I have seen some argue that any relative identity claim can be reduce to an absolute identity claim in the following manner:
1) x and y are the same F ≝ x is an F, y is an F, and x = y.
However, I don’t think this works. Part of the motivation for relative identity is that there may be circumstances like:
2) x and y are the same F, but x and y are not the same G.
But (1) and (2) are not compatible, since we would have to affirm and deny absolute identity between x and y. So the relative identity theorist should reject (1) given his commitment to (2).
Relative identity is not just absolute identity, plus the idea that x and y fall under the same sortal. Moreover, this would be to suggest that relative identity is derivative, and absolute identity is the more primitive notion. I would argue that is it the other way around. So I would define absolute identity in terms of relative identity in the following manner:
4) x = y ≝ for any sortal, S, if x is an S or y is an S, then x and y are the same S.
In other words, the absolute identity between x and y is derived from the fact that for any sortal which belongs to either x or y, it is the case that x and y count as the same S. I say “either x is an S, or y is an S” as opposed to “both x is an S and y is an S” to avoid situations where x can be counted as an S and some y cannot, but they are the same S for any sortal underwhich both can be counted. For there to be absolute identity, it must be the case that all sortals that belong to x must also belong to y. I believe (4) captures this.
So to say x and y are absolutely identical is to say that for any sortal underwhich x or y can be counted, x and y are the same sortal.