In mathematical logic, a signature is a description of 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. They are rarely made explicit in more philosophical treatments of logic.

Formally, a (single-sorted) signature can be defined as a 4-tuple ${\displaystyle \sigma =\left(S_{\operatorname {func} },S_{\operatorname {rel} },S_{\operatorname {const} },\operatorname {ar} \right),}$ where ${\displaystyle S_{\operatorname {func} }}$ and ${\displaystyle S_{\operatorname {rel} }}$ are disjoint sets not containing any other basic logical symbols, called respectively

and a function ${\displaystyle \operatorname {ar} :S_{\operatorname {func} }\cup S_{\operatorname {rel} }\to \mathbb {N} }$ which assigns a natural number called arity to every function or relation symbol. A function or relation symbol is called ${\displaystyle n}$-ary if its arity is ${\displaystyle n.}$ Some authors define a nullary (${\displaystyle 0}$-ary) function symbol as constant symbol, otherwise constant symbols are defined separately.

A signature with no function symbols is called a relational signature, and a signature with no relation symbols is called an algebraic signature. A finite signature is a signature such that ${\displaystyle S_{\operatorname {func} }}$ and ${\displaystyle S_{\operatorname {rel} }}$ are finite. More generally, the cardinality of a signature ${\displaystyle \sigma =\left(S_{\operatorname {func} },S_{\operatorname {rel} },S_{\operatorname {const} },\operatorname {ar} \right)}$ is defined as ${\displaystyle |\sigma |=\left|S_{\operatorname {func} }\right|+\left|S_{\operatorname {rel} }\right|+\left|S_{\operatorname {const} }\right|.}$

The language of a signature is the set of all well formed sentences built from the symbols in that signature together with the symbols in the logical system.

In universal algebra the word type or similarity type is often used as a synonym for "signature". In model theory, a signature ${\displaystyle \sigma }$ is often called a vocabulary, or identified with the (first-order) language ${\displaystyle L}$ to which it provides the non-logical symbols. However, the cardinality of the language ${\displaystyle L}$ will always be infinite; if ${\displaystyle \sigma }$ is finite then ${\displaystyle |L|}$ will be ${\displaystyle \aleph _{0}}$.

As the formal definition is inconvenient for everyday use, the definition of a specific signature is often abbreviated in an informal way, as in:

Sometimes an algebraic signature is regarded as just a list of arities, as in: