In grammar and linguistics, the term complement is used with different meanings, so it is difficult to give a single precise definition and explanation. In a broad general sense however, a complement can be understood as a word, phrase or clause that is necessary to complete the meaning of a given expression. The terms complement and argument overlap in meaning and use. A given complement is therefore often also an argument. Complements are not adjuncts, however.
In many traditional grammars, the terms subject complement and object complement are employed to denote the predicative expressions (e.g. predicative adjectives and nominals) that serve to assign a property to a subject or object, e.g.
Ryan is upset. - Predicative adjective as subject complement
Rachelle is the boss. - Predicative nominal as subject complement
That made Michael lazy. - Predicative adjective as object complement
We call Rachelle the boss. - Predicative nominal as object complement
Although widespread in school grammar, this use of terminology is not employed by many modern theories of syntax. The expressions in bold are viewed as part of the clause predicate, which means they are not complements of the subject or object, but rather they are properties that are predicated of the subject or object.
Complements as arguments
In many modern grammars (for instance in those that build on the X-bar schema), the object argument of a verbal predicate is called a complement. In fact, this use of the term is the one that currently dominates in linguistics. A main aspect of this understanding of complements is that the subject is usually NOT a complement of the predicate, e.g.
He wiped the counter. - the counter is the object complement of the verb wiped.
She scoured the tub. - the tub is the object complement of the verb scoured.
The noun phrases (NPs) the counter and the tub are necessary to complete the meaning of the verbs wiped and scoured, respectively, hence they are complements.
While it is less common to do so, one sometimes extends this reasoning to subject arguments:
He wiped the counter. - He is the subject complement of the verb wiped.
She scoured the tub. - She is the subject complement of the verb scoured.
In these examples, the subject and object arguments are taken to be complements. In this area then, the terms complement and argument overlap in meaning and use. Note that this practice takes a subject complement to be something much different from the subject complements of traditional grammar, which are predicative expressions, as just mentioned above.
Complements broadly construed
Construed in the broadest sense, any time a given expression is somehow necessary in order to render another expression "complete", it can be characterized as a complement of that expression, e.g.
with the class - The NP the class is the complement of the preposition with.
Jim will help. - The main verb help is the complement of the auxiliary verb will.
Chris gave up. - The particle up is the complement of the verb gave.
as a friend - The NP a friend is the complement of the particle of comparison as.
Construed in this broad sense, many complements cannot be understood as arguments. The argument concept is tied to the predicate concept in a way that the complement concept is not.
Complements vs. adjuncts
Adjuncts appear optionally in the clauses and phrases that contain them. They are NOT necessary to complete the meaning; they are therefore NOT complements, e.g.
a. right after the game
b. after the game - The adjunct right is not a complement of the preposition after.
a. fix the bike tomorrow
b. fix the bike - The adjunct tomorrow is not a complement of the verb fix. .
a. the old house
b. the house - The adjunct old is not a complement of the noun house.
a. Stop immediately!
b. Stop! - The adjunct immediately is not a complement of the verb stop.
Since the adjuncts in these examples are not necessary to make the expressions "complete", they cannot be viewed as complements.
^For examples of grammars that employ the terms subject complement and object complement to denote predicative expressions, see Matthews (1981:3ff.), Downing and Locke (1992:64f.), Thomas (1993:46, 49), Brinton (2000:183f.).
^For examples of this "narrow" understanding of complements, see for instance Lester (1971:83), Horrocks (1987:63), Borsley (1991:60ff.), Cowper (1992:67), Burton-Roberts (1997:41), Fromkin et al. (2000:119).
^For examples of theories that take the subject to be a complement of the matrix verb/predicate, see for instance Matthews (1981:101), Pollard and Sag (1994:23), Miller (2011:56).
^See Radford (2004:329) for an explanation of complements along these lines.
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