 Note also that the definitions imply (by pure logic) that EQ, O, and U are all reflexive and symmetric; in addition, EQ is also transitive—an equivalence relation. By contrast, PP and PE are irreflexive and asymmetric, and it follows from (P.2) that both are also transitive—so they are strict partial orderings. Since the following biconditional is also a straightforward consequence of the axioms (specifically, of P.1),