Non-Wellfounded Mereology

Abstract

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,

Sequents for non-wellfounded mereology

“…According to ANW4, fusion is understood a minimal upper bound (with respect to ⊑) of x and y. Notice that ANW4 is weaker than the corresponding axiom Fu 3 of [1], where fusion is a higher-order predicate Fus(z, A) that applies to an individual z and a first-order formula A; it follows that our ANW is weaker that the system presented in [1]. The axiom ANW5 is the transitivity of the proper parthood relation.…”

“…Non-wellfounded mereology has been introduced in [1] to provide a formal account of genuine parthood circularity, namely cases in which an individual is a proper part of itself. Since any wellfounded parthood relation ⊏ rules out such cases, as well as those in which two individuals are proper part of each other, in non-wellfounded mereology parthood is not assumed to be a strict partial order as ⊏ is neither irreflexive nor asymmetric.…”

“…In non-wellfounded mereology ⊏ is just a transitive relation. The question as to whether and how other mereological notions are affected by ⊏ being just transitive is addressed in [1]. In particular, authors argue that the principle of extensionality is invalid in non-wellfounded mereology.…”

“…In particular, authors argue that the principle of extensionality is invalid in non-wellfounded mereology. While in extensional mereologies 1 there cannot be two distinct but indistinguishable individuals, non-wellfounded mereology accounts for situations in which, for instance, two individuals are composed of exactly the same parts without being identical.…”

“…The formal theory of non-wellfounded mereology as developed in [1] is axiomatic. In this paper, which is mainly methodological, we present a sequent calculus approach.…”