Definition: Function, Arity and Constant

let $L$ be a logical calculus, $U$ the domain of discourse, and $I(U,L)$ the corresponding interpretation with the valuation function $[[]]_I$.

A function is a non-empty string over an alphabet $s\in L$ interpreted as a function $U^n\to U$, taking $n$ input arguments from the domain of discourse and mapping them to a new value from the domain of discourse.

The natural number $n\ge 0$ of arguments of the function is called its arity. Special cases of arities are:

$n=0$: Nullary functions (or operations) are usually called constants
$n=1$: Unary functions (or operations)
$n=2$: Binary functions (or operations)
$n=3$: Ternary functions (or operations)
generally $n$-nary functions (or operations)

Examples: 1

Definitions: 1 2 3
Examples: 4

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Bibliography

Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011

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Definition: Function, Arity and Constant

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