How can one determine what an adjectival complement is in an English sentence? Are there are any subcategories to this classification? I'd love concrete examples, to help me better understand this aspect of grammar.
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I must say I always find such terms vague: is it an adjective that functions as a complement, i.e. a complement of an adjectival nature (I am large)? Or a complement to an adjective (I am large of mind)? Some interpret it as the latter; however, most linguistic Google results seem to interpret it as an adjective that is used as a complement, usually a complement to a verb.
A complement, sometimes also known as a value, is a word or phrase that you would expect with a certain verb/predicate. Consider this sentence:
The verb to pick has two complements: a subject I and an object apples. The constituent this morning is not considered a complement, because it is not something you would expect with this verb; that is, you would surely not be surprised if it weren't there. That is called a satellite.
Notice that I is a personal pronoun and apples a noun: none of the complements to the verb in this sentence are of an adjectival (adjective-like) nature. There are no adjectival complements.
Here we have two complements: a subject and what is called a subject complement. Note that this is a different use of the word "complement": the first one is about a verb's having complements, the second is about the subject's complement. We could call the latter a "complement to the subject". The adjective rich is both: it fills in information that you would expect with the verb (she became is not a complete sentence); and it adds information to the subject (now we know the subject is qualified as rich in some way). Adjectival complements usually accompany verbs that can have complements to their subjects (these verbs are called copulae), or verbs that can have complements to their objects.
Whenever an adjective functions as a complement—usually as a complement to a verb, I'd say—, it is mostly called an adjectival complement as mentioned above. Perhaps a clearer name would be adjective-complement, because it is both adjective and complement. The other category, something that functions as a complement to an adjective, could be called a complement to an adjective. I prefer longer names to confusion; but I believe most linguists still use "adjectival complement".
Some adjectives allow, or even require, a further phrase as part of their meaning.
So for example "ready" can be used absolutely, as in "I am ready", but can also take a PP (prepositional phrase), as in "I am ready to go" or "I am ready for anything".
Note that "for anything" is not at the sentential level as would be for example the "with" phrase in "With you, I am ready for anything": it is semantically and syntactically dependent on "ready" - hence it is referred to as a complement to "ready".