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There are indicative as well as non-indicative complements in languages. Non-indicative complements follow the appropriate complementizers. Indicative complements do not follow complementizers but instead are included with special markers and clauses.
However, this use of terminology is avoided by many modern theories of syntax, which typically view the expressions in bold as part of the clause predicate, which means they are not complements of the subject or object but rather are properties that are predicated of the subject or object.
The Cambridge Grammar of the English Language assigns the term "predicative complement" to both uses and shifts the terminological distinction to the verb:
Ed seemed quite competent: -- complex-intransitive verb + predicative complement
She considered Ed quite competent : -- complex-transitive verb + predicative complement 
In many modern grammars (for instance in those that build on the X-bar framework), 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:
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.
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 those examples, the subject and object arguments are taken to be complements. In this area, the terms complement and argument thus overlap in meaning and use. Note that this practice takes a subject complement to be something very different from the subject complements of traditional grammar, which are predicative expressions, as just mentioned above.
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:
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 preposition, as.
Construed in the 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.
In linguistics, an adjunct is an optional, or structurally-dispensable, part of a sentence, clause, or phrase that, when it is removed, will not affect the remainder of the sentence except to discard from it some auxiliary information. A more detailed definition of the adjunct emphasizes its attribute as a modifying form, word, or phrase that depends on another form, word, or phrase, being an element of clause structure with adverbial function. An adjunct is not an argument or a predicative expression, and an argument is not an adjunct. The argument-adjunct distinction is central in most theories of syntax and semantics. The terminology used to denote arguments and adjuncts can vary depending on the theory at hand. Some dependency grammars, for instance, employ the term circonstant (instead of adjunct) and follow Tesnière (1959).
^See Matthews (1981:142f.) and Huddleston (1988:note 2) for good overviews of the different uses of the term complement.
^Hooper, Robin (1993). Studies in Tokelauan Syntax. University of Auckland. pp. 328-332.
^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.
Borsley, R. 1991. Syntactic theory: A unified approach. Cambridge, MA: Blackwell Publishers.
Brinton, L. 2000. The structure of modern English. Amsterdam: John Benjamins Publishing Company.
Burton-Roberts, N. 1997. Analysing sentences: An introduction to English grammar. London: Longman.