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A signature specifies the vocabulary of a formal language by listing constant symbols, function symbols with specified arities, and relation symbols with specified arities. A structure over a signature assigns concrete meaning by providing a non-empty domain and interpretations of all symbols in the language. This separation of abstract syntax from concrete semantics is foundational to model-theoretic analysis.
Before studying model theory, you encountered first-order logic as a formal language with syntax — formulas built from variables, connectives, and quantifiers. But syntax alone does not tell you whether a sentence is true. To evaluate truth, you need a concrete mathematical setting: a domain of objects and specific interpretations of the symbols you use. Model theory makes this step precise by defining two fundamental concepts: the *signature* and the *structure*.
A *signature* (sometimes called a vocabulary or language type) specifies the building blocks of your formal language: which constant symbols, function symbols, and relation symbols are available, together with the *arity* of each function and relation symbol. Arity tells you how many arguments a symbol takes — a binary function symbol takes 2 arguments, a unary relation symbol takes 1. The signature is purely *syntactic*: it is a list of symbols and their types, saying nothing whatsoever about what those symbols mean or which domain they apply to.
A *structure* gives the signature meaning. A structure M over a signature σ consists of two things: a non-empty set |M| called the *domain* or *universe*, and an *interpretation* that assigns to each constant symbol a specific element of |M|, to each n-ary function symbol a specific function |M|ⁿ → |M|, and to each n-ary relation symbol a specific subset of |M|ⁿ. Once you have a structure, every closed formula (sentence) in the language of σ has a determinate truth value in M.
The same signature can be interpreted by many different structures, and this multiplicity is the point. The signature of ordered fields — {0, 1, +, ·, <} — is satisfied by the rationals, the reals, and no-other familiar structure (there are non-standard models too). Group theory uses a smaller signature {·, e}, satisfied by integers under addition, nonzero rationals under multiplication, symmetric groups, and countless others. Model theory studies what all these structures have in common (the consequences of the shared axioms), how they differ (distinguishing properties), and what maps between them preserve (homomorphisms, embeddings, isomorphisms).
This syntax-semantics separation — signature as vocabulary, structure as meaning — is what allows a single formal language to serve as a universal tool for mathematics. When you move to model-interpretation-and-satisfaction, you will make precise what it means for a formula to be *true in* a structure using the satisfaction relation ⊨. Everything in that account rests on the definitions introduced here.