Chapters 2 and 3 of Syntactic Structures answer “how should we define the set of grammatical sentences?” with two negatives first: not meaning, and not probability. Before getting there, Chomsky redefines “grammar” itself with some precision.
The fundamental aim in the linguistic analysis of a language L is to separate the grammatical sequences which are the sentences of L from the ungrammatical sequences which are not sentences of L and to study the structure of the grammatical sequences.
Not “sequences that make sense,” but “grammatical sequences” — this reframing is the starting point for everything that follows.
Neither Meaning Nor Frequency
Start with the familiar pair:
“Colorless green ideas sleep furiously.” “Furiously sleep ideas green colorless.”
Both are nonsense, but English speakers read only the first as a natural sentence. That gap can’t be explained by meaning, since neither means anything.
Nor can it be explained statistically. Both strings had almost certainly never been uttered before. Measured by “how often has this occurred,” the two are equally remote from ordinary English. Yet the feel is entirely different — the first gets read with ordinary sentence intonation and is easy to remember; the second gets read word by word, each one its own isolated phrase.
A Machine That Chains Words Together: The Finite-State Model
So how should grammatical sentences be generated? Chomsky first takes up a finite-state (Markov process) model borrowed from contemporary communication theory: a machine that moves among a finite number of “states,” producing one word at each transition, starting from an initial state and ending at a final one — the resulting sequence of words is a “sentence.”
Add loops and this machine can generate “the old man comes,” “the old old man comes,” and so on without limit. It looks sufficient.
Why the Machine Can’t Generate English
Chomsky proves this is not merely a hard case to describe — it is impossible in principle.
English is not a finite state language.
The reason is a “mirror-image” style of dependency in English. Consider constructions like:
- “if S, then S′”
- “either S, or S′”
- “the man who said that S is arriving today”
“If” demands “then”; “either” demands “or” — a dependency spanning two distant points in the sentence. And crucially, the very same construction can be embedded in the gap between them. Let’s build a concrete example.
Start with something simple: “the cat is on the mat” (S₁). Drop it into the “either” slot: “either the cat is on the mat, or the meeting is cancelled” (S₂). Now drop S₂ into the “the man who said that…” slot: “the man who said that either the cat is on the mat or the meeting is cancelled is arriving today” (S₃). Drop S₃ into the “if… then” slot: “if the man who said that either the cat is on the mat or the meeting is cancelled is arriving today, then we’ll wait at the station.”
This nesting can, in principle, go as deep as you like. Mathematically, it has the same structure as the toy language “ab, aabb, aaabbb, …” — strings where the number of a’s must equal the number of b’s. That toy language provably cannot be generated by any finite-state grammar, because no matter how many states the machine has, it cannot keep count of “how many a’s have I seen” without limit. English constructions of this kind fail for exactly the same reason.
Minimal as a “Machine” — And Still Disqualified
A finite-state grammar, Chomsky notes, is the simplest kind of grammar that can generate infinitely many sentences from finite means. That is precisely why disqualifying it matters: it proves that describing language requires something stronger. The next move is a model that treats phrases, not words, as the basic unit — phrase structure.