Anti-Symmetry and Non-Extensional Mereology

Abstract

I give a formal construction of a non-extensional mereology in which antisymmetry fails. If the notion of 'mereological equivalence' is made explicit, this non-anti-symmetric mereology recaptures all of the structure of classical mereology.In the most recent and extended defence of extensionality principles in mereology, Achille Varzi argues that challenges to extensionality principles are either selfdefeating or unsupported. 1 Varzi claims that his main argument does not presuppose the anti-symmetry of parthood. In this paper, I show that it does presuppose the anti-symmetry of parthood. Towards this end, I develop a non-extensional mereology in which anti-symmetry fails. This new mereology has the additional benefit of recapturing classical mereological structure, given natural constraints.

Does Universalism Entail Extensionalism?

“…There are a few main reasons for dropping antisymmetry. As above, co‐location of material objects and the putative counterexamples to extensionality are one such reason (see Thomson () and Cotnoir ()). Thomson thinks that the statue is constituted by the clay, and that constitution requires mutual parthood: ‘ x constitutes y at t only if x is part of y at t and y is part of x at t ’ (155).…”

Are The Statue and The Clay Mutual Parts?

“…Whereas, Strong Supplementation* concerns some cooked‐up relation expressed in its antecedent, Weak Supplementation* concerns an interesting notion. Cotnoir (, ) distinguishes between two notions of proper parthood: non‐identical parthood and parthood with remainder, claiming that Weak Supplementation* captures the latter notion.…”

“…For example, Cotnoir (: 397) claims that “it is natural to think that a lump of clay and a statue made from it have all the same proper parts”. It will be apparent from §2 that I disagree, siding with Baker () et al in thinking that the statue has parts, such as its head, that the (lump of) clay does not.…”

“…Assuming that Moderate Pluralism is true, we can ask what relations hold between coincident material objects such as the statue and the clay? Many moderate pluralists claim that the clay constitutes the statue. Some moderate pluralists, notably Cotnoir (, , ; see also Cotnoir and Bacon ), Hawthorne (), Hovda (), and Thomson (, 1988), have argued that the statue and the clay are mutual parts, endorsing

Mutual Parthood: If x and y are distinct coincident objects, then x is a part of y and y is a part of x.

…”

Parts of Universals: An Essay on Aristotle's Categories

“…On the face of it, this violates the Weak Supplementation Principle, but Koslicki uses this fact to motivate including a formal component as a proper part of the statue in addition to the lump. For more on how these issues interact with formal issues in mereology, see, e.g., Aaron Cotnoir (2010). have all and only the same proper parts.…”