  According to the invariance criterion, \(\dq{\approx}\) is a logical constant just in case its extension on every domain is invariant under every permutation of that domain. On a domain \(D\) containing no two objects with exactly the same mass, \(\dq{\approx}\) has the same extension as \(\dq{=}\)—the set \(\{ \langle x, x \rangle : x \in D\}\)—and as we have seen, this extension is invariant under every permutation of the domain. Hence, if there is no domain containing two objects with exactly the same mass, \(\dq{\approx}\) counts as a logical constant, and \(\dq{\forall x (x \approx x)}\) as a logical truth.[16] But it seems odd that the logical status of \(\dq{\approx}\) and \(\dq{\forall x (x \approx x)}\) should depend on a matter of contingent fact: whether there are distinct objects with identical mass. Do we really want to say that if we lived in a world in which no two objects had the same mass, \(\dq{\approx}\) would be a logical  constant?[17]