 The last remark is also relevant to the definition of ‘PP’ given above. That is the classical definition used by Leśniewski and by Leonard and Goodman and corresponds verbatim to the intuitive characterization of proper parthood used in the previous section. However, in some treatments (including earlier versions of this  entry[11]),  ‘PP’ is defined directly in terms of ‘P’, without using identity, as per the following variant of (20):