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.
