Mereology (from the Greek μερος, ‘part’) is the theory of parthood relations: of the relations of part to whole and the relations of part to part within a whole.
[1] Its roots can be traced back to the early days of philosophy, beginning with the Presocratics and continuing throughout the writings of Plato (especially the
Parmenides and the
Theaetetus), Aristotle (especially the
Metaphysics, but also the
Physics, the
Topics, and
De partibus animalium), and Boethius (especially
De Divisione and
In Ciceronis Topica). Mereology occupies a prominent role also in the writings of medieval ontologists and scholastic philosophers such as Garland the Computist, Peter Abelard, Thomas Aquinas, Raymond Lull, John Duns Scotus, Walter Burley, William of Ockham, and Jean Buridan, as well as in Jungius's
Logica Hamburgensis (1638), Leibniz's
Dissertatio de arte combinatoria (1666) and
Monadology (1714), and Kant's early writings (the
Gedanken of 1747 and the
Monadologia physica of 1756). As a formal theory of parthood relations, however, mereology made its way into our times mainly through the work of Franz Brentano and of his pupils, especially Husserl's third
Logical Investigation (1901). The latter may rightly be considered the first attempt at a thorough formulation of a theory, though in a format that makes it difficult to disentangle the analysis of mereological concepts from that of other ontologically relevant notions (such as the relation of ontological dependence).
[2] It is not until Leśniewski's
Foundations of the General Theory of Sets (1916) and his
Foundations of Mathematics(1927–1931) that a pure theory of part-relations was given an exact formulation.
[3] And because Leśniewski's work was largely inaccessible to non-speakers of Polish, it is only with the publication of Leonard and Goodman's
The Calculus of Individuals (1940), partly under the influence of Whitehead, that mereology has become a chapter of central interest for modern ontologists and metaphysicians.
[4]
3.1 Supplementation
3.2 Strong Supplementation and Extensionality
3.3 Complementation
3.4 Atomism, Gunk, and Other Options
4. Composition Principles
4.1 Upper Bounds
4.2 Sums
4.3 Infinitary Bounds and Sums
4.4 Unrestricted Composition
4.5 Composition, Existence, and Identity
5. Indeterminacy and Fuzziness
Bibliography
Cited Works
Historical Surveys
Monographs and Collections
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Related Entries
[size=30]1. ‘Part’ and Parthood
A preliminary caveat is in order. It concerns the very notion of ‘part’ that mereology is about, which does not have an exact counterpart in ordinary language. Broadly speaking, in English we can use ‘part’ to indicate any portion of a given entity. The portion may itself be attached to the remainder, as in (1), or detached, as in (2); it may be cognitively or functionally salient, as in (1)–(2), or arbitrarily demarcated, as in (3); self-connected, as in (1)–(3), or disconnected, as in (4); homogeneous or otherwise well-matched, as in (1)–(4), or gerrymandered, as in (5); material, as in (1)–(5), or immaterial, as in (6); extended, as in (1)–(6), or unextended, as in (7); spatial, as in (1)–(7), or temporal, as in (8); and so on.
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| (1) | The handle is part of the mug. |
| (2) | The remote control is part of the stereo system. |
| (3) | The left half is your part of the cake. |
| (4) | The cutlery is part of the tableware. |
| (5) | The contents of this bag is only part of what I bought. |
| (6) | That area is part of the living room. |
| (7) | The outermost points are part of the perimeter. |
| (8) | The first act was the best part of the play. |
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All of these uses illustrate the general notion of ‘part’ that forms the focus of mereology, regardless of any internal distinctions. (For more examples and tentative taxonomies, see Winston
et al. 1987, Iris
et al. 1988, Gerstl and Pribbenow 1995, Pribbenow 2002, Westerhoff 2004, and Simons 2013.) Sometimes, however, the English word is used in a more restricted sense. For instance, it can be used to designate only the cognitively salient relation of parthood illustrated in (1), the relevant notion of salience being determined by Gestalt factors (Rescher and Oppenheim 1955; Bower and Glass 1976; Palmer 1977) or other perceptual and cognitive factors at large (Tversky 2005). Or it may designate only the functional relation reflected in the parts list included in the user's manual of a machine, or of a ready-to-assemble product, as in (2), in which case the parts of an object
x are just its “components”, i.e., those parts that are available as individual units regardless of their actual interaction with the other parts of
x. (A component is
a part of an object, rather than just
part of it; see e.g. Tversky 1989, Simons and Dement 1996.) Clearly, the properties of such restricted relations may not coincide with those of parthood understood more broadly, and it will be apparent that pure mereology is only concerned with the latter.
On the other hand, the English word ‘part’ is sometimes used in a broader sense, too, for instance to designate the relation of material constitution, as in (9), or the relation of mixture composition, as in (10), or the relation of group membership, as in (11):
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| (9) | The clay is part of the statue. |
| (10) | Gin is part of martini. |
| (11) | The goalie is part of the team. |
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The mereological status of these relations, however, is controversial. For instance, although the constitution relation exemplified in (9) was included by Aristotle in his threefold taxonomy of parthood (
Metaphysics, Δ, 1023b), many contemporary authors would rather construe it as a
sui generis, non-mereological relation (see e.g. Wiggins 1980, Rea 1995, Baker 1997, Evnine 2011) or else as the relation of identity (Noonan 1993, Pickel 2010), possibly contingent or occasional identity (Gibbard 1975, Robinson 1982, Gallois 1998). Similarly, the ingredient-mixture relationship exemplified in (10) is of dubious mereological status, as the ingredients may undergo significant chemical transformations that alter the structural characteristics they have in isolation (Sharvy 1983, Bogen 1995, Fine 1995a, Needham 2007). As for cases such as (11), there is disagreement concerning whether teams and other groups should be regarded as genuine mereological wholes, and while there are philosophers who do think so (from Oppenheim and Putnam 1958 to Quinton 1976, Copp 1984, Martin 1988, and Sheehy 2006), many are inclined to regard groups as entities of a different sort and to construe the relation of group membership as distinct from parthood (see e.g. Simons 1980, Ruben 1983, Gilbert 1989, Meixner 1997, Uzquiano 2004, Effingham 2010b, and Ritchie 2013 for different proposals). For all these reasons, here we shall take mereology to be concerned mainly with the principles governing the relation exemplified in (1)–(8), leaving it open whether one or more such broader uses of ‘part’ may themselves be subjected to mereological treatments of some sort.
Finally, it is worth stressing that mereology assumes no ontological restriction on the field of ‘part’. In principle, the relata can be as different as material bodies, events, geometric entities, or spatio-temporal regions, as in (1)–(8), as well as abstract entities such as properties, propositions, types, or kinds, as in the following examples:
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| (12) | Rationality is part of personhood. |
| (13) | The antecedent is the ‘if’ part of the conditional. |
| (14) | The letter ‘m’ is part of the word ‘mereology’. |
| (15) | Carbon is part of methane. |
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This is not uncontentious. For instance, to some philosophers the thought that such abstract entities may be structured mereologically cannot be reconciled with their being universals. To adapt an example from Lewis (1986a), if the letter-type ‘m’ is part of the word-type ‘mereology’, then so is the letter-type ‘e’. But there are two occurrences of ‘e’ in ‘mereology’. Shall we say that the letter is part of the word twice over? Likewise, if
carbon is part of
methane, then so is
hydrogen. But each methane molecule consists of one carbon atom and four hydrogen atoms. Shall we say that
hydrogen is part of
methane four times over? What could that possibly mean? How can one thing be part of another more than once? These are pressing questions, and the friend of structured universals may want to respond by conceding that the relevant building relation is not parthood but, rather, a non-mereological mode of composition (Armstrong 1986, 1988). However, other options are open, including some that take the difficulty at face value from a mereological standpoint (see e.g. Bigelow and Pargetter 1989, Hawley 2010, Mormann 2010, Bader 2013, and Forrest 2013, forthcoming; see also D. Smith 2009: §4, K. Bennett 2013, Fisher 2013, and Cotnoir 2013b: §4, 2015 for explicit discussion of the idea of being part-related “many times over”). Whether such options are viable may be controversial. Yet their availability bears witness to the full generality of the notion of parthood that mereology seeks to characterize. In this sense, the point to be stressed is metaphilosophical. For while Leśniewski's and Leonard and Goodman's original formulations betray a nominalistic stand, reflecting a conception of mereology as an ontologically parsimonious alternative to set theory, there is no necessary link between the analysis of parthood relations and the philosophical position of nominalism.
[5] As a formal theory (in Husserl's sense of ‘formal’, i.e., as opposed to ‘material’) mereology is simply an attempt to lay down the general principles underlying the relationships between an entity and its constituent parts, whatever the nature of the entity, just as set theory is an attempt to lay down the principles underlying the relationships between a set and its members. Unlike set theory, mereology is not committed to the existence of
abstracta: the whole can be as concrete as the parts. But mereology carries no nominalistic commitment to
concreta either: the parts can be as abstract as the whole.[/size]
Whether this way of conceiving of mereology as a general and topic-neutral theory holds water is a question that will not be further addressed here. It will, however, be in the background of much that follows. Likewise, little will be said about the important question of whether one should countenance different (primitive) part-whole relations to hold among different kinds of entity (as urged e.g. by Sharvy 1980, McDaniel 2004, 2009, and Mellor 2006), or perhaps even among entities of the same kind (Fine 1994, 2010). Such a question will nonetheless be relevant to the assessment of certain mereological principles discussed below, whose generality may be claimed to hold only in a restricted sense, or on a limited understanding of ‘part’. For further issues concerning the alleged universality and topic-neutrality of mereology, see also Johnston (2005, 2006), Varzi (2010), Donnelly (2011), Hovda (2014), and Johansson (2015). (Some may even think that there are no parthood relations whatsoever, e.g., because there are there are no causally inert non-logical properties or relations, and parthood would be one such; for a defense of this sort of mereological anti-realism, see Cowling 2014.)
Mereology
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