In response to the paper by Cotnoir and Bacon published in RSL 2/2012, we would like to add some remarks regarding supplementation principles. It is known that in a classical mereology, the Strong Supplementation Principle (SSP) together with antisymmetry enforces the Weak Supplementation Principle (WSP). Instead, in the nonwellfounded mereology, the failure of extensionality causes the failure of antisymmetry (Cotnoir, 2010), hence the investigated model is also nonantisymmetric. Cotnoir supposes that the failure of antisymmetry implies the failure of (WSP) when (...) (PP1) is applied, however gives no explicit argument, which we would like to supply in this paper. Additionally, when (PP2) is applied, (SSP) implies (WSP), hence the failure of antisymmetry does not necessarily imply the failure of (WSP). (shrink)

The new system of axioms we propose is based on the foundational theory of De Giorgi et al. Scuola Normale Superiore di Pisa, Preprints di Matematica 26: 1 (1996) slightly modified. In that paper (which is dedicated to a new axiomatic framework for mathematics, informatics and logic) the authors use two kinds of primitive notions: relations and qualities. Since their system is based on the distribution paradigm, they start from distinction. We propose to shift the perspective and to start from (...) unity and then from within unity to pass to distinction; to this end we apply ideas of Lesniewski, Nijhoff International Philosophy Series 44 (1992). We introduce only one kind of entity as a primitive notion, namely relations, and treat qualities as articulations of relations. The new concept of "primary relation" permits the introduction of a dynamic, non-standard form of identity, which we hope will find application in various fields where self-referential structures are required. (shrink)

In the following work we propose to incorporate the main feature of quantum mechanics, i.e., the concept of indiscernibility. To achieve this goal, first we present two models of set theories: a quasi-set theory and a non-antisymmetric mereology. Next, we show how specific objects of QST—m-atoms—can be defined within NAM. Finally, we introduce a concept of a parthood of indiscernibles and discuss its features in respect to standard notions of indiscernibles and within NAM.