Signature (mathematical logic)

In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes.

Signatures play the same role in mathematics as type signatures in computer programming. They are rarely made explicit in more philosophical treatments of logic.

Formally, a (single-sorted) signature can be defined as a triple σ = (S_func, S_rel, ar), where S_func and S_rel are disjoint sets not containing any other basic logical symbols, called respectively

and a function ar: S_func ${\displaystyle \cup }$ S_rel → ${\displaystyle \mathbb {N} }$ which assigns a non-negative integer called arity to every function or relation symbol. A function or relation symbol is called n-ary if its arity is n. A nullary (0-ary) function symbol is called a constant symbol.

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