Chapter 2. The Nature of Numbers
2.5 Chinese Numerical Expressions
Chinese numerical expressions are similar to English numerical expressions in many ways. They are formed by a combination of multiplicative and additive arithmetical operations, and they are arranged in strings of digit places at successively higher levels. But Chinese numerical
expressions are arranged in strings of four places instead of strings of three places, as in English.
This means that 10,000, pronounced wan, becomes an important multiplier in Chinese, though it does not exist at all as a multiplier in English. Thus the English notated number 23, 417 would be notated 2, 3417 in Chinese, and instead of being pronounced er san qian (“twenty-three thousand”) at the beginning it is pronounced er wan (“two ten-thousand”) at the beginning. The remainder of the number is pronounced san qian (“three thousand”) si bai (“four hundred”) shi qi (“seventeen”). Each Chinese numeral contains four places: a units’ (ge) place, a tens’ (shi) place, a hundreds’ (bai) place, and a thousands’ (qian) place. These four places are repeated at successively higher levels. This illustrated in (6).
(6) Level of ge Level of wan Level of yi
1-10,000 10,000 – 10,000,000 100,000,000 – 100,000,000,000
ge place 0-9 0-9 ten-thousands 0-9 ten-millions of ten thousand
shi place 10-99 10-99 ten-thousands 10-99 ten-millions of ten thousand
bai place 100-999 100-999 ten-thousands 100-999 ten-millions of ten thousand
qian place 1000-9,999 1000-9999 ten-thousands 1000-9999 ten-millions of ten thousand
The most important difference between Chinese and English numerical expressions is that, unlike English, Chinese does not use any conjunction equivalent to “and” at any point in the composition of the expression. There is, however, diachronic evidence in Brainerd and Peng (1968) that in ancient times Chinese did use the morpheme you (or yu) equivalent to “and” in English as a co-ordinate conjunction in numerical expressions. According Liu and Peyraube (1994), grammaticalization first transformed the verb you, meaning “give,” to a preposition and then later to a conjunction. Li and Thompson (1981) observe that you now occurs mostly in pairs, meaning “both . . and” Zhang (2006, p. 180), however, notes that the first you of the construction you . . . you is deletable, suggesting that if you did occur in Chinese numerals, it would do so in the middle of a string. It is also significant to note here that Yang (2005. p. 45) assumes that you is part of the internal logic of the composition of Chinese numerical
expressions. This idea has merit, I believe, and I will develop it in Section 5.2.
The other major difference between Chinese numerical expressions and English numerical expressions is that Chinese sometimes incorporates the morpheme ling, meaning
“zero,” whereas the digit 0 is always phonetically null in English, except when it stands alone.
In Chinese numerals ling always appears in the place of a multiplier when the value of that
multiplier is 0, but ling only appears once in any string of four digits. For example, the number 56, 0000, 0025 is pronounced wu shi liu yi ling er shi wu. There are six zeros in this numeral, but only one ling is pronounced. According to Brainerd and Peng (1968), the ling that is pronounced is the one nearest to the end of the number. It would seem, therefore that the principal function of ling is to hold the place(s) of the absent multipliers. In this example it would be difficult to process and redundant to pronounce the fact that all the multipliers between yi and er shi wu have been omitted. In English the number 56,000,025 is pronounced “fifty-six billion and twenty-five.” Here too mention of the multipliers between “million” and “twenty-five” has been omitted. And yet there is an intervening morpheme, namely the conjunction
“and.” It is tempting, in this example, to equate the English use of “and,” with the Chinese use of ling, because both these morphemes seem to hold the places of intermediate multipliers and to signal that something has been omitted. Nevertheless, we have already seen that the use of “and”
in English numeral expressions appears to be predictably related to the function of co-ordination, and it therefore seems to exhibit phrasal qualities within the framework of X-bar theory. This is not immediately apparent in the case of ling, which is essentially a place holder. I will have more to say about the similarities and differences between “and” and ling when I present my analysis in Section 5.
In the meantime, the phonological rules in (7) may be said to govern the use of ling:
(7) a. Although ling appears medially in Chinese numerals, it is always a number, not a conjunction equivalent to “and,” which appears medially in English numerals.
b. When ling appears in the coefficient position before any multiplier, the multiplier is phonologically null at PF. For example, 305 is pronounced san bai ling wu, not
san bai ling shi wu.
c. When we have more than one consecutive ling, only one of them is pronounced because of phonological haplology. For example, the number 56, 0000, 0025 is pronounced wu shi liu yi ling er shi wu. There are six zeros in this numeral, but only one ling is pronounced.
d. If ling appears at the end of a numeral, it is phonologically null at PF. For example, 1500 is pronounced as yi qian wu bai or yi qian wu, not yi qian wu bai ling ling.
e. If a number ends with a multiplier, pronunciation of the multiplier is optional. Thus, 3500 is pronounced either as san qian wu bai or san qian wu.
CHAPTER 3
HURFORD’S STANDARD ANALYSIS AND MY FIRST RESPONSE
3.1 Hurford’s (1975, 2003) Analysis
The phrase structure rules given by Hurford identify three constituents of numerical expressions:
Phrase, Number, and M(ultiplier). The value of a Phrase is the product of its constituents, and the value of a Number is the sum of its constituents. The value of an M is always 10 or a multiple of 10. Hurford’s structure rules state that both Number and M are recursive. His basic phrase structure rules are given in (8).
(8) DIGIT
NUM →
NUMPHRASE (NUM)
NUMPHRASE → NUM M
In Hurford’s own words (2001, p. 10758): “Here, `NUM' represents the category Numeral itself, the set of possible numeral expressions in a language; `DIGIT' represents any single numeral word up to the value of the base number (e.g., English one, two, . . ., nine); and `M' represents a category of mainly noun-like numeral forms used as multiplicational bases (e.g., English -ty, thousand, and billion). The curly brackets in the rules enclose alternatives; thus a numeral may be either a DIGIT (e.g. eight) or a so-called NUMPHRASE (numeral phrase) followed optionally by another numeral (e.g., eight hundred or eight hundred and eight). If a numeral has two
immediate constituents (i.e., is not just a single word) the value of the whole is calculated by adding the values of the constituents; thus sixty four means 60-4. If a numeral phrase (as distinct
from a numeral) has two immediate constituents the value of the whole is calculated by multiplying the values of the constituents; thus two hundred means 2 – 100.”
There are three problems associated with Hurford’s structure:
1. The first problem is the simple fact that his categories of Phrase, Number, and M are not related to the standard syntactical categories of Noun, Verb, Adjective, and Preposition.
Hurford claims (1975, p. 19) that Phrase, Number, and M are syntactical categories, but all that he specifically mentions in regard to syntax is that Multipliers are always nouns.
2. The second problem is that his phrasal structure incorporates ternary branching to include the conjunction “and.” This violates a major principle of X-bar theory, which states that phrasal structures must be binary branching (Kayne, 1984; Pollard, 1984; Kornai &
Pullum, 1990). This flaw is illustrated in (9) below.
3. The third problem involves the placing of ling, meaning “zero,” when the structure is applied to Chinese numerals. This morpheme is placed in the Phrase category when logically it should appear in the Number category. This problem is illustrated in (10) below.
(9) English numeral 230, 567
NUM
230 567
PHR NUM
NUM M PHR CONJ NUM
PHR CONJ NUM NUM M PHR NUM
NUM M PHR
Two hundred and thirty thousand five hundred and sixty seven
2 100 30 1000 5 100 60 7
The problem of ternary branching is obvious here. This seems to occur because of the attempt to combine nouns and a conjunction syncategorematically through his conjunction insertion rule (p.
50), but this violates the principles of X-bar theory. It is also interesting to note that Hurford makes no attempt to incorporate in his structure the multiplicative composition of two complex Phrases, 30 and 60. Apparently this operation has been left out because it is necessary to pronounce “thirty” and “sixty” as integral parts of the overall numerical expression. In other words, the operations of multiplying the Numbers 3 and 6 by the Multiplier 10 to form the Phrases 30 and 60 are phonologically null. Hurford has omitted these operations in his structure because he is only interested in representing the actual pronunciation of the numerical expression.
(10) Chinese numeral 23, 0567
NUM
23 0567
PHR NUM
PHR PHR NUM
NUM M PHR PHR NUM
PHR NUM M PHR PHR PHR NUM
NUM M NUM M PHR NUM M NUM M NUM
er shi san wan ling wu bai liu shi qi
2 10 3 1,0000 0 5 100 6 10 7
.
The only possible justification for Hurford’s structure here is to say that ling combines with the deleted M qian at LF to form a Phrase and that is what is projected. There seems to be some plausibility for this when we note that in Hurford’s structure of the English numerical expression in (8) he projects both “thirty” and “sixty” as phrases, presumably because the numbers 3 and 6 have already combined with the M 10 and these operations are phonetically null.
3.2 My First Response to Hurford
The phrasal structure I first considered is compatible with Hurford’s generally accepted analysis.
At the same time, my proposal is more precise than Hurford’s in its terminology, and it therefore more accurately describes the ways that numeral expressions are actually generated.
In my proposed analysis all the constituents are nouns, so in the beginning I believed it to be important to distinguish the unique feature of the different kinds of nouns and nominal
compounds used in the proposed phrase structure. This was done in the following ways.
1. A noun that serves as multiplier (10, 100, 1000 and so on) is designated [mN].
2. A simple number (0-9) is designated [sN].
3. A complex number – that is, a number made by multiplication or addition – is designated [cN]. However, all complex numbers are categorized, as in (4) and (5).
4. A complex number that is a product of multiplication is designated [pcN].
5. A complex number that is the sum of addition is designated [scN].
6. A complex number that is combined by [scN] and [mN] is [mP].
Labeling all the numbers and numerical expressions as nouns indicated their major phrasal category, and sub-labeling them according to their functional characteristics indicated the exact and complete ways that they operate in the phrase structure. Hurford’s terminology for nominal phrase structure rules may be more parsimonious; but my terminology was more extensive, precise, and meaningful.
The fundamental phrase structures for the proposed nominal compounds of numerical expressions are given in (11).
(11) a. The Multiplicative Structure pcN
sN mN
b. The Additive Structure scN
pcN/ mP (and) scN/ sN
At first I was willing to accept Hurford’s flat and ternary branching additive structure with the insertion of the conjunction “and”, even though it is difficult to combine this part of the structure with the multiplicative part of the structure. Just the same, I suspected that ternary branching is somehow inappropriate for analyzing the structure of numerical expressions.
The full phrasal structures I first considered for the construction of numerical expressions as nominal compounds in English and Chinese are presented in (12) and (13).
(12) English numeral 15,438,353
scN
scN
mP scN
scN pcN CONJ scN
mP pcN scN
scN mN sN mN pcN sN mN sN mN pcN sN
Fifteen million four hundred thirty eight thousand three hundred and fifty three
(13) Chinese numeral 1543,8353
scN
mP
scN scN
scN scN
scN scN
pcN pcN pcN mN pcN pcN pcN
sN mN sN mN sN mN sN sN mN sN mN sN mN sN
yi qian wu bai si shi san wan ba qian san bai wu shi san
I think that my first revision of Hurford’s phrasal analysis of numerical expressions does have a certain amount of merit. Not only does my revision clarify Hurford’s terminology and make it more exact, but my revision also challenges a basic assumption on which Hurford builds his system. From the outset I have assumed that all numbers are abstract nouns, and therefore they should be regarded as absolutely context free. This assumption represents an important departure from Hurford’s somewhat tentative assumption that small numerals from 0 to 9 behave as adjectives, while larger numbers behave as nouns. For example, in a note on Corbett’s (1978) statement that the higher numbers are, the “nounier” they become, Hurford (1980, p. 247) says,
“I believe that he is right.” And yet Hurford does not incorporate the syntactical category of adjective into his proposed phrasal structure for numerical expressions. The most that he says in regard to syntactic categories is that his Phrases are nouns. I am convinced, however, that all numbers are abstract nouns, and that my precise – though, admittedly, complicated – system for labeling all numerical nouns according to their arithmetical functions is, at the very least, an improvement on Hurford’s nomenclature.
My first response to Hurford’s analysis was that he is not willing to make the attempt to fully integrate the grammar of numerals with the grammar of language. Because of this apparent reluctance Hurford is not able to analyze numerical expressions according to X-bar theory, and, since I am personally committed to acceptance of this theory, I found Hurford’s analysis to be unsatisfactory. I felt that there must be a way to construct a phrasal structure for numerical expressions that did not violate the rule of binary branching simply because it had to incorporate the transformation of conjunction insertion in English. Besides, Chinese numerical expressions do not even exhibit conjunction insertion at PF. The only solution appeared to be the
construction of a binary branching phrase that incorporates conjunctions as part of the phrase.
Therefore, I propose that numerical expressions should be analyzed as additive or cumulative Conjunction Phrases (&P). Throughout the remainder of this thesis I will advance my argument for this interpretation of numerical expressions, providing what I believe is a plausible revision of Hurford’s phrasal analysis.
CHAPTER 4
NUMERICAL EXPRESSIONS AS CONJUNCTION PHRASE (&P) ADJUNCTS
4.1 On Co-ordination
We have seen so far that, in both English and Chinese, in the most fundamental manner possible, numbers are always sums. Therefore, numerical expressions encode, first and foremost, the arithmetical operation of addition. Intuitively, then, we must expect that the most appropriate syntactical operation for the construction of numerical expressions is co-ordination. In my first response to Hurford’s classic analysis I attempted to improve upon the nomenclature of
Hurford’s phrase structure by identifying the various noun phrases involved in the arithmetical operations of numeral expressions, giving these noun phrases specific functional names. But that was not sufficient to really expand or revise Hurford’s analysis. In order to do that we need to interpret numerical expressions according to X-bar theory, but before we proceed in that direction, we need to examine certain matters related to co-ordination in general.
According to Carston and Blakemore (2005), the central issue in the current discourse on co-ordination is the matter of symmetry versus asymmetry. There is an intuitive sense whereby co-ordinated elements possess both similar semantic values and similar syntactical status.
Moreover, the meaning of “and” appears to suggest that in terms of truth-conditional
propositions, P & Q is equivalent to Q & P, though pragmatics suggests that in some cases P &
Q is actually P & then Q. For these reasons, early interpretations of co-ordination, such as that of Jackendorff (1977), tended to be represented by flat structures, either non-headed or multi-headed, with the conjunction mediating between or among symmetrical syntactic elements. This symmetrical interpretation of co-ordination is still favoured by some authors. Even early X-bar theorists sometimes claimed that co-ordination is an exception to the conventions of the X-bar schema. Nevertheless, with the development of Chomskeyan Principles and Parameters theory and Minimalism in the 1990s, most authors have come to accept the phrasal structure of ConjP whereby the two conjuncts of a co-ordinated structure are not symmetrical in that XP is
connected to a constituent formed by the conjunction and YP.
Kubo (2007) points out that it is impossible to ignore the problematic nature of co-ordination in relation to the ambivalence of its symmetrical and asymmetrical features. Of particular interest to Kubo is the fact that co-ordinate constructions are paratactically construed, suggesting that they possess a fundamental symmetry. But the suggestion of symmetry is not the only significant feature of paratactical arrangement. For the purposes of my overall argument in this thesis it is the centrality of the conjunction, particularly “and,” in parataxis that I would like to emphasize. If numerical expressions are co-ordinate structures, as I assume they are, their paratactical construction, in their context-free status as a miniature independent form of
discourse, implies that adjunctive “and” is required in their formation. While I contend that the paratactical construction of co-ordinate structures reveals the centrality and necessity of “and” in numerical expressions, I do not, however, argue that numerical expressions – or, indeed, co-ordinate structures in general – are to be interpreted as phrasally symmetrical. On the contrary, I assume that numerical expressions, like all co-ordinate structures, are best accounted for as asymmetrical phrases following the binary branching and strong endocentricity principles of X-bar theory. In fact, Kubo also points out two additional characteristics that display the
asymmetry of co-ordinate structures: they exhibit c-command relations between the first and second conjuncts, and they exhibit co-ordination internal consistency, as in Ross (1967). I assume that these two features are also evident in numerical expressions.
As Carston and Blakemore (2005) observe, most linguists today accept the asymmetry of co-ordinate structures. This is true even of authors working outside the paradigm of X-bar theory, Principles and Parameters Theory, and Minimalism. Hudson (2003), for example, a proponent of Word Grammar, assumes that in English the conjunction and the second conjunct in a co-ordinate structure form a constituent that is combined with the first conjunct
asymmetrically. Zhang (2006) calls the first conjunct the external conjunct and the combination of the co-ordinator and the second conjunct the internal conjunct. I shall use this convenient terminology throughout the remainder of this thesis. Ross (1967) first proposed the constituency of the conjunction and the second conjunct on phonological grounds, arguing that an intonational pause is possible between the first conjunct and the conjunction, but not between the conjunction and the second conjunct, as in (14):
(14) a. John left, and he didn’t even say good-bye.
b. John left. And he didn’t even say good-bye.
c. *John left and. He didn’t even say good-bye.
Another solid argument for asymmetry in co-ordinate structures comes from pragmatic
processing whereby the logic of co-ordinate truth statements is not always (P & Q) / (Q & P), but sometimes (P & then Q), as in Hudson’s (2003) example, given in (15):
(15) a. She gave him the key, and he unlocked the door.
b. *He unlocked the door, and she gave him the key.
Not only do the conjunction and the second conjunct form a constituent in each of these sentences, but the semantics and pragmatics of the co-ordinated statement indicate that word order cannot be altered without changing the meaning. (P & Q) is not symmetrical with (Q & P).
This simple fact has important implications for the asymmetrical and rigidly ordered structure of numeral expressions, as I will illustrate in Section 5.
This simple fact has important implications for the asymmetrical and rigidly ordered structure of numeral expressions, as I will illustrate in Section 5.