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.