Signature (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. They are rarely made explicit in more philosophical treatments of logic. Formally, a (single-sorted) signature can be defined as a 4-tuple where and are disjoint sets not containing any other basic logical symbols, called respectively function symbols (examples: ), s or predicates (examples: ), constant symbols (examples: ), and a function which assigns a natural number called arity to every function or relation symbol. A function or relation symbol is called -ary if its arity is Some authors define a nullary (-ary) function symbol as constant symbol, otherwise constant symbols are defined separately. A signature with no function symbols is called a , and a signature with no relation symbols is called an . A is a signature such that and are finite. More generally, the cardinality of a signature is defined as The 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 or is often used as a synonym for "signature". In model theory, a signature is often called a , or identified with the (first-order) language to which it provides the non-logical symbols. However, the cardinality of the language will always be infinite; if is finite then will be . As the formal definition is inconvenient for everyday use, the definition of a specific signature is often abbreviated in an informal way, as in: "The standard signature for abelian groups is where is a unary operator." Sometimes an algebraic signature is regarded as just a list of arities, as in: "The similarity type for abelian groups is " Formally this would define the function symbols of the signature as something like (which is binary), (which is unary) and (which is nullary), but in reality the usual names are used even in connection with this convention.