# arity

## Definitions

• n. The number of arguments or operands a function or operation takes. For a relation, the number of domains in the corresponding Cartesian product.

• n. the number of arguments that a function can take

## Etymologies

From -ary, based on a Latin root. Compare adicity and adinity, based on the corresponding Greek root. (Wiktionary)

## Examples

• r is called the arity of the function f, i.e., the number of arguments that it takes.

Computability and Complexity

• Then due to its popul­arity, the dates were extended twice until the end of August.

20th Century German Art Exhibition, London 1938

• A structure contains interpretations of certain predicate, function and constant symbols; each predicate or function symbol has a fixed arity.

Model Theory

• A signature is a set of individual constants, predicate symbols and function symbols; each of the predicate symbols and function symbols has an arity

First-order Model Theory

• In 1870 Peirce published a long paper “Description of a Notation for the Logic of Relatives” in which he introduced for the first time in history, two years before Frege's Begriffschrift a complete syntax for the logic of relations of arbitrary adicity (or: arity).

Nobody Knows Nothing

• Again: rheme (by which Peirce meant a relation of arbitrary adicity or arity) was a first, proposition was a second, and argument was a third.

Nobody Knows Nothing

• These axioms tacitly specify the arity of a combinator as well as their reduction (or contraction) pattern.

Combinatory Logic

• P and Q (or for predicates of higher arity when the variable in their last argument is bound).

Combinatory Logic

• Predicates have a fixed finite arity in FOL, and nothing precludes binding at once a variable in the first argument of one predicate and in the second argument of another predicate.

Combinatory Logic

• Primitive recursion: if f and g are primitive recursive functions of arity k and k+2, respectively, then there is a primitive recursive function, h, of arity k+1 satisfying the following conditions:

Computability and Complexity

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