The syntax-semantics interface maps syntactic structures to semantic representations. The principle of compositionality states that sentence meaning derives from word meanings and how they combine. Formal approaches specify translation rules from syntactic trees to logical forms, handling scope ambiguities, ellipsis, and context-dependent phenomena that syntax alone cannot resolve.
From your study of the syntax-semantics interface, you know the basic puzzle: syntactic structure and semantic interpretation are related but not identical. The same string of words can have multiple syntactic structures (structural ambiguity), and the same structure can sometimes support multiple semantic interpretations (scope ambiguity). Formal models of the interface try to make this mapping explicit — to write precise rules that take a syntactic tree as input and produce a logical form as output.
The foundational principle is compositionality (often called the Fregean principle): the meaning of a complex expression is a function of the meanings of its immediate parts and the way they are syntactically combined. Concretely, this means that if you know what every word means and you know the syntactic structure, you should be able to compute the sentence's meaning by applying compositional rules bottom-up through the tree. A determiner like *every* denotes a function from sets to truth conditions; a noun like *student* denotes a set; combining them compositionally gives a quantifier phrase with a specific logical interpretation. The power of compositionality is that it explains how humans understand infinitely many novel sentences: new meanings are computed, not memorized.
Scope ambiguity is one of the central test cases for formal interface models. The sentence "Every student read a book" has two readings: one where there is a single book that all students read (narrow scope for *a book*), and one where each student potentially read a different book (wide scope for *a book*). The syntactic surface structure is the same for both readings. Formal approaches resolve this by positing a level of Logical Form (LF) — a syntactic level where quantifier phrases have been covertly moved (via "Quantifier Raising") to a position that reflects their scope. The two readings of the sentence correspond to two distinct LF representations, even though the surface strings are identical.
Other phenomena that require formal interface mechanisms include ellipsis (where syntactic material is phonologically absent but semantically present: "John left and Mary did too"), presupposition (background assumptions triggered by certain words: *the* presupposes uniqueness), and binding (anaphors like *himself* must be bound by an antecedent in a specific structural domain). Each of these requires the semantic interpretation to "see" more syntactic structure than is apparent on the surface — which is precisely what the formal interface provides. The broader lesson is that natural language meaning is not simply a matter of stringing word definitions together; it requires a structured, rule-governed mapping between the architecture of syntax and the architecture of logic.