some rank of being, some form of object or some quality in the noun to which it belongs”. The numerals in Shan from one to twenty can be seen in Table 5.

Unlike Ch’ol numeral classifiers, which are derived from verbs or positional roots, Shan classifiers appear to be derived from nominal elements. For example, tǒ, the classifier for animals, also means ‘body’. For some nouns that are typically found as compounds, there is a connection between the form of the classifier and one part of the compound, as in (20). Here the noun compound ton-mâj ‘tree’ and the classifier ton both contain ton. Classifiers obligatorily appear in the presence of a numeral, as shown in (21). Which classifier appears depends on properties of the noun. Table 6 gives some examples of properties that determine which classifier is used.

One interrogative numeral, lǎaj ‘how many’ in (22a), obligatorily appears with classifiers, just like the numerals. Some expressions of quantity, like (22b), do not appear with a classifier.

Shan classifiers are free morphemes derived from nominal elements. We will see in Section 4.3 that classifiers can appear in constructions without numerals.

3.3 Syntax of the classifiers in Ch’ol and Shan

We propose the following two syntactic structures for classifiers in each language. The relevant Ch’ol word order is [ Numeral Classifier Noun ], whereas the Shan order is [ Noun Numeral Classifier ]. We propose different syntactic structures for each language, based on their distinct syntactic properties (see also work by Danon (2012) who argues that there are two types of numeral-noun syntactic structures). We adopt a proposal from Simpson (2005), also argued for by Jenks (2011) for word order in Thai, that the order in Shan comes as a result of phrasal movement of the nominal constituent from its base position to a position above the numeral and classifier.

The order of elements within the noun phrase for each language is given below.

We propose that numerals in Ch’ol are in a specifier position of a functional projection (YP) along the nominal spine (see e.g., Cinque 2005).¹⁰ Importantly, the ordering between numerals and adjectives is strict in Ch’ol. A proposal like Bale et al.’s (2019), in which numerals are adjuncts that adjoin in the same projection as adjectives (nP for them), would predict that numerals and adjectives can be ordered freely, which is not borne out in the language.

Furthermore, ellipsis data in Ch’ol confirms that adjectives and numerals behave differently: adjectives cannot license ellipsis, as in (26b), but numerals can (26c). Both (26b) and (26c) are meant to be follow-ups to (26a). The ellipsis data provides evidence that numerals exhibit distinct syntactic properties from adjectives (cf. Bale et al. 2019: §4.2).

Ellipsis data in Shan also provides evidence for its syntactic structure. In Shan, there is similarly a clear contrast between noun ellipsis when the noun is modified by an adjunct or not. The example in (27a) cannot be followed by (27b) with the noun elided because the noun is modified by an adjective, súk ‘ripe’. In contrast, it can be followed by (27c), indicating that the numeral and classifier do not form a constituent that is a syntactic adjunct of the noun.

While the ellipsis data support a non-adjunct analysis of numeral-classifier constructions in both languages, it does not necessarily show us that these constructions exhibit different structures. We thus turn to A′-extraction facts to show this. Since numeral-classifier constructions in Ch’ol are phrases contained in a specifier position, it is conceivable that they could undergo A′-extraction. On the other hand, Shan’s numeral and classifier do not form a constituent to the exclusion of the NP. It would thus be surprising that they would be able to extract as a constituent without clear motivation for remnant movement. As discussed in Bale et al. (2019) and Little (2020), numerals may extract from their base position to the preverbal focus position, evidence that they form a constituent, shown in (28). Shan numerals and classifiers cannot move to a clause-initial position together, as shown in (29). This is expected if Shan’s numeral and classifier are functional heads along the nominal spine—heads cannot undergo A′-movement.

While this subextraction diagnostic is useful for determining constituency in Ch’ol, it does not follow that all languages with the syntax in (23a) will allow this type of subextraction. As is well known, there is parametric variation regarding which languages allow this type of subextraction from the nominal domain.

We have shown diagnostics for the constituency of the numeral and classifier in Ch’ol (following work in Bale et al. 2019). We refer the reader to Moroney (2021) for further evidence for the structure in (23b) for Shan and to Jenks (2011) and Simpson (2005) for similar arguments for Thai.

3.4 An aside on plural marking

While the focus of this paper is on classifiers and their co-occurrence with nouns/numerals, adopting these different structures yields a welcome result with respect to plural marking. For Borer (2005), plural marking and classifiers are in complementary distribution—which is why they occupy the same functional projection. This accounts for why English does not have classifiers, but has plural marking when appearing with numerals above 1. Mandarin, on the other hand, has classifiers but no plural morphology with numerals. We note that Borer’s analysis can be held for the classifier-for-noun languages, such as Shan, but for classifier-for-numeral languages, we predict that classifiers may be able to co-occur with plural morphology given that the classifier in languages like Ch’ol is not in the same projection as plural marking. This indeed seems to be borne out.

Ch’ol has a plural marker that indexes human- and some animal-denoting nouns. If we were to adopt the account in (23b) for Ch’ol, it would incorrectly predict the incompability of plural markers, which is not the case, as shown in (30).

While Ch’ol plural marking in (30) is optional and it is unclear what governs its optionality, Mi’gmaq, a language analyzed as a classifier-for-numeral language in Bale & Coon (2014), has obligatory plural marking. Two examples with grammatically animate and inanimate nouns are given below from the Mi’gmaq Online Dictionary (MOD, Haberlin et al. 1997–2017).

The syntax provided in (23a) predicts co-occurrence of plural marking, exhibited in (31), where the functional head Y could host number marking.

3.5 Summary

In sum, while Ch’ol and Shan both have numeral classifiers, the initial data seen in this section suggest that they exhibit a different syntax. As schematized in (23), Ch’ol classifiers form a constituent to the exclusion of the noun (Bale et al. 2019), whereas Shan classifiers do not (Moroney 2021). Both these structures capture the strict linearization between numerals and classifiers—they must be adjacent elements. Even though we propose different structures and semantics for classifiers, there are similarities between these functional projections; namely that they are needed to mediate between a numeral and noun. This importantly rules out the numeral from ever combining with the noun first and then the classifier, as the classifier is needed in both theories to mediate between the noun and the numeral, though, as we emphasize below, in different ways.

It is important to mention that the two syntactic structures we provide in (23) contrast with recent work by Dékány (To appear), who proposes that both Ch’ol-style and Shan-style numeral-classifier constructions have a Shan-style syntax (23b). For her, numerals and classifiers are functional heads along the nominal spine. This analysis would not immediately predict the semantically driven phrasal movement of numerals in (28). Furthermore, it would not predict the compatibility of plural marking with the classifiers in Ch’ol (30) and Mi’gmaq (31). In fact, despite different syntactic structures for what we will argue are languages with classifiers-for-numerals (Ch’ol) and classifiers-for-nouns (Shan), our semantic account only rules out the alternative constituency when it comes to classifiers-for-numerals. This is because the numeral modification strategy in classifier-for-noun languages is semantically compatible with a syntactic constituency where the classifier first combines with the numeral (for classifier-for-noun analyses that involve this constituency see, e.g., Zhang 2013; Hall 2019). Tiwa, a Tibeto-Burman language, seems like a candidate for having a Ch’ol-style syntax for numeral-classifier constructions, but the semantic contribution of the classifier parallels that of Shan’s; see discussion in Dawson (To appear) for further details.

Having provided evidence that Ch’ol and Shan numeral-classifier constructions exhibit a different syntax, we now turn to the semantics: while Ch’ol aligns with the predictions of classifier-for-numeral theories, Shan aligns with those of classifier-for-noun theories.

4.1 Prediction 1 (num): Variation in whether a numeral requires a classifier

In classifier-for-numeral theories, classifiers are needed to saturate an argument required by the numeral, represented as a measure function. Since the appearance of the classifier is contingent on the semantics of the numeral, we might expect to find idiosyncrasies in whether a numeral requires a classifier. That is, as argued in Bale & Coon (2014), some numerals might have the measure function encoded in their lexical semantics, and others not. Such idiosyncrasies are observed in Ch’ol, but not in Shan. As noted above, while higher Mayan-based numerals exist, speakers tend to use Mayan-based numerals when counting up to six and Spanish-based numerals for those higher than six. As shown in Bale & Coon (2014), Mayan-based numerals require a numeral classifier, as in (33a), whereas Spanish-based numerals cannot combine with one (34a). Therefore, the semantics of ux ‘three’ would be (33b), first combining with a measure function (the classifier), and the semantics for ocho ‘eight’ would be (34b), where the measure function is already built in.

Further evidence comes from the distinction between Ch’ol and Spanish numerals for measure expressions in (35)–(36).¹¹ We assume that words like ja’ ‘water’ denote non-atomic predicates while words like ts’i’ ‘dog’ denote atomic predicates. For Ch’ol-based numerals, there are two ways to do numeral modification with a non-atomic noun: (i) use a container/measure expression in the classifier position as in (35a) or (ii) use the container in a nominal position with the numeral modifying the container expression as in (35b). We assume that in (35a), the container classifier p’ejty ‘pot’ is directly supplying a measure function compatible with non-atomic expressions to the numeral. In (35b), the container nominal expression is already atomic, so it is compatible with the generic classifier.

For Spanish based numerals, the measure function is already built into the numeral, so it is not possible to have the container p’ejty ‘pot’ in a classifier position. The only way to measure non-atomic nouns is by forming an atomic nominal expression using the container in a nominal position, as in (36).

The contrast here shows that the measure expression in a classifier position is specifying a measure function that is compatible with a non-atomic predicate. Spanish numerals, which already have a measure function built in, must necessarily combine with an atomic noun. The built-in measure function would essentially be equivalent to the meaning of the generic classifier p’ej. We assume that p’ejty functions to create an atomic predicate, which ultimately allows Spanish numerals to combine with non-atomic nouns.

No such idiosyncrasies are found in Shan: all numerals appear with a classifier. Further evidence comes from measure expressions in Shan. Measure functions are not part of the numeral semantics, as is expected in classifier-for-noun theories, where the numeral exhibits the denotation in (10), repeated in (37).

There are specific classifiers—standard measures or container classifiers—which can appear with mass nouns. In (38), nâm ‘water’ appears with the classifier kɔ́k ‘cup’, which is derived from the noun of the same form.

In Shan, classifiers that are compatible with atomic nouns tend to be incompatible with non-atomic nouns. The generic classifier is similarly incompatible with non-atomic nouns, as (39a) shows. While the noun-specific classifier is preferred, the generic classifier sounds better with a noun that has an atomic denotation, as (39b) shows, than it does with one with a non-atomic denotation, as in (39a). Since the classifier quantizes the noun and does not provide a measure function to the numeral, we do not find the same alternative way that Ch’ol has to measure non-atomic nouns, like lɤt ‘blood’. It is possible to make a compound noun, e.g., thǒŋ lɤt ‘bag of blood’, but this refers to the container more than the contents and still requires the container classifier thǒŋ to be felicitous.

4.2 Prediction 2 (noun): Variation in whether a noun requires a classifier

For classifier-for-noun theories, we may expect to find the opposite of prediction 1. That is, if a classifier is required to atomize the set denoted by the noun, then there might be variation in whether nouns require atomization (and thus a classifier). For instance, it is possible that a subset of nouns in a language denote only a set of atoms, as argued in Simpson (2005) and Simpson & Ngo (2018) for Vietnamese and other East/Southeast Asian languages. In such cases, the plurality-forming numeral given above in (37) could combine directly with a noun, circumventing the necessity for a classifier.

Ch’ol does not exhibit idiosyncrasies with respect to nouns. While Shan includes some cases that appear to fit this description, as in (40), this is probably best analyzed as a reduced repeater classifier construction, which occurs when the noun is omitted in certain contexts when the form of the noun and classifier are phonologically identical.

See Jenks (2011) for discussion of similar data in Thai. These cases require a more complex analysis than simply saying the classifier is absent, but the repeater classifier constructions and reduced repeater constructions are based on the noun used, rather than the numeral. For this diagnostic, we will use the more straightforward data from Simpson & Ngo (2018) on Vietnamese.

In Vietnamese, a language which, as far as we can tell, patterns with classifier-for-noun languages, some nouns never combine with a classifier, as in (41a). Vietnamese has both nouns with optional classifier and nouns with no classifiers (Simpson & Ngo 2018). The optional classifiers have a different form than the noun, making them easily identifiable. We would expect words like màu ‘color’ in (41a) to denote a set of atoms, as in (41b), which is why they do not need a classifier.

It would be difficult for a classifier-for-numeral theory to account for this pattern of data. It is clearly the noun that is conditioning this variation, so we would need to propose null classifiers for nouns with this feature. In the classifier-for-numeral theory, the function of the classifier is to specify the measure function for the numeral. In the case of this heterogeneous class of nouns, if they had ‘null’ classifiers, the measure function would have to be some default measure. However, such nouns in Vietnamese are not compatible with the classifier used for inanimate nouns (Simpson & Ngo 2018), which one might consider to have the same semantics. Similarly, in Shan, the repeater classifier constructions are not compatible with the generic inanimate classifier ʔǎn.

4.3 Prediction 3 (noun): Classifiers found beyond numerals

For classifier-for-noun theories, if a classifier is used to create an atomic set from the noun predicate, we might expect to find it in environments other than with numerals. That is, it is conceivable that other modifiers or constructions in classifier-for-noun languages require the noun to be atomized, resulting in the presence of the classifier. This has been noted previously for some classifier languages by, for example, Simpson (2005). As shown below, this is the case in Shan, which can have a classifier occur with quantifiers (42a), demonstratives (42b), and relative clauses (42d), even in the absence of a numeral. Furthermore, while nouns are number-neutral in Shan, in (42b) the classifier appears to atomize the noun, giving rise to an obligatorily singular interpretation. The contrast between (42b) and (42c) demonstrates that the number neutrality of nouns in Shan is conditioned by the presence of the classifier, even when there are structurally higher elements, such as a demonstrative.

This is not the case in Ch’ol. Classifiers in this language only ever appear with numerals or the interrogative numeral jay- ‘how many’. They are ungrammatical in other contexts, as shown in the example with a demonstrative in (43a) (see also (19) above).

Again, in order to maintain a classifier-for-numeral account for this type of data, it would be necessary to propose that a null numeral ‘one’ appears, for example, in (42a) and (42b). It is not acceptable for an overt ‘one’ to appear in these contexts. As (44a) and (44b) demonstrate, the numeral ‘one’ is not acceptable in these contexts. The distributive quantifier ku ‘every’ generally does not appear with a numeral in expressions with nouns, and the demonstrative is not able to appear with the numeral ‘one’. In contrast to (44b), the Ch’ol example in (43b) is only possible with an overt ‘one’. Rather than positing these null classifiers and numerals to fit these two types of classifier expressions into one schema, we propose to separate them.

4.4 Prediction 4 (num): Classifiers can appear with numerals even in the absence of nouns

Finally, we discuss a novel diagnostic regarding theories of numeral classifiers. If a classifier is a measure function which (i) is generally required by a numeral and (ii) forms a constituent with the numeral to the exclusion of the noun, then we may expect it to always appear with that numeral. That is, even when there is no noun, the classifier could occur if it is required by that numeral. This is the case in Ch’ol: classifiers are always required, even when counting, as in (45), or when referring directly to the number, as in (46), which describes a context in which a teacher is pointing to a number.

In these instances, we propose that the generic classifier is counting abstract points on a number line (indeed the generic classifier appears to count abstract nouns such as ‘belief’ (Bale et al. 2019)). Even though there is no overt noun, it is inferred that there is still some abstract object being counted (e.g., abstract points on a number line). This is further supported by the fact that in multiplication the classifier -sujtyel attaches to the first numeral denoting ‘times’ then the second numeral appears as a possessor to ‘number’ (‘the number of three’). The example in (47a) refers to generic contexts, e.g., in an educational setting. The classifier changes to -kojty if we are multiplying the number of cows in (47b), e.g., in a context where there are two trucks, being able to fit three cows each.¹²

This is not a Mayan-only feature: classifiers in sixteenth century Nahuatl (Uto-Aztecan) also must always occur in the presence of numerals even with no overt noun (Herrera 2022).

In contrast, Shan classifiers are not always required with numerals when a noun is not present. For instance, they are degraded when counting, as in (48), and are unacceptable when referring to the number itself, as in (49).

Before summarizing, we address a caveat of this diagnostic. Reviewers raised the possibility that numerals are affixal in Ch’ol but not in Shan, leading to the requirement that Ch’ol numerals have a classifier even in these contexts. If this were the case, we might expect to see numerals without classifiers in counting contexts in classifier-for-numeral languages that have numerals that are not affixal. We agree that this is possible. In that case, whatever type-shifting takes place to go from modifier numerals to counting would work slightly differently with respect to the semantic contribution of the classifier. While morphophonology plays a role in Ch’ol, we also believe that the analysis we have proposed captures the fact that the classifier does appear even in counting constructions: the classifier in numeral-classifier languages is integral to the interpretation of the numeral. The choice of the classifier still has semantic consequences, as when counting cows, or in mathematical equations. Finally, we refer back to the A’-movement data in (28), where numerals and classifiers can front to preverbal positions. This again supports the fact that the numeral and classifier in Ch’ol form a constituent, regardless of their morphophonological properties—if it were a (fused) head it would not be able to undergo phrasal movement. Our claim is that if a language has obligatory classifiers in counting contexts, it is evidence that the language is a classifier-for-numeral language, but if it does not, it does not provide evidence either way.

4.5 Summary

We began this section with the proposal in (50).

While Ch’ol shows evidence for classifier-for-numeral theories, Shan shows evidence for classifier-for-noun theories. The predictions of each proposal are summarized in Table 8.

Clearly, Shan and Ch’ol numeral classifiers behave differently. Ch’ol numeral classifiers are required by the numeral and therefore always appear with numerals. Shan classifiers are semantically connected to the meaning of the noun, so they can appear in the absence of a numeral. We believe that the different predictions laid out above can be used as diagnostics to identify the type of classifier exhibited in a language.

The literature has largely concentrated on the function of classifiers, and here we have provided evidence that we should distinguish at least two types of numeral classifiers cross-linguistically. We proposed that this follows from variation in the numeral modification strategy, which we took as our point of departure. The semantics of the numeral and classifier will co-vary based on the strategy for how numeral modification proceeds. If the numeral functions to restrict the noun denotation to the pluralities of a certain number of entities, a classifier-for-numeral is more likely to appear. If the numeral functions to form pluralities from a quantized set, a classifier-for-noun is more likely to be used to generate that quantized set. While the constituency in classifier-for-noun languages is not fixed, the constituency in classifier-for-numeral languages is. Classifiers-for-numerals and numerals always form constituents.¹³

The full derivations for the Ch’ol and Shan equivalents of ‘two dogs’ are given below in (51) and (52). While derivationally distinct, both languages end up with equivalent interpretations after the numeral, classifier, and noun combine. In (51), a Ch’ol-based numeral first combines with a classifier so that the classifier supplies the measure function. The resulting measure phrase then measures entities denoted by the noun, in this case those which consist of two atoms.

In Shan, the proposed semantics is given in (52), abstracting away from word order. A classifier first combines with a noun. This is required given the denotation of the numeral. Numerals in Shan, unlike Ch’ol, do not measure entities. Therefore, the classifier is needed to atomize the noun.