Extensionally

Extensionally is an adverb used mainly in philosophy, logic, and related disciplines to describe analysis, comparison, or description that centers on extension—the actual objects a term or predicate applies to—rather than its sense or internal content (intension). The extension of a term is the set of all objects that fall under it. For example, the extension of the term dog consists of all dogs.

In extensional discussions, two terms are extensionally equivalent if they have the same extension; two predicates or functions are extensionally equal if they yield the same results for every input. This contrasts with intensional or semantic analyses that emphasize the mental content, criteria, or possible-world considerations behind terms. Classic examples include the morning star and the evening star, which are extensionally identical as Venus though purportedly different senses.

In mathematics and logic, extensionality principles formalize these ideas. Set theory treats two sets as equal when they contain exactly the same members; in higher-order logic and functional programming, two functions are considered extensionally equal if they produce identical outputs for all inputs. Extensionally oriented approaches are common in analysis of language, logic, and mathematics, where concrete referents or observable behavior are central.

Critics note that extension can overlook essential properties, structure, or necessity that intensional analyses capture, particularly in natural language with belief, obligation, or modality. Consequently, extensionally focused discussions complement but do not fully replace intensional analyses.