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. 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 = (Sfunc, Srel, ar), where Sfunc and Srel are disjoint sets not containing any other basic logical symbols, called respectively and a function ar: Sfunc  ∪ {displaystyle cup }  Srel → N {displaystyle mathbb {N} } which assigns a natural number 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. 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 Sfunc and Srel are finite. More generally, the cardinality of a signature σ = (Sfunc, Srel, ar) is defined as |σ| = |Sfunc| + |Srel|. 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 σ is often called a vocabulary, or identified with the (first-order) language L to which it provides the non-logical symbols. However, the cardinality of the language L will always be infinite; if σ is finite then |L| will be ℵ0.