There are two major kinds of non-verbal predicates in English:
- predicate adjectives ("adjectival predicates", if you prefer that construction)
- predicate nouns ("nominal predicates").
Both types require an auxiliary be, though this often gets wiped by Whiz Deletion or some other clause-reduction rule. In addition, count nouns serving as predicates require an article.
- He is tired.
- He is a doctor.
- This stuff is just mud.
- He got tired = He came to be tired (inchoative of be)
- They made him a teacher = They caused him to come to be a teacher (causative of be)
There are many kinds of constructions, like the last two, where non-verbal predicates can occur without an auxiliary be, but they always refer back to a full form which contains be.
Predictable auxiliaries like be are used just to hold the tense, so in an untensed infinitive or gerund, or in a reduced relative clause, where tense is either irrelevant or already specified by the matrix verb, the be usually gets deleted.
Especially for adjective predicates, this is the norm, and it's reasonable to assume that any attributive adjective is reduced from a relative clause with a predicate adjective by Whiz-Deletion and adjective-preposing.
- the old man ~ the man
Wh- is old