 (See e.g. Goodman 1951: 35; Eberle 1967: 272; Simons 1991a: 286; Casati and Varzi 1999: 36; Niebergall 2011: 274). Similarly for ‘PE’. In M the difference is immaterial, since the relevant definientia are provably equivalent. But the equivalence in question depends crucially on Antisymmetry. Absent (P.3), the second definition is strictly stronger: any two things that are mutually P-related would count as proper parts of each other according to (20) but not, obviously, according to (20′), which forces PP to be asymmetric. Indeed, in the presence of (P.1) and (P.2) the latter definition is still strong enough to deliver a strict partial ordering, whereas (20) does not even yield a transitive relation unless (P.3) is  assumed.[12]  Another important difference is that, absent (P.3), the biconditional in (24) continues to hold only if ‘PP’ is defined as in (20); if (20′) is used instead, the left-to-right direction fails whenever x and y are distinct mutual parts. In view of the above remarks concerning the doubtful status of (P.3), it is therefore convenient to work with the weaker definition. Standardly it makes no difference, but some of the definitions and results presented below would not extend to non-well-founded mereology if (20′) were used instead. (See e.g. Cotnoir 2010 and Gilmore 2016.) Furthermore, since both definitions force PP to be irreflexive, it should be noted that the only way to develop a non-well-founded mereology that allows for strict mereological loops, i.e., things that are proper part of themselves, is to rely on yet another definition or else take ‘PP’ as a primitive (as in Cotnoir and Bacon 2012, where PP is axiomatized as transitive but neither irreflexive nor asymmetric).