Definedness

Definedness is a concept used across mathematics, logic, and computer science to describe whether an expression, term, or value has a meaningful and unambiguous interpretation within a given formal system. An object is defined if its value or truth can be determined under the specified rules; it is undefined if no such value can be assigned.

In mathematics, definedness often concerns partial functions and expressions with domain restrictions. A function f may be partial, meaning it is only defined for inputs in a certain subset of its apparent domain. The statement “f(x) is defined” is true precisely when x lies in that subset. For example, 1/x is undefined when x equals zero, and the real logarithm log(x) is undefined for x ≤ 0. Some expressions are indeterminate or require extension to other domains, such as extending sqrt to complex numbers or interpreting 0^0 in a chosen convention.

In logic and formal languages, definedness distinguishes statements that have a determinate truth value from those that do not. Some logics introduce a third value, undefined, to handle partiality, while others use explicit definedness predicates or domain conditions to ensure formulas are meaningful before evaluation.

In computer science, definedness is central to semantics and error handling. Programs may produce undefined results if they access uninitialized variables or divide by zero. Languages and type systems address this with option or Maybe types, total versus partial functions, and strict or lazy evaluation strategies, all aiming to make undefinedness explicit and manageable.

A statement or object is said to be well-defined when its value or truth is unambiguous and independent of representation or method. Definedness thus underpins rigorous reasoning, computability, and the safe application of mathematical and logical rules.