Identity: Two terms or ideas A and B are identical iff anything A represents or refers to is represented by B and conversely. In a logical formula, with R for the relation of representing: (C)(ARC iff BRC) iff A=B.

Two things are identical iff they have all their properties (of a certain kind) in common. In a logical formula: (F)(Fa iff Fb) iff a=b.

This is Leibniz's definition, and it should not be confused with the former sense, and noted that the quantified properties F tend to involve no referents to the terms for a or for b, nor to ideas involving them.

In either case, the normal formal properties for identities - reflexiveness, symmetry, transitivity, and substitutability in certain conditions - hold by reason of the defining equivalences.

For five problems with identity see under equals.