The phrase "relation of identity" is correct and can be used in written English.
It refers to the connection or correlation between two people or things that are the same or identical. Example: The two suspects have been arrested for fraud, and the police are now investigating their relation of identity to determine if they are working together.
It follows by transitivity from Replacement being identical with Original, and Original being identical with Reassembly, that Replacement is identical with Reassembly only if a single relation of identity is involved.
The relation of identity or sameness is represented by the sign '='.
Each thing trivially stands in the relation of identity to itself.
If it does not, it is unclear why it should be taken to be a relation of identity.
Note that we are here concerned with identity in the sense of individuality rather than the relation of identity.
So is the set of all sets containing at least two letters, and the relation of identity, which holds between each letter and itself.
Before that, we ought to get a better picture of what it means to say that an object depends upon something for its identity, that is, we should clarify the relation of identity-dependence.
Relations of identity, when compared with the other forms of relationship already described, are comparatively rare, occurring most frequently in traditions about the soul.
In general, where the notion of soul-stuff predominates, relations of identity are prevalent; where the notion of a death soul is present, the traditions are more closely akin to relations of transformation (see above Relationships of transformation).
When logicians, in discussing FOL=, speak of "real identity" — and they often do (see Enderton 2000 or Silver 1994, for example) — they do not mean a relation of universal identity, since the universal set does not exist.
Some, however, deny that a relation of absolute identity exists.
