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.
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),}
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 \sigma =\left(S_{\operatorname {func} },S_{\operatorname {rel} },S_{\operatorname {const} },\operatorname {ar} \right)}
{\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
is often called a vocabulary, or identified with the (first-order) language
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: Formally this would define the function symbols of the signature as something like
(which is nullary), but in reality the usual names are used even in connection with this convention.
In mathematical logic, very often symbols are not allowed to be nullary,[citation needed] so that constant symbols must be treated separately rather than as nullary function symbols.
However, this only serves to complicate matters, especially in proofs by induction over the structure of a formula, where an additional case must be considered.
Any nullary relation symbol, which is also not allowed under such a definition, can be emulated by a unary relation symbol together with a sentence expressing that its value is the same for all elements.
This translation fails only for empty structures (which are often excluded by convention).
If nullary symbols are allowed, then every formula of propositional logic is also a formula of first-order logic.
to formalize expressions and equations about a vector space over an infinite scalar field
denotes the unary operation of scalar multiplication by
This way, the signature and the logic can be kept single-sorted, with vectors being the only sort.
[2] In the context of first-order logic, the symbols in a signature are also known as the non-logical symbols, because together with the logical symbols they form the underlying alphabet over which two formal languages are inductively defined: The set of terms over the signature and the set of (well-formed) formulas over the signature.
In a structure, an interpretation ties the function and relation symbols to mathematical objects that justify their names: The interpretation of an
For many-sorted logic and for many-sorted structures, signatures must encode information about the sorts.
The most straightforward way of doing this is via symbol types that play the role of generalized arities.