extensional

Extensional describes approaches that treat terms by their extension—the set of things they denote—or by observable behavior, rather than by internal structure, sense, or intention. It is used across philosophy, logic, mathematics, linguistics, and computer science to emphasize external features over inward content.

In set theory, the axiom of extensionality states that two sets are equal if and only if they have exactly the same elements. For example, the sets {1, 2, 3} and {3, 2, 1} are identical because they contain the same elements, regardless of any apparent order.

In logic and mathematics, extensionality also applies to functions: two functions are equal if they yield the same outputs for every possible input (functional extensionality). If f(x) = g(x) for all x, then f and g are considered the same function, even if their definitions differ.

In philosophy of language and metaphysics, extensionalism is the view that the identity or reference of terms and statements is determined by their extension (the objects they refer to or the truth conditions they express). Intensional considerations focus on sense, content, or mode of presentation. The classic example is that the terms “the morning star” and “the evening star” refer to the same object but may carry different cognitive content in certain contexts.

In linguistics and computer science, extensional semantics concentrates on denotation and observable meaning, while intensional contexts concern meaning that depends on how something is described. In programming, extensional equality often aligns with observable behavior, as opposed to internal code structure.

See also: extensionality, intensionality.