How does syntax differ from semantics in a formal language?

In the context of metalogic, syntax and semantics represent two distinct ways of analyzing a formal language.

Syntax is concerned exclusively with the mechanical structure of a language, independent of any meaning. It involves:
* Formation rules: A specification of primitive symbols and the inductive rules used to construct well-formed formulas.
* Proof theory: The manipulation of symbols through axioms and rules of inference to derive theorems.

Semantics, by contrast, deals with meaning, truth, and denotation. It involves:
* Interpretation: Assigning meaning to symbols by relating them to a domain of objects, which determines the truth-value of sentences.
* Model theory: The study of these interpretations and the structures that satisfy the formal system.

While syntax focuses on the "grammar" and deductive structure of the system, semantics provides the "content" or interpretation that allows us to evaluate whether statements are true or false.12