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Count noun

A count noun is a noun which is itself counted, or the units which are used to count it. This differs from a mass noun in that a count noun is countable, whereas a mass noun refers to an uncounted group of objects (e.g., a corporation). This is where the situation gets tricky. To take an example from Mandarin Chinese, which marks count(ed) nouns with a noun classifier:
Na4 ren2 chi1 wan2 le, can equally well mean "That person has eaten" or "Those people have eaten" - you're not counting them, so you don't need a classifier, and Mandarin doesn't distinguish singular vs. plural
Na4 wei4 ren2 chi1 wan2 le, means "That (one) person has eaten"
Na4 san1 wei4 ren2 chi1 wan2 le, means "Those three people have eaten"

A classifier, therefore, implies that the object(s) referred to are countable in the sense that the speaker intends them to be enumerated, rather than considered as a unit (regardless of quantity).

An example that is more familiar comes from British vs. American English, which treat mass nouns differently when the noun refers to a group of people:

American English: "I am bankrupt", "We are bankrupt", "Enron Corp. is bankrupt", "The members of the board are bankrupt"
British English: "I am bankrupt", "We are bankrupt", "Enron Corp. are (or is) bankrupt", "The members of the board are bankrupt"

Note that of the above examples, only Enron Corp. is a mass noun; all others are count nouns. In American English, mass nouns referring to groups of people take a singular verb form; in British English, they usually take a plural verb form in speech, but singular in more formal writing.

On the other hand, words such as "milk" or "rice" are not count nouns, but they can be counted with an appropriate unit of measure (e.g. "glasses of milk" or "spoonfuls of rice"). This leads to another example from Mandarin to illustrate some further points about count nouns:

In both cases, the word "book" is a count noun, and in Mandarin take the classifier "ben3"

This use of a classifier is similar to, but not identical with, the use of units of measure to count groups of objects in English. For example, in "three shelves of books", "shelves" is used as a unit of measurement, and books is indeed a mass noun, since the speaker is not counting individual books - she is counting shelves of books. By contrast, in the sentence "At 10 books per shelf, you have 30 books," both instances of "books" are an example of a count noun, and require a "measure word" in Chinese (or, as linguists sometimes call it, a "noun classifier", of which measure words are one type).

Different languages may treat "measured nouns" differently from "count nouns"; some, like Mandarin, will require a classifier before the unit of measure, while others may not require them at all. San1 bei bei1-zi mian4 "Three [classifier] cups (of) noodles" versus San1 kuai4 mian4 "Three [classifier] noodles''. (Notice that the classifier changes as what is counted (cups vs. noodles) changes!)

In Mandarin and languages with similar rules, English speakers can "fake it" by automatically putting a classifier between a number and a noun or unit of measure. While it's not completely fail-safe ("a/an" vs. "the" vs. "one" vs. {no word / skipped word}, as well as exceptions, can trip people up), it's a good rule of thumb. Read the article on mass nouns for examples of how count and mass nouns may be distinguished in English grammar.

Additionally, certain words of quantity vary depending on whether the quantity is that of a count or mass noun. For example:

The choice of words signals whether the speaker is indicating an unspecified quantity of a mass noun or a count noun. Also, mass nouns cannot take plurals.

Note that the word "some" can be applied to both. However, there is a difference in implicational meaning between I like some apples (implied: but not all kinds) and I like some water (implied: but not too much). The additional implied meaning depends on whether some attaches to a count or to a mass noun. There is, in effect, a mass~count paradigm operating where the following pairs contrast: much~many; a little~some; some~a few, as well as others.

Additional examples:
CountMass
Many watersMuch water
A few piesSome pie
Some bananasSome banana