This is related to a previous question on first-order logic without equality. In that question, I asked what the axioms for equality relations are in first-order logic without equality, with a single binary relation $R$. They are the same as the axioms for equivalence relations. In this questions, I want to ask what the theory of diversity relations, that is, the complement of equality, is, in first-order logic without equality. I suspect that the axioms are the same as apartness relations:
- $\forall x \,\neg xRx$
- $\forall x \forall y (xRy \rightarrow yRx)$
- $\forall x \forall y \forall z(xRy \rightarrow xRz \lor yRz)$
Is this correct?