THE LOGIC NOTES

Arity Glossary


Definition

The arity or adicity of a predicate or function is the number of arguments (i.e. inputs or parameters) it takes.

Comments

Each function is taken to have a fixed arity. This can lead to a little artificiality, as for instance 'the sum of' applied to three numbers has to be a different function from 'the sum of' applied to four numbers, or else it has to be construed as a function of the set of numbers.

The arity is always finite, but it may be zero. A zero-ary function symbol is a name. A zero-ary predicate is just a sentence.

It doers not matter whether we refer to symbols (or the functions they denote) as unary, binary, ternary, etc. or as monadic, dyadic, triadic, etc. The choice between those two vocabularies is a matter of taste.

Examples

  1. 1-ary functions: 'the father of' in the domain of people; 'the successor of' in the domain of integers.
  2. 2-ary functions: 'plus', 'minus', 'times' in the domain of numbers.
  3. 3-ary predicate: 'is between' in the domain of places.
  4. Sentence containing a 5-ary predicate: 'X is the intersection of lines AB and CD' in plane geometry.

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