Syntax takes the units morphology defines and studies their grammar of combination, and its first difficulty is that the obvious unit, the sentence, is surprisingly hard to define. Psychological definitions treat it as a thought expressed in words; Logical ones as a judgement; Semantic ones as a complete unit of meaning, and grammatical ones as a subject with a predicate or a finite verb with its required elements. The Duden formulations combine such criteria, and each fails in some case, as with one-word utterances like Hilfe! Two criteria survive across the attempts: that a sentence expresses a complete predication and that it is well formed, and together they separate grammaticality from meaningfulness. Chomsky’s example “colourless green ideas sleep furiously” is grammatical and nonsensical at once, which shows that form and meaning are independent levels.
Between word and sentence lies the phrase, a unit that is syntactically saturated and behaves as one piece, as shown by tests such as substitution and questioning. Phrases give sentences a hierarchical rather than a linear structure, which can be shown as bracketing or as a tree. The same intuition is made mathematically explicit in formal languages, where an alphabet, a set of words, and a set of rewriting rules define what counts as a valid string. Context-free grammars generate the phrase structure of a fragment of language. The Chomsky hierarchy orders grammars by expressive power and automata class.