Quantifiers In Linguistics Research Paper

The basic semantic structure of quantification is bipartite, consisting of the quantifier itself plus the expression that it quantifies. For example, in a sentence such as John saw three cats, three is the quantifier, and cats the quantified expression.

Quantification has traditionally been of great interest to semanticists, logicians and philosophers of language, due at least in part to the perceived ‘logical’ or ‘mathematical’ nature of the meanings involved. As such, it is striking to observe how such basic and seemingly immutable meanings may, in different languages, be expressed with very different morphosyntactic strategies, exhibiting a great degree of cross-linguistic variation.

Quantifiers may be classified internally, with respect to their morphosyntactic structure and basic semantic properties. In addition, they may be classified externally, with regard to the syntactic and semantic relationships between the quantifiers and the quantified expressions and/or the remainder of the sentences in which they occur.

2. Internal Classifications

Internal classifications of quantification make reference to a variety of fundamental morphosyntactic and semantic properties.

2.1 Mass And Count Quantifiers

One of the most basic distinctions is between mass and count quantifiers (Pelletier 1979, Allan 1980). Mass quantifiers constitute expressions which denote an undifferentiated homogeneous mass; for example, English much is a mass quantifier, because it forms expressions such as much milk, much cake. In contrast, count quantifiers constitute expressions which refer to one or more countable units of characteristic size and shape; for example, English many is a count quantifier, because it forms expressions such as many cats, many cakes. Whereas many quantifiers belong to just one of the two types, mass or count, some quantifiers are undifferentiated with respect to the mass count distinction, and may appear in expressions of either kind.

An interesting case is that of the so-called ‘classifier languages,’ many of which are located in the East Asian linguistic region (Craig 1986; see example (4) below). In such languages, many or all of the count quantifiers cannot occur in immediate construction with the noun that they quantify; instead, the quantifier must occur in construction with a classifier, and the quantifier-plus-classifier constituent may then occur in construction with the noun. Typically, the classifier makes reference to various characteristics— size, shape, function, and so forth—of the quantified noun. Accordingly, different nouns often require different classifiers. However, in some cases, different classifiers may occur with the same noun, sometimes resulting in subtle differences in meaning.

2.2 Existential And Universal Quantifiers

Of particular interest to logicians are the existential and universal quantifiers (Vendler 1967, Heim 1982, Gil 1995). Existential quantifiers, such as English some and a(n), form expressions denoting at least a minimal, non-zero amount or number of the quantified expression, for example some water, a boy. Universal quantifiers, such as English all and every, form expressions referring to an exhaustive amount or number of the quantified expression, for example all the water, every boy. Existential and universal quantifiers are semantically related to each other through negation by means of the classical ‘square of opposition’: thus, in English, Not a boy came is equivalent to All the boys did not come, and Not all the boys came is equivalent to A boy did not come.

Existential quantification, in many languages, is inextricably intertwined with singular number and/or indefiniteness: in some cases they are expressed identically, while in other cases they are expressed by means of the same marker occurring in different constructions, or by means of similar though not identical markers. Universal quantification, too, is expressed in a variety of different ways across languages. Some languages, such as English, do have ‘dedicated’ universal quantifiers, that is to say, words such as all and every whose primary or exclusive function is the expression of universal quantification. However, many other languages lack such words, instead making use of more complex circumlocutions if and when the expressive need arises.

Given the importance of existential and universal quantifiers in mathematical logic, it is worthy of note how relatively unimportant these quantifiers are within the grammatical systems of languages. It is indeed not all that common to find a single form, in any language, whose interpretation corresponds precisely and unproblematically to either the existential or the universal quantifier. Moreover, there is probably no language in which expressions of existential and universal quantification constitute a natural grammatical class to the exclusion of other expressions. The diversity of morphosyntactic expression of existential and universal quantifiers across the worlds languages suggests that, although greatly valued by logicians, they may not be endowed with privileged status vis a vis the structure of language.

2.3 Mid-Range Quantifiers

‘In between’ the existential and the universal quantifiers lie a variety of mid-range quantifiers, such as, in English, few, several, many, most, and so forth. Among the mid-range quantifiers, a particularly privileged category is that of cardinal numerals—quantifiers which refer to natural numbers, for example, one, two, thirteen, one hundred and twenty seven (Menninger 1969, Hurford 1975). Complex phrasal numerals often make use of syntactic patterns available elsewhere in the language; however, in addition, they sometimes exhibit unique structural features not found in other grammatical domains

2.4 Other Internal Classifications

Quantifiers are sometimes classified as either strong (alternatively referred to as definite) or weak (indefinite) (Milsark 1977). Strong quantifiers, such as English all and most, are inherently definite in their meaning, whereas weak quantifiers, such as English some and three, are not. Strong and weak quantifiers often exhibit different syntactic behaviour; for example, strong quantifiers are typically awkward or ungrammatical in existential sentences, e.g.,? There is all the water on the floor.

Some further semantic distinctions are captured by the categories of increasing (alternatively referred to as upward entailing) and decreasing (or downward entailing) (Keenan and Stavi 1986). A quantifier is increasing if a sentence containing it would be logically true in a situation in which the amount or number of quantified expressions involved is larger than that explicitly stipulated by the quantifier, for example, at least three. Conversely, a quantifier is decreasing if a sentence containing it would be logically true in a situation in which the amount or number of quantified expressions involved is smaller than that explicitly stipulated by the quantifier, for example, at most three. Other quantifiers are neither increasing nor decreasing, for example exactly three.

An interesting problem is posed by simple monomorphemic quantifiers such as three. At first blush, such quantifiers appear to be neither increasing nor decreasing: if somebody says There are three cats on the bed, the mental picture that is conjured up is, of course, one of three cats, not two, or four, or any other number. However, a moment’s reflection will reveal an important asymmetry. If in fact there are exactly two cats on the bed, then the above sentence is clearly false. In contrast, if there are exactly four cats on the bed, then, in most contexts, the above sentence would obviously be misleading and inappropriate—but it would not, strictly speaking, be false. Thus, while the pragmatic appropriateness conditions of three resemble those of exactly three, the actual truth conditions of three are identical to those of at least three. Hence, three is, in fact, increasing.

3. External Classifications

External classifications of quantification make reference to the morphosyntactic and semantic relationships which obtain between the quantifiers and the quantified expressions and/or the remainder of the sentences in which they occur.

3.1 Nominal And Verbal Quantifiers

One basic semantic classification relates to the category of the quantified expression, specifically whether it is of a nominal or verbal nature (Gil 1993). For example, in the English sentence John saw three cats, the numeral three quantifies a noun; accordingly this sentence illustrates nominal quantification. In contrast, in the sentence John sang three times, the numeral phrase three times quantifies a verb; hence this sentence instantiates verbal quantification.

Up to a point, nominal and verbal quantification exhibit parallel grammatical properties. Obvious, but still worth noting, is the fact that in both of the preceding sentences, the same numeral, three, is used. However, nominal and verbal quantification also differ in many ways; typically, verbal quantification is more highly marked than its nominal counterpart.

3.2 Continuous And Discontinuous Quantifiers

A distinct albeit related syntactic classification pertains to the position of the quantifier with respect to the quantified expression, specifically, whether or not the two form a continuous constituent (Gil 1993, Bach et al. 1995). Sentences (1)–(4) below all instantiate nominal quantification, in which a numeral quantifies an expression referring to an animal; however, each of the examples exhibits a different syntactic configuration.

• H_EBREW

raiti                              [salos arnavot]

see-_PAST1: _SG      three-_F rabbit-_PL: _F

‘I saw three rabbits’

• T_AGALOG

Tatlo    [ang      nakita

three                _{TOP                   PAT}._TOP: _PFV-see

kong     baboy]

_DIR:1: _SG–_LIG pig

‘I saw three pigs’

(‘The pigs seen by me were three’)

(3)        K_UTENAL (Matthew Dryer, p.c.)

hu                    [qa sa               wu: kati]          cupqa

1: _SG     three-_PRW          see-_IND deer

‘I saw three deer’ (‘I three-saw deer’)

(4)        J_APANESE

Watasi wa                    inu       o

1: _{SG                 TOP}                   dog      _ACC

sanbiki mita

three-_CLF see-_PFV

‘I saw three dogs’ (‘I saw dogs threely’)

In Modern Hebrew, (1), as in its English gloss, the quantifier groups with the quantified expression, salos arnavot, to the exclusion of the remainder of the sentence; in Tagalog, (2), the quantifier tatlo stands alone as a sister to the remainder of the sentence, which forms a constituent; in Kutenai, (3), the quantifier occurs in construction with the verb, qa sa wu: kati, to the exclusion of the remainder of the sentence, while in Japanese, (4), the quantifier sanbiki is a sentential adverb not forming an obvious constituent with any part of the remainder of the sentence.

Constructions (1) and (2) exemplify continuous quantification, in which the quantifier and the quantified expression form a constituent. In contrast, constructions (3) and (4) exemplify discontinuous quantification, in which the quantifier occurs apart from the quantified expression. Examples (1)–(4) above illustrate the distinction between continuous and discontinuous quantification with respect to nominal quantification. In principle, a similar distinction may also be drawn with respect to verbal quantification, though examples of discontinuous verbal quantification are more hard to come by.

3.3 Scope And Scopal Quantifiers

When two quantified expressions are present in the same construction, a variety of semantic relations may obtain between the two expressions (Jackendoff 1971, Gil 1982b). The following English sentence, containing two numerically quantified NPs, may, at least potentially, be interpreted in any or all of the following four ways.

(5)        Three boys saw two girls

(a) ‘Each of three boys saw each of two girls’ not scope differentiated: strong symmetric.

(b) ‘Three boys saw two girls between them’ not scope differentiated: weak symmetric.

(c) ‘Three boys saw two girls each’ scope differentiated: wide scope for three boys distributive key: three boys; distributive share: two girls.

(d) ‘Two girls were seen by three boys each’ scope differentiated: wide scope for two girls distributive key: two girls; distributive share: three boys.

Interpretations (5a) and (5b) both involve a single set of three boys and a single set of two girls: each of the two NPs has independent reference. These interpretations are accordingly characterized as non-scope- differentiated, or symmetric. Interpretations (5c) and (5d) present a more complex picture. In (5c) there is a single set of three boys; however, each of the three boys is associated with a different set of two girls. Thus, while the subject NP has independent reference, the direct-object NP is referentially dependent on the subject NP. In this interpretation, then, the subject NP has scope over the direct-object NP. More precisely, the two NPs are in a relationship of distributivity, with the subject NP as distributive key, and the direct-object NP as distributive share. In (5d), a mirror-image situation obtains. Here, there is a single set of two girls; however, each of the two girls is associated with a different set of three boys. Accordingly, while the direct-object NP has independent reference, the subject NP is referentially dependent on the direct-object NP; the direct-object NP has scope over the subject NP. Here too then, a relation of distributivity obtains; this time, though, it is the direct object NP which is the distributive key and the subject NP which is the distributive share.

The above four interpretations are not equally readily available for speakers of English. Specifically, the symmetric interpretations (5a) and (5b) are more easily available than the asymmetric interpretations (5c) and (5d). Moreover, between the latter two interpretations, (5c) with wide scope for the subject NP, is more readily available than (5d) with wide scope for the direct-object NP. These preferences are not accidental; rather, they are particular consequences of more general principles governing the assignment of quantifier scope across languages.

Given the general preference for symmetric interpretations, languages typically have at their disposal various lexical and morphosyntactic devices whose function is to induce the more marked asymmetric interpretations, when these are required. Consider the following English sentences:

(6)        (a) All the men carried three suitcases.

(b) Every man carried three suitcases.

Sentences (6a) and (6b) differ in the lexical choice of quantifier within the subject NP. Whereas in (6a), with all, a variety of scope relationships may obtain between the two NPs, and the number of suitcases may vary between three in total and three per man, in (6b), with every, the subject NP has scope over the direct object NP, and the suitcases number three per man. Whereas all, like three, is a simple quantifier, every is a scopal quantifier, that is to say, a quantifier endowed with an additional denotational component, one that forces a particular scope relationship to obtain between the NP in which it occurs, and some other expression. In particular, it is a distributive-key quantifier, inducing a relationship of distributivity, in which the NP containing the quantifier in question is the distributive key, or element with wide scope (Gil 1995).

A different kind of scopal quantifier is exemplified by the following pair of Maricopa sentences:

(7)        M_ARICOPA

(a) ipac                      ii          xmokm

man-_PL: _NOM    stick 3-three-_SG–_DS

upaa k

3-carry-_PL–_REAL

‘The men carried three sticks’

(b) ipac                      ii          xmokxperm

man-_PL:_NOM    stick 3-three-_SG–_DIST._SHARE–_DS

upaa k

3-carry--

(i) ‘The men carried three sticks each’

(ii) ‘The men carried the sticks in threes’

Sentences (7a) and (7b) differ only in the form of the numeral within the direct-object expression. Sentence (7a) contains a simple numeral, xmok ‘three,’ and is ambiguous in roughly the same ways as its English gloss. In contrast, sentence (7b) contains the same numeral plus the suffix -xper, whose effect is to mark the numeral xmokand the expression containing it as distributive share. The ambiguity exhibited by sentence (7b) results from different expressions being selected as the distributive key counterpart. Under interpretation (7b(i)), the subject NP ipac ‘men’ is chosen as distributive key: the sticks accordingly number three per man. Under interpretation (7b(ii)), the verb upaavk ‘carried’ is chosen as distributive key: here the sticks accordingly number three per carrying. Cross-linguistically, distributive-share numerals are formed by a variety of formal processes, including affixation (as in the above example), periphrasis, reduplication, and suppletion (Gil 1982a).

3.4 Quantifier As A Grammatical Category

Quantifier is primarily a semantic category: an expression is a quantifier only if it bears certain meaning properties. One of the most striking characteristics of quantifiers is the absence of any distinctive shared formal morphosyntactic features, which might justify the positing of a corresponding grammatical category of quantifier. Within languages, different quantifiers often display different grammatical properties. And across languages, the ‘same’ quantifier may exhibit different morphosyntactic behavior as well. Thus, in terms of their formal properties, quantifiers constitute a heterogeneous collection, exhibiting a great amount of morphological and syntactic diversity.

Looking within individual languages, there is probably no language within which there is a formal grammatical category consisting exactly of all quantifiers but no other expressions. Rather, in most or all languages, different quantifiers exhibit different arrays of grammatical properties, some grouping together with expressions belonging to one category, others patterning together with expressions belonging to some other category, yet others exhibiting idiosyncratic morphological or syntactic behaviour.

Looking across languages, quantifiers with the same meaning often exhibit quite different morphological and syntactic behaviour. Examples (1)–(4) above showed how the numeral ‘three,’ as a nominal quantifier, may occur, in different languages, in a variety of different constructions, exhibiting continuous and discontinuous quantification of various kinds. Even if attention is restricted to the most well-known type, that of continuous NP-internal quantification, as in (l), a great amount of cross-linguistic variation may be observed. In particular, in different languages, nominal quantifiers in attributive NP internal position may resemble, to various degrees, nouns, adjectives, and/or verbs.

4. Delimitation Of Quantification

The discussion in the preceding sections was based on a bipartite definition of quantifiers as ‘free-standing expressions’ whose meanings ‘involve the notion of quantity.’ Like many similar definitions, this one is useful, but not unproblematical. Specifically, in addition to prototypical quantifiers such as those discussed above, there are numerous expressions and construction types for which it is difficult to adjudicate whether or not they may appropriately be characterized as quantificational.

The first part of the definition is formal: in order for an expression to be considered as a quantifier, it must be free standing—a word or a phrase, rather than an affix or some other bound morphological unit. Accordingly, various kinds of clitics, particles, and other intermediate units smaller than words but larger than affixes may provide borderline cases straddling the boundary between what is a quantifier and what is not. The distinction between free and bound forms is what underlies the distinction between quantification and the semantically-related grammatical category of number. Specifically, whereas quantifiers are free standing, number markers are bound forms. However, in some instances it may be difficult to adjudicate whether a form associated with a quantificational meaning is free or bound, and hence whether it constitutes an instance of quantification or of number. In order to determine whether forms such as these are quantifiers or number markers, it may be necessary to make use of additional diagnostic criteria distinguishing quantifiers from number markers.

The second part of the definition of quantifier is the obvious semantic condition: its meaning must involve the notion of quantity. While in many instances this is quite straightforward, in other cases, quantificational meanings may shade off gradually into other kinds of meanings not generally considered to be quantificational. One class of expressions related in meaning to nominal quantifiers are those denoting size and other scalar properties. An important class of expressions related to verbal quantifiers are those marking aspectual categories, such as perfect and progressive. More generally, different quantifiers in different languages may be related to an extremely broad range of non-quantificational concepts, synchronically and/or diachronically.

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