Linguistic structure can be represented using symbols, variables, and operators (e.g., S → NP VP). These symbolic systems allow linguists to state generalizations about word order, feature agreement, and other patterns concisely.
From your introduction to formal linguistics, you've encountered the idea that language has structure that can be described systematically. Symbolic representation is the toolkit that makes this description precise. Instead of writing out every possible sentence in full, linguists use symbols — standing in for categories — and rules that specify how those categories combine. This is the same move that algebra makes when it replaces "a specific number" with "x": abstraction over instances to reveal the underlying pattern.
The most familiar example is the phrase structure rule: S → NP VP. Read this as "a sentence consists of a noun phrase followed by a verb phrase." The arrow means "can be expanded as" or "consists of." The symbols on the right — NP, VP — are themselves categories that can be expanded by further rules: NP → Det N, VP → V NP, and so on. Together, a set of such rules constitutes a grammar in the formal sense — not a list of dos and don'ts, but a generative system that specifies which combinations of symbols are well-formed structures of the language. The tree diagrams you may have seen in syntax are simply the visible record of which rules were applied in which order to build a given sentence.
The power of symbolic representation is that it captures generalizations. The rule NP → Det N applies to "the dog," "a tree," "every student" — an infinite class of phrases. Without symbolic abstraction, you would need to list each phrase individually, which is both impossible and misses the point. Symbols let linguists express what all these phrases have in common: they are all headed by a noun and optionally preceded by a determiner. Similarly, feature notation (writing [+plural], [−animate], or [3sg] on category labels) allows agreement and selection restrictions to be stated symbolically rather than as lists of exceptions.
Symbolic representation becomes the foundation for everything more complex that follows — typed feature structures, formal grammars (context-free, context-sensitive), and the derivational accounts in minimalism and HPSG all depend on this basic notation. When you encounter a rule like S → NP VP or a feature matrix like [CASE: nom, NUM: pl], you are reading the symbolic vocabulary that lets linguists make predictions, state crosslinguistic generalizations, and build computational models. Learning to read and write in this notation is not just a technical drill — it is learning the formal language that linguistics uses to communicate structural claims with precision.