A Practical Theory of Equality Is Relative

"Equality" is a "problematic" concept due to a lack of nuance when applied to practical matters such as human affairs.

Take, for example, the standard axioms of equality theory:

x=x
if x=y then y=x
if x=y and y=z then x=z

In human affairs, many people extend equality theory with one more axiom:

∀x,y(x=y)

That is, everyone is equal to everyone else.

This doesn't get one very far in practice.

Now does it?

On the other hand, let's talk about relative equality theory with a new notation:

"x(y=z)" which means  "x regards y as the same as z".

Reformulating the standard axioms of equality theory:

x(y=y)
if x(y=z) then x(z=y)
if x(y=z) and x(z=w) then x(y=w)

Now, we're in a far more interesting domain of discourse, aren't we?

For example, let:

x = "US Constitution"
y = "Some White Guy"
z = "Some Black Guy"

We have:

US Constitution(Some White Guy = Some Black Guy)

In contrast, if we let:

x = "Race"
y = "Some White Guy"
z = "Some Black Guy"

We have:

Race(Some White Guy ≠ Some Black Guy)

(Yes, I know, I didn't introduce the axioms for "≠" yet... so ban me from Facebook.)

PS: I can't claim credit for this, very powerful, notion of equality. See Tom Etter's paper, "Three-place Identity".