An Extensional Mereology for Structured Entities

Erkenntnis https://doi.org/10.1007/s10670-020-00305-5 ORIGINAL RESEARCH An Extensional Mereology for Structured Entities Ilaria Canavotto1 · Alessandro Giordani2 Received: 11 October 2019 / Accepted: 27 July 2020 © The Author(s) 2020 Abstract In this paper, we present an extensional system of mereology suitable to account for the intuitive distinction between heaplike and non-heaplike entities. Since the need to capture this distinction has been a key motivation for non-extensional mereologies, we first assess the main non-extensional systems advanced in the last years and highlight some mereological and metaphysical difficulties they involve. We then advance a novel program, according to which the distinction between heaplike and non-heaplike entities can be accounted for by bringing together the parthood relation characterized by classical extensional mereology and an Aristotelian extensional notion of potential parthood. Thus, while rejecting the thesis of mereological monism, our proposal is consistent with the thesis of mereological extensionalism. We show that within this framework it is possible to characterize the above-mentioned distinction, to define the relation of material constitution, and to capture three fundamental standpoints in metaphysics. 1 Introduction In Metaphysics Z 17 Aristotle introduces a primitive distinction between concrete heaplike composites, like a bunch of bricks or a pile of sand, and concrete non-heaplike composites, like a house or a clay pot. This distinction is based on the intuitive judgment that, although the entities in the two groups have parts, and so are * Ilaria Canavotto Alessandro Giordani 1 Institute for Logic, Language and Computation, University of Amsterdam, Amsterdam, The Netherlands 2 Department of Philosophy, Catholic University of Milan, Milan, Italy 13 Vol.:(0123456789) I. Canavotto, A. Giordani composite, they differ both in the way in which they are unified and in the way in which they can be composed and decomposed. To illustrate, consider the entities depicted in Fig. 1.1 Hardly anyone would take it as problematic to classify what we see in Fig. 1a as a puzzle and what we see in Fig. 1b as a bunch of puzzle pieces. In addition, most of us would certainly be prepared to claim that both the puzzle and the bunch of pieces have parts, even if the pieces in the puzzle, unlike those in the bunch, are unified in an appropriate way, partly dependent on the form of the pieces. We would also typically classify what we see in Fig. 2a as a complete puzzle and what we see in Fig. 2b as a puzzle lacking a piece.2 But it would strike us as odd that a bunch of puzzle pieces can lack a piece: we would typically classify what we obtain by removing one of the pieces in a bunch as a new bunch rather than as an incomplete bunch. Hence, while a puzzle can be decomposed in such a way that it makes sense to say that it is no more a complete puzzle, a bunch of puzzle pieces is different in this respect. Finally, in order to produce a puzzle, the pieces need to be assembled in a specific way, while no particular way of composition is required in order to produce a bunch of puzzle pieces. In fact, we tend to say that a puzzle has a certain structure and that it is in virtue of this that it has a unity allowing us to judge whether it is complete or not. So, unlike the bunch of pieces, the puzzle is what it is insofar as the pieces are arranged according to a specific structure. In light of this, we can also say that the bunch of pieces (in general, any heaplike entity) is a mere sum, while the puzzle (in general, any non-heaplike entity) is a structured sum. Making the framework underlying these intuitions explicit, by assuming that there are both heaplike and non-heaplike entities, Aristotle’s view on composition implies that (i) there exist composed entities; (ii) some composed entities are mere sums of their components; (iii) some composed entities are not mere sums of their components. Claims (i)–(iii) represent a substantive standpoint in the metaphysics of composition. While claim (i) is the negation of mereological nihilism, the viewpoint according to which no composed entity exists (Rosen and Dorr 2007; Sider 2013), the recognition of different kinds of composites, as expressed by claims (ii) and (iii), is the negation of either mereological extensionalism, the viewpoint according to which no two composed entities can have the same proper parts (Lewis 1991; Varzi 2008), or 1 The following examples are introduced to highlight the pre-philosophical intuitions motivating the distinction between heaplike and non-heaplike entities, not to justify them. In discussing the examples, we thus take the perspective, say, of a kid looking at the pictures and answering questions about what she sees. 2 In fact, if you ask a kid whether the puzzles in the figure are finished, you should not be surprised if she answers “yes, it is” in the first case, and “no, a piece is still missing” in the second case. 13 An Extensional Mereology for Structured Entities Fig. 1  a A puzzle. b A bunch of pieces mereological monism, the viewpoint according to which there is only one parthood relation and one corresponding way of composition (Lando 2017; Lewis 1991). In the current literature, this view has led to the development of both non-extensional and non-monistic mereologies. On the one hand, some scholars maintain mereological monism, allowing for theories of parthood in which either different sums can result from composing the same proper parts in the same way—against mereological extensionalism (Cotnoir 2010; Cotnoir and Bacon 2012; Thomson 1983, 1998)—or different sums can result from composing the same proper material parts by adding a formal part to one of them—in accordance with mereological extensionalism (Koslicki 2008). On the other hand, other scholars renounce to both mereological monism and mereological extensionalism, proposing theories of parthood in which the same proper parts, composed either in the same or in different ways, can give rise to different sums (Armstrong 1997; Fine 1999, 2010; McDaniel 2009). Interestingly, in the Aristotelian perspective, the distinction between kinds of sums is related to a distinction between parthood relations and corresponding ways of composition. This implies a negation of mereological monism but not necessarily of mereological extensionalism: according to Aristotle, while a collection of entities can, by itself, give rise to one and only one mere sum, it can give rise to one and only one structured sum only if it is unified in a substance by a formal principle (not to be intended, pace some Neo-Aristotelian philosophers including, e.g. Fine (1999) and Koslicki (2008), as a formal part3). The senses in which the entities in the given collection will be proper parts of the resulting sums will then differ accordingly. By following the Aristotelian lead, the aim of the present pap (...truncated)