# Numerical Expressions in Chinese: Syntax and Semantics

# Numerical Expressions in Chinese: Syntax and Semantics

- Chuansheng HeChuansheng HeHunan University
- and Min ZhangMin ZhangHunan University

### Summary

Numerical expressions are linguistic forms related to numbers or quantities, which directly reflect the relationship between linguistic symbols and mathematical cognition. Featuring some unique properties, numeral systems are somewhat distinguished from other language subsystems. For instance, numerals can appear in various grammatical positions, including adjective positions, determiner positions, and argument positions. Thus, linguistic research on numeral systems, especially the research on the syntax and semantics of numerical expressions, has been a popular and recurrent topic.

For the syntax of complex numerals, two analyses have been proposed in the literature. The traditional constituency analysis maintains that complex numerals are phrasal constituents, which has been widely accepted and defended as a null hypothesis. The nonconstituency analysis, by contrast, claims that a complex numeral projects a complementative structure in which a numeral is a nominal head selecting a lexical noun or a numeral-noun combination as its complement. As a consequence, additive numerals are transformed from full NP coordination. Whether numerals denote numbers or sets has aroused a long-running debate. The number-denoting view assumes that numerals refer to numbers, which are abstract objects, grammatically equivalent to nouns. The primary issue with this analysis comes from the introduction of a new entity, numbers, into the model of ontology. The set-denoting view argues that numerals refer to sets, which are equivalent to adjectives or quantifiers in grammar. One main difficulty of this view is how to account for numerals in arithmetic sentences.

### Keywords

### Subjects

- Linguistic Theories
- Semantics
- Syntax

### 1. Introduction: Numeral System in Chinese

Chinese numerals employ the decimal system of counting. Eleven simple numerals, that is, *ling* ‘zero’, *yi* ‘one’, *er* ‘two’, *san* ‘three’, *si* ‘four’, *wu* ‘five’, *liu* ‘six’, *qi* ‘seven’*, ba* ‘eight’, *jiu* ‘nine’, and *shi* ‘ten’, and usually five numerical bases, that is, *shi* ‘ten’, *bai* ‘hundred’, *qian* ‘thousand’, *wan* ‘ten thousand’, and *yi* ‘hundred million’, constitute the foundation of counting, based on which, two kinds of complex numerals are formed: multiplicative numerals and additive numerals. A multiplicative numeral is generated by combining a numeral with a numerical base, as shown in (1a). It can be further combined with a larger base to form a high-level multiplicative numeral, as shown in (1b).

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The generation of *shi bai* ‘ten hundred’, *shi qian* ‘ten thousand’, and *bai qian* ‘hundred thousand’ is ungrammatical in Chinese, which can be excluded by Hurford’s Packing Strategy (1975, 2007).

An additive numeral is generated by conjoining at least two numerals, disallowing the occurrence of coordinators. The juxtaposed numerals cannot have the same base and are always in a high-low base order, as shown in (2).

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When the numerals are not adjacent in powers, the numeral *ling* ‘zero’ is needed to fill in the missing power as a placeholder for the number 0 (see He, 2015, for more discussions of *ling*). But *ling* is not allowed between two adjacent powers, as shown in (3).

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China was one of the earliest countries to use both integer numerals and decimal fractions, which can be found in *The Nine Chapters on the Mathematical Art* around the 1st century. Decimal fractions originate from measurement. The ancient length measure words include *fen* ‘deci’, *li* ‘centi’, *hao* ‘milli’, *si* ‘decimilli’, and so forth, which gradually developed to be numerical bases for decimal fractions during the Song and Yuan Dynasties (Teng, 2010). Among these numerical bases, *fen* stands for a tenths unit (10^{−1}), *li* a hundredth unit (10^{−2}), and so on. In ancient Chinese, a decimal fraction ‘3.1415’ can be expressed as in (4a). Due to the influence of the Arabic notation, units behind the decimal point are omitted in modern Chinese, as seen in (4b).

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### 2. Syntax of Chinese Numerical Expressions

Syntactic analysis is the first step of grammatical studies since semantics is composed based on constituent analysis. Thus, in linguistic studies of numerical expressions, it is necessary to figure out their syntax first. The internal syntax of numerical expressions mainly concerns complex numerals but not simple numerals because the latter are unanalyzable words. The key questions for the internal syntax of complex numerical expressions such as multiplicative numerals, additive numerals, and decimal fractions include issues like whether complex numerical expressions form constituents and what the proper analysis of the phrase structure of complex numerical expressions is. The external syntax of numerals concerns both complex and simple numerals in construction with other elements, for example when numerals appear in larger nominal constructions, or classifier-nominal constructions as in the case of Chinese. The issue of external syntax of numerals emerges when two opposing analyses are available in the literature, especially when it comes to Chinese. The controversy lies in whether numerals form constituents with classifiers or not. This section examines current studies on the internal syntax of complex numerals (including multiplicative numerals, additive numerals, and decimal fractions) and the external syntax of numerals in larger numeral-classifier constructions (including verbal classifiers) in Chinese.

#### 2.1 Internal Syntax of Complex Numerals

Two accounts of the internal syntax of complex numerals are currently available in the literature. The traditional constituency analysis, which has been widely adopted by a number of scholars (Borer, 2005; Corver & Zwarts, 2006; Hurford, 1975; Jackendoff, 1977; Kayne, 2010; Selkirk, 1977; among others), assumes that complex numerals form phrasal constituents. It has, however, been recently challenged by the nonconstituency analysis (Ionin & Matushansky, 2006, 2018) which, mainly based on the case marking data in Russian, assumes that a complex numeral does not form an immediate constituent but projects a complementative structure, as bracketed in (5). A numeral is treated as a nominal head selecting a lexical noun or a numeral-noun combination as its complement. Consequently, there are no numeral phrases like *two hundred* in the grammar, and the underlying forms of any numeral phrases are actually noun phrases.

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Furthermore, an additive numeral expression, for example, *two hundred twenty books*, is derived from a full NP coordination in which the head noun is either right-node-raised or PF-deleted, as illustrated in (6). That means *two hundred twenty books* is interpreted as a sum of two hundred books and twenty books.

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The nonconstituency analysis, however, may not be applicable to Chinese numerals, as argued in some recent works (He, 2015; He et al., 2017). A phrase structure of complex numerals is accordingly provided under the traditional constituency analysis, as exemplified in (7).

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In this phrase structure, numerals, either simple or complex, are constituents and form numeral phrases. The numerical bases determine the hierarchies of a phrase structure. Simple numerals directly project one-digit NumeralP_{0}; the base *shi* projects two-digit NumeralP_{1}; the base *bai* projects three-digit NumeralP_{2}; and the like.

A multiplicative numeral forms a phrase with a NumeralP as its specifier and a base as its functional head. As the specifier NumeralP can be simple or complex, a multiplicative NumeralP can be one-level, for example, [*wu bai*] ‘five hundred’, or two-level, for example, [[*wu bai*] *wan*] ‘five hundred ten-thousand’. An additive numeral phrase is formed by conjoining two NumeralPs with an implicit coordinator CONJ. However, the numeral coordinator may be explicitly spelled out and may be (but not necessarily, see He et al., 2017) morphologically the same as nominal coordinators in some languages, for example, *and* in English. The power of an additive numeral phrase is the same as the first NumeralP (the same subscript). For example, [[*wu shi*] *yi*] ‘five ten one’ projects a NumeralP_{2} the same as [*wu shi*] ‘five ten’.

In addition to multiplicative and additive numerals, decimal fractions also project complete phrase structures, as schematized in (8b); *dian* ‘point’ is treated as a numeral coordinator and numerals following *dian* are coordinated by covert numeral coordinators.

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A phrase structure, as evidenced by He (2015), can accommodate Chinese numerals more satisfactorily than a complementative structure. The constituency analysis of complex numerals in Chinese may have theoretical implications cross-linguistically.

#### 2.2 External Syntax of Numeral-Classifier Combinations

Numerals can occur with two types of classifiers in Chinese: nominal classifiers and verbal classifiers. The former is used as a unit to count the object denoted by the succeeding noun, for example, *san ben shu* ‘three books’, whereas the latter counts events or actions denoted by the co-occurring verb phrase, for example, *san ci qu Beijing* ‘go to Beijing three times’. This section reviews the syntax of numeral-classifier combinations that include the [Num-CL_{n}] and [Num-CL_{v}] constructions. The former construction contains a numeral, a nominal classifier, and a noun, while the latter is formed by a numeral and a verbal classifier. The central question is whether numeral-classifier combinations in Chinese form constituents.

For the [Num-CL_{n}-N] constructions, it is generally agreed in traditional Chinese linguistics that a numeral forms an immediate constituent with a nominal classifier, then combining with a head noun in a left-branching structure (Chao, 1968; Hu & Zhang, 1984; Li, 2000; Ma, 1990; Wang & Lu, 1981; Xing, 1996). The traditional view seems intuitively correct. Generative linguists, however, have divergent views on this matter. Some follow the traditional view (Gao, 1994; Hsieh, 2008; Huang, 1982; Tang, 1990a; Tsai, 2011), while many argue that a numeral combines with a noun in a right-branching structure (Cheng & Sybesma, 1998, 1999; Li, 1998; Tang, 1990b; among others). Proponents of the right-branching approach adopt this view mainly based on theoretical considerations, for example, in line with Abney’s DP Hypothesis (1987). This view has remained popular in generative linguistics. Thus, there are two syntactic possibilities of the [Num-CL_{n}-N] construction: left-branching [[Num CL_{n}] N] and right-branching [Num [CL_{n} N]].

The constituency issue has been further complicated by the split account (Li, 2011; Zhang, 2011, 2013), which proposes that both left-branching and right-branching structures exist in Chinese. Zhang (2011) argued that container measure classifiers, standard measure classifiers, partitive classifiers, and collective classifiers (henceforth, measure words) involve a left-branching structure, and individual or individuating classifiers have a right-branching one. The split account is argued against by more recent works (He, 2016, 2021; Her, 2012, 2017) that defend for the left-branching structure.

For the [Num-CL_{v}] constructions, most research considers a numeral and a verbal classifier to be a constituent, though these analyses may differ in details (Ernst, 1996; Matthews & Yip, 1999; Shao, 1996; Sybesma, 1999; Wu, 2011; Zhou, 2007). Some scholars do not accept the constituency analysis but maintain that numerals and CL_{v}s have a spec-head relation of a functional projection UnitP (Liu, 2009; Zhang, 2017).

In general, two syntactic structures have been sketched for the numeral-classifier combinations in Chinese. Subsections 2.2.1 and 2.2.2 present a brief introduction to Zhang’s analyses (2011, 2017) and the counterevidence from He (2016) and He and Tan (2019), with the former being the most influential advocate of the right-branching approach and the latter being proponents of the left-branching approach.

##### 2.2.1 Right-Branching Approach

On the right-branching approach, Zhang (2011, 2017) unified the syntax of the numeral-classifier combinations (except measure words) with a complementative structure. For nominal classifiers, measure words have the left-branching structure, as illustrated in (9a), while individual and individuating classifiers embrace a right-branching structure, as shown in (9b).

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The split account is claimed to best capture the different performances between measure words and individual classifiers. For example, a strong semantic selection is generally considered to exist between individual classifiers and nouns; however, other types of classifiers do not show the selectional restrictions, as shown, respectively, in (10a) and (10b).

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The differences in semantic selection are explained by assuming that individual classifiers c-command the associated nouns while measure words don’t. The difference can be represented by different structures: [*san* [*pi ma*]] and [[*san chexiang*] *ma*].

Similar to nominal classifiers, a verbal classifier is treated as a functional head that selects a numeral as its specifier and a verb phrase as its complement, as illustrated in (11).

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As discussed in detail in He (2016) and He and Tan (2019), the right-branching approach faces some empirical problems.

##### 2.2.2 Problems of the Right-Branching Approach

###### Problem 1: Independent [Num-CL] Combinations

A consequence of the right-branching approach is that the numeral-classifier combinations such as *san ge* or *san ci* can never occur alone, which means the superficially independent [Num-CL] combination, if there is one, is actually followed by an elliptical or implicit noun for CL_{n}s, and a VP for CL_{v}s. However, the [Num-CL] combinations in Chinese can occur alone in various environments, and other elements cannot be inserted as the complement of classifiers without leading to ungrammaticality, as seen in (12) and (13).

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**Problem 2: LF-Movement of [Num-CL] Combinations**

He and Pan (2014) argued that the fractional or even integer number phrases in average sentences must move in the logical form (see details in section 3.2). Along this line, *3.5 ge* and *san ci* in (14a–b) must form constituents, since LF-movement can only apply to constituents. Besides, a right-branching structure like [*san* [*ci* [*zuoyou*]]] is semantically uninterpretable.

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###### Problem 3: Semantic Composition

The right-branching approach to the [Num-CL] combinations faces various difficulties in semantic composition. Firstly, the right-branching structure, that is, [Num [CL_{n} N]] will lead to wrong interpretations when CL_{n}s are followed by *ban* ‘half’ or *duo* ‘many’. For instance, the expression *shi mu duo di* ‘ten acres of land or more’ refers to a range of 10–11 acres of land, not 13 or 14 acres, thus *duo* is not semantically related to *shi mu*, but to (*yi) mu*. Supposing *duo* expresses 0.5 acre here, two possible right-branching structures (15a–b) and a left-branching (15c) can be ruled out because they either are uninterpretable or yield wrong interpretations. Example (15d) is the only possible structure here (see details in section 4).

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Secondly, in the right-branching approach, the [Num-CL_{v}] constructions, for example, *san ci*, cannot obtain the meaning of ‘three times’ since *san ci* is not a complete component (see He & Tan, 2019, section 5.2 for more discussion).

Thus, the [Num-CL] combinations in Chinese form constituents, and their syntactic and semantic features are better captured by the left-branching approach.

### 3. Semantics of Chinese Numerical Expressions

Current investigations of Chinese numerals shed light on the two competing lines of research on the denotation of numerals, and the compositional semantics of complex numerals and larger nominal constructions in Chinese can be derived according to their structures. The structure-interpretation mapping theory of the numeral system, sketched by He (2021) with Chinese as the representative language, may provide implications for the theoretical basis for natural language numeral systems.

#### 3.1 Denotation of Numerals

Numerals are special in that they can appear in various grammatical positions, that is, adjective positions, determiner positions, and argument positions, as shown in (16), (17), and (18).

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In extensional semantics, the meaning of a linguistic form is assigned to an entity in the model of ontology, such as objects, sets, and truth values. However, the distribution of numerals here cannot be easily accounted for in the existing model. At present, there are two main views, that is, numerals refer to numbers (noun-like), or to sets (adjective-like or quantifier-like).

##### 3.1.1 Numerals Denote Numbers

The first view assumes that numerals refer to numbers, a kind of abstract object, grammatically equivalent to nouns. The underlying philosophical basis of this view is realism (or Platonism), which has been prevalent in the philosophy of mathematics since the 19th century. Realism believes that numbers do exist as an abstract object outside of time, space, or human thinking.

Frege (1884/1960) first introduced numbers into the model as the denotation of numerals. He maintained that the argument uses of numerals in (18) are best analyzed as referring to numbers. For example, (18d) is an identity statement in which *the number of moons of Jupiter* and *four* are both referring terms having the same denotation, the number 4. Frege’s view is supported or adopted by many linguistic studies (Balcerak Jackson, 2013; Brogaard, 2007; Corver & Zwarts, 2006; Hackl, 2001; Krifka, 1995; Qi & He, 2019; Snyder, 2017).

This account can easily explain the argument uses of numerals but faces difficulties when it comes to the modifier uses in (16) and (17). Numbers denoted by numerals and sets denoted by nouns will constitute a sortal mismatch such that the meaning cannot be composed. Two solutions have been proposed to adjust the sortal mismatch. The first solution, mostly adopted by linguistic studies, assumes that there is an invisible element following the numerals (Hackl, 2001; Krifka, 1995). For instance, the expression *five apples* is actually *five + X apples* in which X type-shifts a number to a set. The second solution, proposed by Frege, argues that numerals are ambiguous and can refer to both numbers and sets.

There is empirical evidence that lends support to the Fregean view. Qi and He (2019) investigated numerals in Shuhi, a Qiangic language of the Tibeto-Burman branch in Chinese, and reported that the Shuhi language features two kinds of numerals for the numeral ‘one’, the deficient numeral *ʥi*^{33} and full-fledged numeral *ʥĩ*^{35}. They argued that although most numerals directly refer to numbers, there are numerals denoting sets in natural languages, such as the numeral *ʥĩ*^{35} in Shuhi and probably *lia* ‘two-Cl’ and *sa* ‘three-Cl’ in Mandarin Chinese. It is further suggested that, like most numerals in Shuhi (except *ʥĩ*^{35}), English numerals also directly refer to numbers. The difference is that numerals in Shuhi have overt classifiers in syntactic projection, while English classifiers are implicit (see Krifka, 1995; Snyder, 2017, for similar analysis).

Though appealing, one disadvantage of the Fregean view is the need to make an ontological commitment to numbers. Philosophers are usually unwilling to take this step for fear of unexpected paradoxes. For this reason, many studies claim that numerals can refer only to sets.

##### 3.1.2 Numerals Denote Sets

The second view argues that numerals refer to sets, which are equivalent to adjectives or quantifiers in grammar (Felka, 2014; Hofweber, 2005; Kim, 2013; Knowles, 2015; Moltmann, 2013). This view is based on nominalism, which originated in the 11th century. Nominalism believes that numbers do not exist in the objective world because they are not concrete things. This account coincides with the principle of Occam’s Razor because numbers are repelled from the existing model.

The set-denoting view can be further divided into two accounts according to the members of the set. One account assumes that the members of the set are plural individuals (Felka, 2014; Knowles, 2015). A numeral, for example, *five*, expresses a property of cardinality and denotes a set in which plural individuals are composed of five elements, illustrated as〚five〛= λX[|X| = 5]. The other account claims that numerals have the same semantic type with quantifiers, that is, <et, <et, t>>, expressing properties of property. A numeral *five* is defined as〚five〛= λPλQ |P∩Q| = 5. This account was first proposed in the Generalized Quantifier Theory (Barwise & Cooper, 1981) and has been ruled out by Ionin and Matushansky (2006), since quantifier phrases cannot combine with determiners but numeral phrases can, for example, *the every student/the three students.

Along this line, modifier uses of numerals can be easily accounted for, while the argument uses need explanations. Many scholars defend the set-denoting view by rejecting Frege’s identity statement analysis of (18d), which is the critical evidence that Frege introduces numbers into the model. Issues have also been raised on the numerals in (18a) *Two and two is four*. For example, Hofweber (2005) claimed that numerals in arithmetical equations are bare determiners of which the type <et <et, t>> can be lowered to type e through cognitive type coercion. But currently there is no research on numerals in sentences such as (18c) *Seven is a prime number*, which poses serious challenges to the set-denoting view (see He, 2018, for more discussion).

To summarize, the Fregean view that numerals have two denotations is relatively more economical because ambiguity is a common phenomenon in language. The set-denoting view, however, must employ certain invisible mechanisms to some extent when dealing with numerals in arithmetic sentences. The treatment of an invisible mechanism faces difficulties in child language acquisition, and, more important, it is hard to verify and impossible to be falsified.

#### 3.2 Compositional Semantics of Numerical Expressions

The formal semantics of Chinese numerical expressions is rarely explored under the generative framework, though there is a much descriptive research in traditional grammar. A structure-interpretation mapping theory was proposed by He (2021) to represent the syntax and compositional semantics of numerical expressions in Chinese.

The syntactic structure determines how the meanings of parts of a compound expression are to be combined. In syntax, a complex numeral forms a numeral phrase first, and thus its meaning can be generated within itself. A numeral phrase and a classifier constitute a complete component, and, finally, the [Num-CL] constituent forms a larger phrase with a noun, for example, [[[*san shi*] *ben*] *shu*], ‘three ten CL_{n} book’, as reviewed in the previous sections.

Following Krifka (1995), He (2021) assumed that the semantic type of simple numerals is d and defined the numerical bases to have the meaning of predicates. Under this assumption, the meaning of a multiplicative numeral, for example, [[*san bai*] *wan*], ‘three hundred ten-thousand’, can be derived as follows.

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Additive numerals in Chinese are formed by the juxtaposition of two or more numerals, conjoined by implicit numeral coordinators. The semantic relation between additive numerals is that of arithmetic addition encoded by the implicit numeral coordinators, as defined in (20). The meaning of an additive numeral, for example, [[*san shi*] *wu*] ‘three ten five’ can be composed as shown in (21).

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Decimal fractions in Chinese are special additive numerals; *dian* ‘point’ is treated as an overt numeral coordinator, and thus the compositional meaning of a decimal fraction is as shown in (22).

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A decimal fraction usually appears in average sentences with a nominal or verbal classifier, as seen in (23).

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Noun phrases in average sentences, for example, *the average man*, seem unlikely to be assigned to any counterparts in the objective world. Therefore, average sentences have been considered to pose a challenge to truth-conditional semantics that is established on the correspondence between language and the world (Chomsky, 2000). Analysis of Chinese decimal fractions in average sentences may provide a new thought for solving this puzzle. He and Pan (2014) argued that the constituent of a numeral and a classifier is not interpreted in situ and must be moved out of the original place in the logical form. Thus, the [Num-CL-N] phrase, for example, *1.43 ge haizi* ‘1.43 children’, does not really exist in the logical form. Accordingly, the fractional or even integer noun phrases in average sentences cannot refer to individuals or quantities.

In average sentences, *pingjun* ‘average’ is proposed to encode the arithmetic division and one-to-one mapping relation, and is treated as a universal quantifier that requires three arguments: a measure function *f*, a plural individual X, and a quantity d, as shown in (24).

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In (24), f expresses a numerical relation, f = λnλX [X have n]. Therefore, an average sentence (25a) can be analyzed as in (25b), which can be paraphrased as ‘Every student corresponds to a numerical value that is obtained by dividing the total number of extracurricular books owned by all students by the number of students, that is, 1.5’.

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With respect to the compositional semantics of [Num-CL-N] constructions in Chinese, He (2021) maintained that classifiers in Chinese are type-shifters that fix the sortal mismatch between numerals (d type) and bare nouns (<e, t> or e type), as shown in (26a). The semantic type of bare nouns depends on whether they are treated as referring to sets or kinds. Classifiers can be generally defined as in (26b), in which the property A varies with specific classifiers. For example, the classifier *tiao*, which is used to describe long objects, is translated as in (26c).

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Therefore, the meaning of the [Num-CL-N] phrase, for example, [[*san tiao*] *yu*], ‘three fish’, can be derived as follows. In (27), the application of existential closure is optional since *san tiao yu* can directly refer to a set composed of plural individuals.

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### 4. Numerical Particles *Duo* and *Ban*

This section presents the syntax and semantics of two special numerical particles, *duo* and *ban* in Chinese.

In Chinese, numerals and [Num-CL] phrases can be followed by some particles for numerical approximation such as *duo* ‘many’ and *lai* ‘approach’. The syntactic distribution of the numerical particle *duo* has been extensively surveyed in the literature (Lü, 1967/2002; Xing, 1993, 2003; Yang, 1993; Ying & Wang, 2014; Zhang, 2001; Zhu, 1958; Zong & Zhang, 2008), although few formal studies can be found (He et al., 2020). Numeral phrases that can be attached to by *duo* include one-level multiplicative numerals, for example, *wu shi duo* ‘fifty or more (51~55)’; higher-level multiplicative numerals, for example, *san qian wu bai duo* ‘three thousand five hundred or more (3501~3550)’; decimal fractions, for example, *san dian yi duo* ‘three point one or more (3.11~3.15)’; and fractional numerals, for example, *shi fen zhi yi duo* ‘one tenth or more (0.11~0.15)’. Simple numerals cannot occur with *duo*, for example, **san duo* ‘3 or more (3.1~3.5)’.

He et al. (2020) offered a unified account of the distribution and interpretation of *duo* in various syntactic environments. As assumed in more recent research (Xing, 2003; Zhang, 2001), *duo* is analyzed as a particle conveying numerical approximation, instead of a real numeral. In addition, *duo* is syntactically attached to its preceding numeral phrase as a whole instead of the numerical base alone, as illustrated in (28).

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With the quantity denoted by *duo* within a range of 10%–50% of its left-adjacent numerals (Chao, 1968), the meaning of *duo* was defined by He et al. (2020) as in (29).

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The symbol n represents the number denoted by the numeral phrase that directly combines with *duo*. The exponential function *MINexponent* (n) yields the smallest power of n, and *MAXexponent* (n) yields the largest power. The compositional meaning of (28) can be derived as in (30).

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The syntax-semantics interface analysis of *duo* can also apply to the particles for numerical approximation in minority languages in South China (see He, 2021 for more discussion). The general applicability of analysis of *duo* indicates that the structure-interpretation mapping analysis proposed for Chinese may apply to other languages such as English.

As for the numerical particle *ban* ‘half’, it is generally believed to be both a numeral and a classifier in traditional grammar (Liu et al., 2004; Xing, 1993). When combined with a classifier, *ban* is a numeral that means 0.5, for example, *ban jin* ‘half jin’ (*jin* is a measure word that means half a kilo); when combined with a numeral, *ban* is a classifier that means a half of the amount. In this case, Chinese features only two expressions, that is, *yi ban* ‘a half’ and *liang ban* ‘two halves’. The *ban* following a classifier is considered to be a numeral the same as the *ban* in *ban jin*. Consequently, *san jin ban* ‘three and a half jin’ is derived from *san jin ban jin*.

This traditional view has recently been challenged by He’s analysis (2021) that *ban* following a classifier is a numerical particle similar to *duo*. With *ban* as a suffixed particle, *san jin* and *ban* form a constituent. So far, there are three uses of *ban* in Chinese: functioning as a numeral (*ban jin*), a classifier (*yi ban*), and a numerical particle (*san jin ban*). The numerical particle *ban* is sketched in (31).

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### 5. Conclusion

Mandarin Chinese features a simple and uniform numeral system, studies of which may shed light on the studies of the numeral systems from a cross-linguistic perspective. This article presented an overview of two major issues in the formal studies of Chinese numerical expressions as well as the analyses that have been advanced in the literature. The first issue concerns the internal syntax of complex numerals and external syntax of numerals occurring in larger numeral-classifier constructions. The second issue concerns the denotation of numerals and compositional semantics of complex numerals and [Num-CL-N] constructions. Research on the syntax and semantics of two special numerical particles, *duo* and *ban*, was introduced.

#### Further Reading

- Brogaard, B. (2007). Number words and ontological commitment.
*The Philosophical Quarterly*,*57*, 1–20. - Corver, N., & Zwarts, J. (2006). Prepositional numerals.
*Lingua*,*116*(6), 811–835. - Dummett, M. (1991).
*Frege: Philosophy of mathematics*. Cambridge, MA: Harvard University Press. - He, C. S. (2021).
*Shuci de jufa yuyi jiemian yanjiu*[A study on the syntax-semantics interface of numerals]. Shanghai, China: Shanghai Education Publishing House. - Her, O.-S. (2017). Structure of numerals and classifiers in Chinese: Historical and typological perspectives and cross-linguistic implications.
*Language and Linguistics*,*18*(1), 26–71. - Hofweber, T. (2005). Number determiners, numbers, and arithmetic.
*Philosophical Review*,*114*(2), 179–225. - Hu, Y. S., & Zhang, B. (1984).
*Shuci he liangci*[Numerals and classifiers]. Shanghai, China: Shanghai Education Publishing House. - Hurford, J. (1975).
*The linguistic theory of numerals*. Cambridge, UK: Cambridge University Press. - Ionin, T., & Matushansky, O. (2018).
*Cardinals: The syntax and semantics of cardinal-containing expressions*. Linguistic Inquiry Monographs 79. Cambridge, MA: MIT Press. - Menninger, K. (1969).
*Number words and number symbols: A cultural history of numbers*. Cambridge, MA: MIT Press. - Moltmann, F. (2013). Reference to numbers in natural language.
*Philosophical Studies*,*162*(3), 499–536. - Xing, F. Y. (1993). Xiandai hanyu shuliangci xitong zhong de “ban” he “shuang” [
*Ban*and*shuang*in modern Chinese numeral-classifier system].*Yuyan Jiaoxue Yu Yanjiu*[Language teaching and linguistic studies],*4*, 36–56. - Zhang, N. N. (2013).
*Classifier structures in Mandarin Chinese*. Berlin, Germany: Mouton de Gruyter.

#### References

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*The English noun phrase in its sentential aspect*[Doctoral dissertation]. MIT. - Balcerak Jackson, B. (2013). Defusing easy arguments for numbers.
*Linguistics and Philosophy*,*36*(6), 447–461. - Barwise, J., & Cooper, R. (1981). Generalized quantifiers and natural language.
*Linguistics and Philosophy*,*4*(2), 159–219. - Borer, H. (2005).
*Structuring sense: Volume 1. In name only*. Oxford, UK: Oxford University Press. - Brogaard, B. (2007). Number words and ontological commitment.
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