nonidentity

Nonidentity is the logical or philosophical relation expressed by the negation of identity. In ordinary language, to say that two objects are non-identical is to say that they are not the same object. The formal counterpart is the inequality relation, often written as ≠ or not equal, which is the negation of the identity predicate =.

In standard first-order logic with identity, identity is a primitive, binary relation that is reflexive, symmetric, and transitive. Nonidentity (a ≠ b) simply states that a and b do not denote the same object. It is irreflexive (a ≠ a is false) and its truth can depend on the existence and interpretation of the terms involved, especially in logics that permit non-referring terms or empty or partially defined domains.

Philosophically, nonidentity helps distinguish numerical identity from qualitative similarity. Two objects can be identical in the sense of Leibniz’s law if they share all properties, but discussions of nonidentity lead to questions about whether any two distinct objects can be completely indistinguishable in all qualitative respects. Some positions introduce haecceities or primitive thisness to account for apparent non-identities among otherwise indiscernible objects.

Practical usage appears in mathematics and computer science as a straightforward notion of inequality: numbers such as 2 ≠ 3, or different symbolic references in a data structure. In semantics and linguistics, nonidentity clarifies referential problems and distinctions between co-referential terms.

See also: identity, equality, inequality, indiscernibility, haecceity.