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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.

Abstract. This paper is a systematic exploration of non-wellfounded mereology. Motivations and applications suggested in the literature are considered. Some are exotic like Borges' ALEPH, and the TRINITY; other examples are less so, like TIME TRAVELING BRICKS, and even Geach's TIBBLES THE CAT. The authors point out that the transitivity of non-wellfounded parthood is inconsistent with extensionality. A non-wellfounded mereology is developed with careful consideration paid to rival notions of supplementation and fusion. Two equivalent axiomatizations are given, and are compared to classical mereology. We provide a class of models with respect to which the non-wellfounded mereology is sound and complete.This paper explores the prospects of non-wellfounded mereology. An order < (in this case proper parthood) on a domain is said to be wellfounded if every nonempty subset of that domain has a <-minimal element. We say that x is a <-minimal element of a set S if there is no y in S such that y < x. Wellfoundedness rules out any infinite descending <-chains. There are atomless mereologies, sometimes called gunky, in which proper parthood chains are all infinite. 1 This is one interesting and important case of a nonwellfounded mereology. But notice, wellfoundedness also rules out structures in which for some x, x < x; likewise, it rules out cases in which there is some x and y such that x < y and y < x. That is, wellfoundedness rules out parthood loops. In this paper, we explore a non-wellfounded mereology that allows for both these sorts of parthood loops.In §1, we briefly survey some applications for non-wellfounded mereology that have been suggested in the literature. In §2, we consider difficulties with the classical definitions of parthood and proper parthood; we discuss extensionality principles in mereology, and argue that extensionality is inconsistent with the transitivity of parthood in certain nonwellfounded scenarios. In §3, we examine supplementation principles and rival notions of fusion for non-wellfounded mereology. §4 examines the relationship between classical mereology and non-wellfounded mereology. We show that the latter is a simple generalization of the former. Finally, we give a class of models for which non-wellfounded mereology is sound and complete in §5. §1. Why? Why would one consider a mereology according to which there could be proper parthood loops? After all, there appears to be a consensus that it is a conceptual truth that parthood is a partial order. Simons (1987) writes,

The dominant theory of parts and wholes – classical extensional mereology – has faced a number of challenges in the recent literature. This article gives a sampling of some of the alleged counterexamples to some of the more controversial principles involving the connections between parthood and identity. Along the way, some of the main revisionary approaches are reviewed. First, counterexamples to extensionality are reviewed. The ‘supplementation’ axioms that generate extensionality are examined more carefully, and a suggested revision is considered. Second, the paper considers an alternative approach that focuses the blame on antisymmetry but allows us to keep natural supplementation axioms. Third, we look at counterexamples to the idempotency of composition and the associated ‘parts just once’ principle. We explore options for developing weaker mereologies that avoid such commitments.