Mereology (from the Greek μερος,‘part’) is the theory of parthood relations: of therelations 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 writingsof Plato (especially theParmenides and theTheaetetus), Aristotle (especially theMetaphysics,but also thePhysics, theTopics, andDepartibus animalium), and Boethius (especiallyDeDivisione andIn Ciceronis Topica). Mereology occupies aprominent role also in the writings of medieval ontologists andscholastic philosophers such as Garland the Computist, Peter Abelard,Thomas Aquinas, Raymond Lull, John Duns Scotus, Walter Burley, Williamof Ockham, and Jean Buridan, as well as in Jungius'sLogicaHamburgensis (1638), Leibniz'sDissertatio de artecombinatoria (1666) andMonadology (1714), and Kant'searly writings (theGedanken of 1747 and theMonadologiaphysica of 1756). As a formal theory of parthood relations,however, mereology made its way into our times mainly through the workof Franz Brentano and of his pupils, especially Husserl's thirdLogical Investigation (1901). The latter may rightly beconsidered the first attempt at a thorough formulation of a theory,though in a format that makes it difficult to disentangle the analysisof mereological concepts from that of other ontologically relevantnotions (such as the relation of ontological dependence).^[2] It is not until Leśniewski'sFoundations of the GeneralTheory of Sets (1916) and hisFoundations of Mathematics(1927–1931) that a pure theory of part-relations was given anexact formulation.^[3] And because Leśniewski's work was largely inaccessible tonon-speakers of Polish, it is only with the publication of Leonard andGoodman'sThe Calculus of Individuals (1940), partly underthe influence of Whitehead, that mereology has become a chapter ofcentral interest for modern ontologists and metaphysicians.^[4]

In the following we focus mostly on contemporary formulations ofmereology as they grew out of these recenttheories—Leśniewski's and Leonard and Goodman's. Indeed,although such theories come in different logical guises, they aresufficiently similar to be recognized as a common basis for mostsubsequent developments. To properly assess the relative strengths andweaknesses, however, it will be convenient to proceed in steps. Firstwe consider some core mereological notions and principles. Then weproceed to an examination of the stronger theories that can be erectedon that basis.

• 1. ‘Part’ and Parthood
• 2. Core Principles
  • 2.1 Parthood as a Partial Ordering
  • 2.2 Other Mereological Concepts
• 3. Decomposition Principles
  • 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
• Other Internet Resources
• Academic Tools
• Related Entries

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 anexact counterpart in ordinary language. Broadly speaking, in Englishwe can use ‘part’ to indicate any portion of a givenentity. The portion may itself be attached to the remainder, as in(1), or detached, as in (2); it may be cognitively or functionallysalient, 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), orgerrymandered, as in (5); material, as in (1)–(5), orimmaterial, as in (6); extended, as in (1)–(6), or unextended,as in (7); spatial, as in (1)–(7), or temporal, as in (8); andso on.

All of these uses illustrate the general notion of ‘part’that forms the focus of mereology, regardless of any internaldistinctions. (For more examples and tentative taxonomies, see Winstonet al. 1987, Iriset al. 1988, Gerstl and Pribbenow1995, Pribbenow 2002, Westerhoff 2004, and Simons 2013.) Sometimes,however, the English word is used in a more restricted sense. Forinstance, it can be used to designate only the cognitively salientrelation of parthood illustrated in (1), the relevant notion ofsalience being determined by Gestalt factors (Rescher and Oppenheim1955; Bower and Glass 1976; Palmer 1977) or other perceptual andcognitive factors at large (Tversky 2005). Or it may designate onlythe functional relation reflected in the parts list included in theuser's manual of a machine, or of a ready-to-assemble product, as in(2), in which case the parts of an objectx are just its“components”, i.e., those parts that are available asindividual units regardless of their actual interaction with the otherparts ofx. (A component isa part of an object,rather than justpart of it; see e.g. Tversky 1989, Simonsand Dement 1996.) Clearly, the properties of such restricted relationsmay not coincide with those of parthood understood more broadly, andit will be apparent that pure mereology is only concerned with thelatter.

On the other hand, the English word ‘part’ is sometimesused in a broader sense, too, for instance to designate the relationof material constitution, as in (9), or the relation of mixturecomposition, as in (10), or the relation of group membership, as in(11):

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 authorswould rather construe it as asui generis, non-mereologicalrelation (see e.g. Wiggins 1980, Rea 1995, Baker 1997, Evnine 2011) orelse as the relation of identity (Noonan 1993, Pickel 2010), possiblycontingent or occasional identity (Gibbard 1975, Robinson 1982,Gallois 1998). Similarly, the ingredient-mixture relationshipexemplified in (10) is of dubious mereological status, as theingredients may undergo significant chemical transformations thatalter the structural characteristics they have in isolation (Sharvy1983, Bogen 1995, Fine 1995a, Needham 2007). As for cases such as(11), there is disagreement concerning whether teams and other groupsshould be regarded as genuine mereological wholes, and while there arephilosophers who do think so (from Oppenheim and Putnam 1958 toQuinton 1976, Copp 1984, Martin 1988, and Sheehy 2006), many areinclined to regard groups as entities of a different sort and toconstrue 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 differentproposals). For all these reasons, here we shall take mereology to beconcerned mainly with the principles governing the relationexemplified in (1)–(8), leaving it open whether one or more suchbroader uses of ‘part’ may themselves be subjected tomereological treatments of some sort.

Finally, it is worth stressing that mereology assumes no ontologicalrestriction on the field of ‘part’. In principle, therelata can be as different as material bodies, events, geometricentities, or spatio-temporal regions, as in (1)–(8), as well asabstract entities such as properties, propositions, types, or kinds,as in the following examples:

This is not uncontentious. For instance, to some philosophers thethought that such abstract entities may be structured mereologicallycannot be reconciled with their being universals. To adapt an examplefrom Lewis (1986a), if the letter-type ‘m’ is part of theword-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 theword twice over? Likewise, ifcarbon is part ofmethane, then so ishydrogen. But each methanemolecule consists of one carbon atom and four hydrogen atoms. Shall wesay thathydrogen is part ofmethane four timesover? What could that possibly mean? How can one thing be part ofanother more than once? These are pressing questions, and the friendof structured universals may want to respond by conceding that therelevant building relation is not parthood but, rather, anon-mereological mode of composition (Armstrong 1986, 1988). However,other options are open, including some that take the difficulty atface value from a mereological standpoint (see e.g. Bigelow andPargetter 1989, Hawley 2010, Mormann 2010, Bader 2013, and Forrest2013, forthcoming; see also D. Smith 2009: §4, K. Bennett 2013,Fisher 2013, and Cotnoir 2013b: §4, 2015 for explicit discussionof the idea of being part-related “many times over”).Whether such options are viable may be controversial. Yet theiravailability bears witness to the full generality of the notion ofparthood that mereology seeks to characterize. In this sense, thepoint to be stressed is metaphilosophical. For while Leśniewski'sand Leonard and Goodman's original formulations betray a nominalisticstand, reflecting a conception of mereology as an ontologicallyparsimonious alternative to set theory, there is no necessary linkbetween the analysis of parthood relations and the philosophicalposition of nominalism.^[5] As a formal theory (in Husserl's sense of ‘formal’, i.e.,as opposed to ‘material’) mereology is simply an attemptto lay down the general principles underlying the relationshipsbetween an entity and its constituent parts, whatever the nature ofthe entity, just as set theory is an attempt to lay down theprinciples underlying the relationships between a set and its members.Unlike set theory, mereology is not committed to the existence ofabstracta: the whole can be as concrete as the parts. Butmereology carries no nominalistic commitment toconcretaeither: the parts can be as abstract as the whole.

Whether this way of conceiving of mereology as a general andtopic-neutral theory holds water is a question that will not befurther addressed here. It will, however, be in the background of muchthat follows. Likewise, little will be said about the importantquestion of whether one should countenance different (primitive)part-whole relations to hold among different kinds of entity (as urgede.g. by Sharvy 1980, McDaniel 2004, 2009, and Mellor 2006), or perhapseven among entities of the same kind (Fine 1994, 2010). Such aquestion will nonetheless be relevant to the assessment of certainmereological principles discussed below, whose generality may beclaimed to hold only in a restricted sense, or on a limitedunderstanding of ‘part’. For further issues concerning thealleged universality and topic-neutrality of mereology, see alsoJohnston (2005, 2006), Varzi (2010), Donnelly (2011), Hovda (2014),and Johansson (2015). (Some may even think that there are no parthoodrelations whatsoever, e.g., because there are there are no causallyinert non-logical properties or relations, and parthood would be onesuch; for a defense of this sort of mereological anti-realism, seeCowling 2014.)

2. Core Principles

With these provisos, and barring for the moment the complicationsarising from the consideration of intensional factors (such as timeand modalities), we may proceed to review some core mereologicalnotions and principles. Ideally, we may distinguish here between (a)those principles that are simply meant to fix the intended meaning ofthe relational predicate ‘part’, and (b) a variety ofadditional, more substantive principles that go beyond the obvious andaim at greater sophistication and descriptive power. Exactly where theboundary between (a) and (b) should be drawn, however, or even whethera boundary of this sort can be drawn at all, is by itself a matter ofcontroversy.

2.1 Parthood as a Partial Ordering

The usual starting point is this: regardless of how one feels aboutmatters of ontology, if ‘part’ stands for the generalrelation exemplified by (1)–(8) above, and perhaps also(12)–(15), then it stands for a partial ordering—areflexive, transitive, antisymmetric relation:

As it turns out, most theories put forward in the literature accept(16)–(18). Some misgivings are nonetheless worth mentioning thatmay, and occasionally have been, raised against these principles.

Concerning reflexivity (16), two sorts of worry may be distinguished.The first is that many legitimate senses of ‘part’ justfly in the face of saying that a whole is part of itself. Forinstance, Rescher (1955) famously objected to Leonard and Goodman'stheory on these grounds, citing the biologists' use of‘part’ for the functional subunits of an organism as acase in point: no organism is a functional subunit of itself. This isa legitimate worry, but it appears to be of little import. Takingreflexivity (and antisymmetry) as constitutive of the meaning of‘part’ simply amounts to regarding identity as a limit(improper) case of parthood. A stronger relation, whereby nothingcounts as part of itself, can obviously be defined in terms of theweaker one, hence there is no loss of generality (see Section 2.2below). Vice versa, one could frame a mereological theory by takingproper parthood as a primitive instead. As already Lejewski (1957)noted, this is merely a question of choosing a suitable primitive, sonothing substantive follows from it. (Of course, if one thinks thatthere are or might be objects that are not self-identical,for instance because of the loss of individuality in the quantumrealm, or for whatever other reasons, then such objects would not bepart of themselves either, yielding genuine counterexamples to (16).Here, however, we stick to a notion of identity that obeys traditionalwisdom, which is to say a notion whereby identity is an equivalencerelation subject to Leibniz's law.) The second sort of worry is moreserious, for it constitutes a genuine challenge to the idea that (16)expresses a principle that is somehow constitutive of the meaning of‘part’, as opposed to a substantive metaphysical thesisabout parthood. Following Kearns (2011), consider for instance ascenario in which an enduring wall, W, is shrunk down to the size of abrick and eventually brought back in time so as to be used to build(along with other bricks) the original W. Or suppose wall W isbilocated to my left and my right, and I shrink it to the size of abrick on the left and then use it to replace a brick from W on theright. In such cases, one might think that W is part of itself in asense in which ordinary walls are not, hence that either parthood isnot reflexive or proper parthood is not irreflexive. For anotherexample (also by Kearns), if shapes are construed as abstractuniversals, then self-similar shapes such as fractals may very well besaid to contain themselves as parts in a sense in which other shapesdo not. Whether such scenarios are indeed possible is by itself acontroversial issue, as it depends on a number of backgroundmetaphysical questions concerning persistence through time, locationin space, and the nature of shapes. But precisely insofar as thescenarios are not obviouslyimpossible, the generality andmetaphysical neutrality of (16) may be questioned. (Note that thosescenarios also provide reasons to question the generality of manyother claims that underlie the way we ordinarily talk, such as theclaim that nothing can belarger than itself, ornext to itself, orqualitatively different fromitself. Such claims might be even more entrenched in common sense thanthe claim that proper parthood is irreflexive, and parthood reflexive;yet this is hardly a reason to hang on to them at every cost. Itsimply shows that our ordinary talk does not take into accountsituations that are—admittedly—extraordinary.)

Similar considerations apply to the transitivity principle, (17). Onthe one hand, several authors have observed that many legitimatesenses of ‘part’ are non-transitive, fostering the studyof mereologies in which (17) may fail (Pietruszczak 2014). Exampleswould include: (i) a biological subunit of a cell is not a part of theorgan(ism) of which that cell is a part; (ii) a handle can be part ofa door and the door of a house, though a handle is never part of ahouse; (iii) my fingers are part of me and I am part of the team, yetmy fingers are not part of the team. (See again Rescher 1955 alongwith Cruse 1979 and Winstonet al. 1987, respectively; forother examples see Iriset al. 1988, Moltmann 1997, Hossack2000, Johnston 2002, 2005, Johansson 2004, 2006, and Fiorinietal. 2014). Arguably, however, such misgivings stem again from theambiguity of the English word ‘part’. What counts as abiological subunit of a cell may not count as a subunit, i.e., adistinguished part of the organ, but that is not to say thatit is not part of the organ at all. Similarly, if there is a sense of‘part’ in which a handle is not part of the house to whichit belongs, or my fingers not part of my team, it is a restrictedsense: the handle is not afunctional part of the house,though it is a functional part of the door and the door a functionalpart of the house; my fingers are notdirectly part of theteam, though they are directly part of me and I am directly part ofthe team. (Concerning this last case, Uzquiano 2004: 136–137,Schmitt 2003: 34, and Effingham 2010b: 255 actually read (iii) as areductio of the very idea that the group-membership relationis a genuine case of parthood, as mentioned abovead (11).)It is obvious that if the interpretation of ‘part’ isnarrowed by additional conditions, e.g., by requiring that parts makea functional or direct contribution to the whole, then transitivitymay fail. In general, ifx is a φ-part ofy andy is a φ-part ofz,x need not be aφ-part ofz: the predicate modifier ‘φ’may not distribute over parthood. But that shows the non-transitivityof ‘φ-part’, not of ‘part’, and within asufficiently general framework this can easily be expressed with thehelp of explicit predicate modifiers (Varzi 2006a; Vieu 2006; Garbacz2007). On the other hand, there is again a genuine worry that,regardless of any ambiguity concerning the intended interpretation of‘part’, (17) expresses a substantive metaphysical thesisand cannot, therefore, be taken for granted. For example, it turns outthat time-travel and multi-location scenarios such as those mentionedin relation to (16) may also result in violations of the transitivityof both parthood (Effingham 2010a) and proper parthood (Gilmore 2009;Kleinschmidt 2011). And the same could be said of cases that involveno suchexotica. For instance, Gilmore (2014) bringsattention to the popular theory of structured propositions originatedwith Russell (1903). Already Frege (1976: 79) pointed out that if theconstituents of a proposition are construed mereologically as (proper)parts, then we have a problem: assuming that Mount Etna is literallypart of the proposition that Etna is higher than Vesuvius, eachindividual piece of solidified lava that is part of Etna would also bepart of that proposition, which is absurd. The worse for Russell'stheory of structured propositions, said Frege. The worse, one couldreply, for the transitivity of parthood (short of claiming that theargument involves yet another equivocation on ‘partof’).

Concerning the antisymmetry postulate (18), the picture is even morecomplex. For one thing, some authors maintain that the relationshipbetween an object and the stuff it is made of provides a perfectlyordinary counterexample of the antisymmetry of parthood: according toThomson (1998), for example, a statue and the clay that constitutes itare part of each other, yet distinct. This is not a popular view: asalready mentioned, most contemporary authors would either deny thatmaterial constitution is a relation of parthood or else treat it asimproper parthood, i.e., identity, which is triviallyantisymmetric (and symmetric). Moreover, those who regard constitutionas a genuine case of proper parthood tend to follow Aristotle'shylomorphic conception and deny that the relation also holds in theopposite direction: the clay is part of the statue but not vice versa(see e.g. Haslanger 1994, Koslicki 2008). Still, insofar as Thomson'sview is a legitimate option, it represents a challenge to the putativegenerality of (18). Second, one may wonder about the possibility ofunordinary cases of symmetric parthood relationships. Sanford(1993: 222) refers to Borges's Aleph as a case in point: “I sawthe earth in the Aleph and in the earth the Aleph once more and theearth in the Aleph …”. In this case, a plausible reply issimply that fiction delivers no guidance to conceptual investigations:conceivability may well be a guide to possibility, but literaryfantasy is by itself no evidence of conceivability (van Inwagen 1993:229). Perhaps the same could be said of Fazang's Jeweled Net of Indra,in which each jewel has every other jewel as part (Jones 2012).However, other cases seem harder to dismiss. Surely the Scholasticswere not merely engaging in literary fiction when arguing that eachperson of the Trinity is a proper part of God, and yet also identicalwith God (see e.g. Abelard,Theologia christiana, bk. III).And arguably time travel is at least conceivable, in which case again(18) could fail: if time-traveling wall W ends up being one of thebricks that compose (say) its own bottom half, H, then we have aconceivable scenario in which W is part of H and H is part of W whileW ≠ H (Kleinschmidt 2011). Third, it may be argued thatantisymmetry is also at odds with theories that have been foundacceptable on quite independent grounds. Consider again the theory ofstructured propositions. IfA is the proposition that theuniverse exists—where the universe is something of whicheverything is part—and ifA is true, then on such atheory the universe would be a proper part ofA; and sinceA would in turn be a proper part of U, antisymmetry would beforfeit (Tillman and Fowler 2012). Likewise, ifA is theproposition thatB is true, andB the propositionthatA is contingent, then againA andBwould be part of each other even thoughA ≠B(Cotnoir 2013b). Finally, and more generally, it may be observed thatthe possibility of mereological loops is to be taken seriously for thesame sort of reasons that led to the development of non-well-foundedset theory, i.e., set theory tolerating cases of self-membership and,more generally, of membership circularities (Aczel 1988; Barwise andMoss 1996). This is especially significant in view of the possibilityof reformulating set theory itself in mereological terms—apossibility that is extensively worked out in the works of Bunt (1985)and especially Lewis (1991, 1993b) (see also Burgess 2015 and Hamkinsand Kikuchi forthcoming). For all these reasons, the antisymmetrypostulate (18) can hardly be regarded as constitutive of the basicmeaning of ‘part’, and some authors have begun to engagein the systematic study of “non-well-founded mereologies”in which (18) may fail (Cotnoir 2010; Cotnoir and Bacon 2012; Obojska2013).

In the following we aim at a critical survey of mereology asstandardly understood, so we shall mainly confine ourselves totheories that do in fact accept the antisymmetry postulate along withboth reflexivity and transitivity. However, the above considerationsshould not be dismissed. On the contrary, they are crucially relevantin assessing the scope of mereology and the degree to which itsstandard formulations and extensions betray intuitions that may befound too narrow, false, or otherwise problematic. Indeed, they arecrucially relevant also in assessing the ideal desideratum mentionedat the beginning of this section—the desideratum of a neatdemarcation between core principles that are simply meant to fix theintended meaning of ‘part’ and principles that reflectmore substantive theses concerning the parthood relation. Classicalmereology takes the former to include the threefold claim that‘part’ stands for a reflexive, transitive andantisymmetric relation, but this is not to say that “anyone whoseriously disagrees with them had failed to understand the word”(Simons 1987: 11), just as departure from the basic principles ofclassical logic need not amount to a “change of subject”(Quine 1970: 81). And just as the existence of widespread anddiversified disagreement concerning the laws of logic may lead one toconclude that “for all we know, the only inference left in theintersection of (unrestricted)all logics might be theidentity inference: FromA to inferA” (Beall and Restall 2006: 92), so one might take theabove considerations and the corresponding development ofnon-classical mereologies to indicate that there may be “noreason to assume that any useful core mereology […] functionsas a common basis for all plausible metaphysical theories”(Donnelly 2011: 246).

2.2 Other Mereological Concepts

It is convenient at this point to introduce some degree offormalization. This avoids ambiguities stemming from ordinary languageand facilitates comparisons and developments. For definiteness, weassume here a standard first-order language with identity, suppliedwith a distinguished binary predicate constant, ‘P’, to beinterpreted as the parthood relation.^[6] Taking the underlying logic to be the classical predicate calculuswith identity,^[7] the requisites on parthood discussed in Section 2.1 may then beregarded as forming a first-order theory characterized by thefollowing proper axioms for ‘P’:

(Here and in the following we simplify notation by dropping allinitial universal quantifiers. Unless otherwise specified, allformulas are to be understood as universally closed.) We may call sucha theoryCore Mereology—M for short^[8]—since it represents the common starting point of all standard theories.

Given (P.1)–(P.3), a number of additional mereologicalpredicates can be introduced by definition. For example:

An intuitive model for these relations, with ‘P’interpreted as spatial inclusion, is given in Figure 1.

Figure 1. Basic patterns of mereological relations.(Shaded cells indicate parthood).

Note that ‘Uxy’ is bound to hold if one assumesthe existence of a “universal entity” of which everythingis part. Conversely, ‘Oxy’ would always hold ifone assumed the existence of a “null item” that is part ofeverything. Both assumptions, however, are controversial and we shallcome back to them below.

Note also that the definitions imply (by pure logic) that EQ, O, and Uare all reflexive and symmetric; in addition, EQ is alsotransitive—an equivalence relation. By contrast, PP and PE areirreflexive and asymmetric, and it follows from (P.2) that both arealso transitive—so they are strict partial orderings. Since thefollowing biconditional is also a straightforward consequence of theaxioms (specifically, of P.1),

it should now be obvious that one could in fact use proper parthood asan alternative starting point for the development of classicalmereology, using the right-hand side of (24) as a definiens for‘P’. This is, for instance, the option followed in Simons(1987), as also in Leśniewski’s original theory (1916),where the partial ordering axioms for ‘P’ are replaced bythe strict ordering axioms for ‘PP’.^[9] Ditto for ‘PE’, which was in fact the primitive relationin Whitehead's (1919) semi-formal treatment of the mereology of events(and which is just the converse of ‘PP’). Other optionsare in principle possible, too. For example, Goodman (1951) used‘O’ as a primitive and Leonard and Goodman (1940) used its opposite:^[10]

However, the relations corresponding to such predicates are strictlyweaker than PP and PE and no biconditional is provable inM that would yield a corresponding definiens of‘P’ (though one could define ‘P’ in terms of‘O’ or ‘D’ in the presence of further axioms;see belowad (61)). Thus, other things being equal,‘P’, ‘PP’, and ‘PE’ appear to bethe only reasonable options. Here we shall stick to ‘P’,referring to J. Parsons (2014) for further discussion.

Finally, note that identity could itself be introduced by definition,due to the following obvious consequence of the antisymmetry postulate(P.3):

Accordingly, theoryM could be formulated in a purefirst-order language by assuming (P.1) and (P.2) and replacing (P.3)with the following variant of the Leibniz axiom schema for identity(where φ is any formula in the language):

One may in fact argue on these grounds that the parthood relation isin some sense conceptually prior to the identity relation (as inSharvy 1983: 234), and since ‘EQ’ is not definable interms of ‘PP’ or ‘PE’ alone except in thepresence of stronger axioms (see belowad (27)), the argumentwould also provide evidence in favor of ‘P’ as the mostfundamental primitive. As we shall see in Section 3.2, however, thelink between parthood and identity is philosophically problematic. Inorder not to compromise our exposition, we shall therefore keep to alanguage containing both ‘P’ and ‘=’ asprimitives. This will also be convenient in view of the previousremarks concerning the controversial status of Antisymmetry, on which(26) depends.

The last remark is also relevant to the definition of ‘PP’given above. That is the classical definition used by Leśniewskiand by Leonard and Goodman and corresponds verbatim to the intuitivecharacterization of proper parthood used in the previous section.However, in some treatments (including earlier versions of this entry^[11]), ‘PP’ is defined directly in terms of ‘P’,without using identity, as per the following variant of (20):

(See e.g. Goodman 1951: 35; Eberle 1967: 272; Simons 1991a: 286;Casati and Varzi 1999: 36; Niebergall 2011: 274). Similarly for‘PE’. InM the difference is immaterial,since the relevant definientia are provably equivalent. But theequivalence in question depends crucially on Antisymmetry. Absent(P.3), the second definition is strictly stronger: any two things thatare mutually P-related would count as proper parts of each otheraccording to (20) but not, obviously, according to (20′), whichforces PP to be asymmetric. Indeed, in the presence of (P.1) and (P.2)the latter definition is still strong enough to deliver a strictpartial ordering, whereas (20) does not even yield a transitiverelation unless (P.3) is assumed.^[12] Another important difference is that, absent (P.3), the biconditionalin (24) continues to hold only if ‘PP’ is defined as in(20); if (20′) is used instead, the left-to-right directionfails wheneverx andy are distinct mutual parts. Inview of the above remarks concerning the doubtful status of (P.3), itis therefore convenient to work with the weaker definition. Standardlyit makes no difference, but some of the definitions and resultspresented below would not extend to non-well-founded mereology if(20′) were used instead. (See e.g. Cotnoir 2010 and Gilmore2016.) Furthermore, since both definitions force PP to be irreflexive,it should be noted that the only way to develop a non-well-foundedmereology that allows for strict mereological loops, i.e., things thatare proper part of themselves, is to rely on yet another definition orelse take ‘PP’ as a primitive (as in Cotnoir and Bacon2012, where PP is axiomatized as transitive but neither irreflexivenor asymmetric).

3. Decomposition Principles

M is standardly viewed as embodying the common coreof any mereological theory. Not just any partial ordering qualifies asa part-whole relation, though, and establishing what furtherprinciples should be added to (P.1)–(P.3) is precisely thequestion a good mereological theory is meant to answer. It is herethat philosophical issues begin to multiply, over and above thegeneral concerns mentioned in Section 2.1.

Generally speaking, such further principles may be divided into twomain groups. On the one hand, one may extendM bymeans ofdecomposition principles that take us from a wholeto its parts. For example, one may consider the idea that wheneversomething has a proper part, it has more than one—i.e., thatthere is always somemereological difference (a“remainder”) between a whole and its proper parts. Thisneed not be true in every model forM: a world withonly two items, only one of which is part of the other, would be acounterexample, though not one that could be illustrated with the sortof geometric diagram used in Figure 1. On the other hand, one mayextendM by means ofcomposition principlesthat go in the opposite direction—from the parts to the whole.For example, one may consider the idea that whenever there are somethings, there exists a whole that consists exactly of thosethings—i.e., that there is always amereological sum(or “fusion”) of two or more parts. Again, this need notbe true in a model forM, and it is a matter of muchcontroversy whether the idea should hold unrestrictedly.

3.1 Supplementation

Let us begin with the first sort of extension. And let us start bytaking a closer look at the intuition according to which a wholecannot be decomposed into a single proper part. There are various waysin which one can try to capture this intuition. Consider the following(from Simons 1987: 26–28):

The first principle, (P.4_a), is a literal rendering of theidea in question: every proper part must be accompanied by another.However, there is an obvious sense in which (P.4_a) onlycaptures the letter of the idea, not the spirit: it rules out theunintended model mentioned above (see Figure 2, left) but not, forexample, an implausible model with an infinitely descending chain inwhich the additional proper parts do not leave any remainder at all(Figure 2, center).

The second principle, (P.4_b), is stronger: it rules outboth models as unacceptable. However, (P.4_b) is still tooweak to capture the intended idea. For example, it is satisfied by amodel in which a whole can be decomposed into several proper parts allof which overlap one another (Figure 2, right), and it may be arguedthat such models do not do justice to the meaning of ‘properpart’: after all, the idea is that the removal of a proper partshould leave a remainder, but it is by no means clear what would beleft ofx oncez (along with its parts) isremoved.

Figure 2. Three unsupplemented models. (Here and inthe diagrams below, connected lines going downwards represent properextension relationships, i.e., the inverse of proper parthood. Thus,in all diagrams parthood behaves reflexively and transitively.)

It is only the third principle, (P.4), that appears to provide a fullformulation of the idea that a whole cannot be decomposed into asingle proper part. According to this principle, every proper partmust be “supplemented” by another,disjoint part,and it is this last qualification that captures the notion of aremainder. Should (P.4), then, be incorporated intoMas a further fundamental principle on the meaning of‘part’?

Most authors (beginning with Simons himself) would say so. Yet herethere is room for genuine disagreement. In fact, it is not difficultto conceive of mereological scenarios that violate not only (P.4), butalso (P.4_b) and even (P.4_a). A case in pointwould be Brentano's (1933) theory of accidents, according to which amind is a proper part of a thinking mind even though there is nothingto make up for the difference. (See Chisholm 1978, Baumgartner andSimons 1993.) Similarly, in Fine's (1982) theory ofqua-objects, every basic object (John) qualifies as the onlyproper part of its incarnations (Johnqua philosopher, Johnqua husband, etc.). Another interesting example is providedby Whitehead's (1929) theory of extensive connection, where noboundary elements are included in the domain of quantification: onthis theory, a topologically closed region includes its open interioras a proper part in spite of there being no boundary elements todistinguish them—the domain only consists of extended regions.(See Clarke 1981 for a rigorous formulation, Randellet al.1992 for developments.) Finally, consider the view, arguably held byAquinas, according to which the human person survives physical deathalong with her soul (see Brown 2005 and Stump 2006,paceToner 2009). On the understanding that persons are hylomorphiccomposites, and that two things cannot become one, the view impliesthat upon losing her body a person will continue to exist,pre-resurrection, with only one proper part—the soul. (This isalso the view of some contemporary philosophers; see e.g. Oderberg2005 and Hershenov and Koch-Hershenov 2006.) Indeed, any case ofmaterial coincidence resulting from mereological diminution, as in theStoic puzzle of Deon and Theon (Sedley 1982) and its modern variant ofTibbles and Tib (Wiggins 1968), would seem to be at odds withSupplementation: after the diminution, there is nothing that makes upfor the difference between what was a proper part and the whole withwhich it comes to coincide, short of holding that the part has becomeidentical to the whole (Gallois 1998), or has ceased to exist (Burke1994), or did not exist in the first place (van Inwagen 1981). One mayrely on the intuitive appeal of (P.4) to discard all of the abovetheories and scenarios as implausible. But one may as well turn thingsaround and regard the plausibility of such theories as a good reasonnot to accept (P.4) unrestrictedly, as argued e.g. by D. Smith (2009),Oderberg (2012), and Lowe (2013). As things stand, it therefore seemsappropriate to regard such a principle as providing a minimal butsubstantive addition to (P.1)–(P.3), one that goes beyond thebasic characterization of ‘part’ provided byM. We shall label the resulting mereological theoryMM, forMinimal Mereology.

ActuallyMM is now redundant, as Supplementationturns out to entail Antisymmetry so long as parthood is transitive andreflexive: ifx andy were proper parts of eachother, contrary to (P.3), then everyz that is part of onewould also be part of—hence overlap—the other, contrary to(P.4). For ease of reference, we shall continue to treat (P.3) as anaxiom. But the entailment is worth emphasizing, for it explains whySupplementation tends to beexplicitly rejected by those whodo not endorse Antisymmetry, over and above the more classicalexamples mentioned above. For instance, whoever thinks that a statueand the corresponding lump of clay are part of each other will findSupplementation unreasonable: after all, such parts are coextensive;why should we expect anything to be left over when, say, the clay is“subtracted” from the statue? (Donnelly 2011: 230).Indeed, Supplementation has recently run into trouble alsoindependently of its link with Antisymmetry, especially in the contextof time-travel and multilocation scenarios such as those alreadymentioned in connection with each of (P.1)–(P.3) (see Effinghamand Robson 2007, Gilmore 2007, Eagle 2010, Kleinschmidt 2011, Daniels2014). As a result, a question that is gaining increasing attention iswhether there are any ways of capturing the supplementation intuitionthat are strong enough to rule out the models of Figure 2 and yetsufficiently weaker than (P.4) to be acceptable to those who do notendorse someM-axiom or other—be itAntisymmetry, Transitivity, or Reflexivity.

Two sorts of answer may be offered in this regard (see e.g. Gilmore2016). The first is to weaken the Supplementation conditional bystrengthening the antecedent. For instance, one may simply rephrase(P.4) in terms of the stricter notion of proper parthood defined in(20′), i.e., effectively:

InM this is equivalent to (P.4). Yet it is logicallyweaker, and it is easy to see that this suffices to block theentailment of (P.3) even in the presence of (P.1)–(P.2) (justconsider a two-element model with mutual parthood, as in Figure 3,left). Still, (P.4_c) is sufficiently stronger than(P.4_a) and (P.4_b) to rule out all three patternsin Figure 2, and it obviously preserves the spirit of (P.4)—ifnot the letter. The second sort of answer is to weaken Supplementationby adjusting the consequent. There are various ways of doing this, themost natural of which appears to be the following:

Again, this principle is stronger than (P.4_a) and(P.4_b), since it rules out all patterns in Figure 2, and inM it is equivalent to (P.4). Indeed,(P.4_d) says, literally, that if something has a properpart, then it has at leasttwo disjoint parts, which Simons(1987: 27) takes to express the same intuition captured by (P.4). Yet(P.4_d) is logically weaker than (P.4), since it admits thenon-antisymmetric model in Figure 3, middle, and for that reason itmay be deemed more suitable in the context of theories that violate(P.3). Note also that (P.4_d) does not admit the symmetricmodel on Figure 3, left, so in a way it is stronger than(P.4_c). In another way, however, it is weaker, since itadmits the model in Figure 3, right, which (P.4_c) rules out(and which someone who thinks that, say, the clay is part of thestatue, but not vice versa, might want to retain).

Figure 3. More unsupplemented patterns.

There are other options, too. For instance, in some standardtreatments, the Supplementation principle (P.4) is formulated using‘PP’ also in the consequent:

InM this is once again equivalent to (P.4), but theequivalence depends on Reflexivity and Symmetry. Absent (P.1) or(P.2), (P.4′) is logically stronger. Yet again one may rely onthe alternative definition of ‘PP’ to obtain variants of(P.4′) that are stronger than (P.4_c) and weaker than(P.4). Similarly for (P.4_d), which may be further weakenedor strengthened by tampering with the parthood predicates occurring inthe antecedent and in the consequent.

3.2 Strong Supplementation and Extensionality

We may also ask the opposite question: Are there any stronger ways ofexpressing the supplementation intuition besides (P.4)? In classicalmereology, the standard answer is in the affirmative, the maincandidate being the following:

Intuitively, this says that if an objectfails to includeanother among its parts, then there must be a remainder, somethingthat makes up for the difference. It is easily seen that, givenM, (P.5) implies (P.4), so anyM-theory violating (P.4) willa fortioriviolate (P.5). For instance, on Whitehead's boundary-free theory ofextensive connection, a closed region is not part of its interior eventhough each part of the former overlaps the latter. More generally,the entailment holds as long as parthood is antisymmetric (see againFigure 3, center, for a non-antisymmetric counterexample). However,the converse is not true. The diagram in Figure 4 illustrates anM-model in which (P.4) is satisfied, since eachproper part counts as a supplement of the other; yet (P.5) isfalse.

Figure 4. A supplemented model violating StrongSupplementation.

The theory obtained by adding (P.5) to (P.1)–(P.3) is thus aproper extension ofMM. We label this stronger theoryEM, forExtensional Mereology, the attribute‘extensional’ being justified precisely by the exclusionof countermodels that, like the one in Figure 4, contain distinctobjects with the same proper parts. In fact, it is a theorem ofEM thatno composite objects with the sameproper parts can be distinguished:

(The analogue for ‘P’ is already provable inM, since P is reflexive and antisymmetric.) This goesfar beyond the intuition that lies behind the basic Supplementationprinciple (P.4). Does it go too far?

On the face of it, it is not difficult to envisage scenarios thatwould correspond to the diagram in Figure 4. For example, we may takex andy to be the sets {{z}, {z,w}} and {{w},{z,w}}, respectively(i.e., the ordered pairs ⟨z,w⟩ and⟨w,z⟩), interpreting ‘P’as the ancestral of the improper membership relation (i.e., of theunion of ∈ and =). But sets are abstract entities, and theancestral relation does not generally satisfy (P.4) (the singleton ofthe empty set, for instance, or the singleton of any urelement, wouldhave only one proper part on the suggested construal of‘P’). Can we also envisage similar scenarios in the domainof concrete, spatially extended entities, granting (P.4) in itsgenerality? Admittedly, it is difficult topicture twoconcrete objects mereologically structured as in Figure 4. It isdifficult, for example, to draw two extended objects composed of thesame proper parts because drawing somethingis drawing itsproper parts; once the parts are drawn, there is nothing left to bedone to get a drawing of the whole. Yet this only proves that picturesare biased towards (P.5). Are there any philosophical reasons toresist the extensional force of (P.5) beyond the domain of abstractentities, and in the presence of (P.4)?

Two sorts of reason are worth examining. On the one hand, it issometimes argued that sameness of proper parts is notsufficient for identity. For example, it is argued that: (i)two words can be made up of the same letters (Hempel 1953: 110;Rescher 1955: 10), two tunes of the same notes (Rosen and Dorr 2002:154), and so on; or (ii) the same flowers can compose a nice bunch ora scattered bundle, depending on the arrangements of the individualflowers (Eberle 1970: §2.10); or (iii) two groups can haveco-extensive memberships, say, the Library Committee and thePhilosophy Department football team (Simons 1987: 114; Gilbert 1989:273); or (iv) a cat must be distinguished from the correspondingamount of feline tissue, for the former can survive the annihilationof certain parts (the tail, for instance) whereas the latter cannot bydefinition (Wiggins 1968; see also Doepke 1982, Lowe 1989, Johnston1992, Baker 1997, Meirav 2003, Sanford 2003, and Crane 2012,interalia, for similar or related arguments). On the other hand, it issometimes argued that sameness of parts is notnecessary foridentity, as some entities may survive mereological change. If a catsurvives the annihilation of its tail, then the tailed cat (before theaccident) and the tailless cat (after the accident) are numericallythe same in spite of their having different proper parts (Wiggins1980). If any of these arguments is accepted, then clearly (27) is toostrong a principle to be imposed on the parthood relation. And since(27) follows from (P.5), it might be concluded thatEM is on the wrong track.

Let us look at these objections separately. Concerning the necessityaspect of mereological extensionality, i.e., the left-to-rightconditional in the consequent of (27),

it is perhaps enough to remark that the difficulty is not peculiar toextensional mereology. The objection proceeds from the considerationthat ordinary entities such as cats and other living organisms (andpossibly other entities as well, such as statues and ships) surviveall sorts of gradual mereological change. This a legitimate thought,lest one be forced into some form of “mereologicalessentialism” (Chisholm 1973, 1975, 1976; Plantinga 1975;Wiggins 1979). However, the same can be said of other types of changeas well: bananas ripen, houses deteriorate, people sleep at night andeat at lunch. How can we say that they are the same things, if theyare not quite the same? Indeed, (28) is essentially an instance of theidentity axiom schema

and it is well known that this axiom schema runs into trouble when‘=’ is given a diachronic reading. (See the entries onchange andidentity over time.) The problem is a general one. Whatever the solution, it willtherefore apply to the case at issue as well, and in this sense theabove-mentioned objection to (28) can be disregarded. For example, theproblem would dissolve immediately if the variables in (28) were takento range over four-dimensional entities whose parts may extend in timeas well as in space (Heller 1984, Lewis 1986b, Sider 2001), or ifidentity itself were construed as a contingent relation that may holdat some times or worlds but not at others (Gibbard 1975, Myro 1985,Gallois 1998). Alternatively, on a more traditional, three-dimensionalconception of material objects, the problem of change is oftenaccounted for by relativizing properties and relations to times,rewriting (ID) as

(This may be understood in various ways; see e.g. the papers inHaslanger and Kurtz 2006, Part III.) If so, then again the specificworry about (28) would dissolve, as the relativized version of (P.5)would only warrant the following variant of the conditional inquestion:

(See Thomson 1983, Simons 1987: §5.2, Masolo 2009, Giaretta andSpolaore 2011; see also Kazmi 1990 and Hovda 2013 for tensed versionsof this strategy.) The need to relativize parthood to time, andperhaps to other parameters such as space, possible worlds, etc., hasrecently been motivated also on independent grounds, from theso-called “problem of the many” (Hudson 2001) to materialconstitution (Bittner and Donnelly 2007), modal realism (McDaniel2004), vagueness (Donnelly 2009), relativistic spacetime (Balashov2008), or the general theory of location (Gilmore 2009, Donnelly2010). One way or the other, then, such revisions may be regarded asan indicator of the limited ontological neutrality of extensionalmereology. But their independent motivation also bears witness to thefact that controversies about (28) stem from genuine and fundamentalphilosophical conundrums and cannot be assessed by appealing to ourintuitions about the meaning of ‘part’.

The worry about the sufficiency aspect of mereological extensionality,i.e., the right-to-left conditional in the consequent of (27),

is more to the point. However, here too there are various ways ofresponding on behalf ofEM. Consider counterexample(i)—say, two words made up of the same letters, as in‘else’ and ‘seel’. If these are taken asword-types, a lot depends on how exactly one construes such thingsmereologically, and one might simply dismiss the challenge byrejecting, or improving on, the dime-store thought that word-types areletter-type composites (see abovead (14)). Indeed, if theywere, then word-types would not only violate extensionality, hence theStrong Supplementation principle (P.5); they would violate the basicSupplementation principle (P.4), since ‘seel’ (forinstance) would contain a proper part (the string ‘ee’)that consists of a single proper part (the letter ‘e’). Onthe other hand, if the items in question are taken as word-tokens,then presumably they are made up of distinct letter-tokens, so againthere is no violation of (29), hence no reason to reject (P.5) onthese grounds. Of course, we may suppose that one of the twoword-tokens is obtained from the other by rearranging the sameletter-tokens. If so, however, the issue becomes once again one ofdiachronic non-identity, with all that it entails, and it is notobvious that we have a counterexample to (29). (See Lewis 1991: 78f.)What if our letter-tokens are suitably arranged so as to form bothwords at the same time? For example, suppose they are arranged in acircle (Simons 1987: 114). In this case one might be inclined to saythat we have a genuine counterexample. But one may equally well insistthat we have got just one circular inscription that, curiously, can beread as two different words depending on where we start. Compare: Idraw a rabbit that to you looks like a duck. Have I thereby made twodrawings? I write ‘p’ on my office glass door; from theoutside you read ‘q’. Have I therefore produced twoletter-tokens? And what if Mary joins you and reads it upside down;have I also written the letter ‘b’? Surely then I havealso written the letter ‘d’, as my upside-down office mateJohn points out. This multiplication of entities seems preposterous.There is just one thing there, one inscription, and what it looks (ormean) to you or me or Mary or John is irrelevant to what that thingis. Similarly—it may be argued—there is just oneinscription in our example, a circular display of four letter-tokens,and whether we read it as an ‘else’-inscription or a‘seel’-inscription is irrelevant to its mereologicalstructure. (Varzi 2008)

Case (ii)—the flowers—is not significantly different. Thesame, concrete flowers cannot compose a nice bunch and a scatteredbundleat the same time. Similarly for many other cases ofthis sort that may come to mind, including much less frivolousprima facie counterexamples offered by the naturalsciences—from the different phases of matter (solids, liquids,and gases) to the different possibilities of chemical binding; seee.g. Harré and Llored (2011, 2013) and Sukumar (2013). (Not allcases are so easily dismissed, though. In particular, severalauthors—from Maudlin 1998 to Krause 2011—have argued thatthe world of quantum mechanics provides genuine type-(ii)counterexamples to extensionality. A full treatment of such argumentsgoes beyond the scope of this entry, but see e.g. Calosietal. 2011 and Calosi and Tarozzi 2014 for counter-arguments.)

Case (iii) is more delicate, as it depends on one's metaphysics ofsuch things as committees, teams, and groups generally. If one deniesthat the relevant structural relation is a genuine case of parthood(see Section 1,ad (11)), then of course the counterexamplemisfires. If, on the other hand, one takes groups to bebonafide mereological composites—and composites consisting ofenduring persons as opposed to, say, person-stages, as in Copp(1984)—then a lot depends on one's reasons to treat groups withco-extensive memberships as in fact distinct. Typically such reasonsare just taken for granted, as if the distinctness were obvious. Butsometimes informal arguments are offered to the effect that, say, thecoextensive Library Committee and football team must be distinguishedinsofar as they have different persistence conditions, or differentproperties broadly understood. For instance, the players of the teamcan change even though the Committee remains the same, or one groupcan be dismantled even though the other continues to operate, or onegroup has different legal obligations than the other, and so on (seee.g. Moltmann 1997). If so, then case (iii) becomes relevantly similarto case (iv). There, too, the intuition is that a living animal suchas a cat is something “over and above” the mere lump offeline tissue that constitutes its body—that they have differentsurvival conditions and, hence, different properties—so itappears that here we have a genuine counterexample to mereologicalextensionality (via Leibniz's Law). It is for similar reasons thatsome philosophers are inclined to treat a vase and the correspondinglump of clay as distinct in spite of their sharing the same properparts—possibly even the same improper parts, contrary to (P.3),as seen in Section 2.2.^[14] Two responses may nonetheless be offered in such cases on behalf ofEM (besides rejecting the intuition in question onthe basis of a specific metaphysics of persistence).

Focusing on (iv), the first response is to insist that, on the face ofit, a cat and the corresponding lump of feline tissue (or a statue andthe lump of clay that constitutes it) do not share the same properparts after all. For, on the one hand, if one believes that at leastone such thing,x, is part of the other,y, then itmust be a proper part; and insofar as nothing can be a proper part ofitself, it follows immediately that such things do not in factconstitute a counterexample to (29). (This would also follow fromSupplementation, as emphasized e.g. in Olson 2006, since theassumption thatx andy have the same proper partsentails that no part ofy is disjoint fromx, atleast so long as parthood is reflexive; but there is no need to invoke(P.4) here.) On the other hand, if one believes that neitherx nory is part of the other, then presumably thesame belief will also apply to some of their proper parts—say,the cat's tail and the corresponding lump of tissue. And if the tailis not part ofthat lump, then presumably it is also not partof the larger lump of tissue that constitutes the whole cat (asexplicitly acknowledged by some anti-extensionalists, e.g. Lowe 2001:148 and Fine 2003: 198, n. 5, though see Hershenov 2008 formisgivings). Thus, again, it would appear thatx andy do not have the same proper parts after all and do not,therefore, constitute a counterexample to (29). (For more on this lineof argument, see Varzi 2008.)

The second and more general response on behalf ofEMis that the appeal to Leibniz's law in this context is illegitimate.Let ‘Tibbles’ name our cat and ‘Tail’ itstail, and grant the truth of

There is, indeed, an intuitive sense in which the following is alsotrue:

However, this intuitive sense corresponds to ade dictoreading of the modality, where the definite description in (31) hasnarrow scope:

On this reading, (31) is hardly negotiable. Yet this is irrelevant inthe present context, for (31a) does not amount to an ascription of amodal property and cannot be used in connection with Leibniz's law.(Compare: 8 is necessarily even; the number of planets might have beenodd; hence the number of planets is not 8.) On the other hand,consider ade re reading of (31), where the definitedescription has wide scope:

On this reading, the appeal to Leibniz's law would be legitimate(modulo any concerns about the status of modal properties) and onecould rely on the truth of (30) and (31) (i.e., (31b)) to concludethat Tibbles is distinct from the relevant lump of feline tissue.However, there is no obvious reason why (31) should be regarded astrue on this reading. That is, there is no obvious reason to supposethat the lump of feline tissue that in the actual world constitutesTail and the rest of Tibbles's body—that lump of felinetissue that is now resting on the carpet—cannot survive theannihilation of Tail. Indeed, it would appear that any reason in favorof this claimvis-à-vis the truth of (30) would havetopresuppose the distinctness of the entities in question,so no appeal to Leibniz's law would be legitimate todetermine the distinctess (on pain of circularity). This isnot to say that the putative counterexample to (29) is wrong-headed.But it requires genuine metaphysical work to establish it and it makesthe rejection of extensionality, and with it the rejection of theStrong Supplementation principle (P.5), a matter of genuinephilosophical controversy. (Similar remarks would apply to anyargument intended to reject extensionality on the basis of competingmodal intuitions regarding the possibility of mereologicalrearrangement, rather than mereologicalchange, aswith the flowers example. On ade re reading, the claim thata bunch of flowers could not survive rearrangement of theparts—while the aggregate of the individual flowers composing itcould—must be backed up by a genuine metaphysical theory aboutthose entities. For more on this general line of defense on behalf of(29), see e.g. Lewis 1971: 204ff, Jubien 1993: 118ff, and Varzi 2000:291ff. See also King's 2006 reply to Fine 2003 for a more generaldiagnosis of the semantic mechanisms at issue here.)

3.3 Complementation

There is a way of expressing the supplementation intuition that iseven stronger than (P.5). It corresponds to the following thesis,which differs from (P.5) in the consequent:

This says that ify is not part ofx, there existssomething that comprises exactly those parts ofy that aredisjoint fromx—something we may call thedifference or relativecomplement betweenyandx. It is easily checked that this principle implies(P.5). On the other hand, the diagram in Figure 5 shows that theconverse does not hold: there are two parts ofy in thisdiagram that do not overlapx, namelyz andw, but there is nothing that consists exactly of such parts,so we have a model of (P.5) in which (P.6) fails.

Figure 5. A strongly supplemented model violatingComplementation.

Any misgivings about (P.5) may of course be raised against (P.6). Butwhat if we agree with the above arguments in support of (P.5)? Do theyalso give us reasons to accept the stronger principle (P.6)? Theanswer is in the negative. Plausible as it may initially sound, (P.6)has consequences that even an extensionalist may not be willing toaccept. For example, it may be argued that although the base and thestem of this wine glass jointly compose a larger part of the glassitself, and similarly for the stem and the bowl, there is nothingcomposed just of the base and the bowl (= the difference between theglass and the stem), since these two pieces are standing apart. Moregenerally, it appears that (P.6) would force one to accept theexistence of a wealth of “scattered” entities, such as theaggregate consisting of your nose and your thumbs, or the aggregate ofall mountains higher than Mont Blanc. And since V. Lowe (1953), manyauthors have expressed discomfort with such entitiesregardless of extensionality. (One philosopher who explicitlyaccepts extensionality but feels uneasy about scattered entities isChisholm 1987.) As it turns out, the extra strength of (P.6) istherefore best appreciated in terms of the sort of mereologicalaggregates that this principle would force us to accept, aggregatesthat are composed of two or more parts of a given whole. This suggeststhat any additional misgivings about (P.6), besides its extensionalimplications, are truly misgivings about matters of composition. Weshall accordingly postpone their discussion to Section 4, where weshall attend to these matters more fully. For the moment, let ussimply say that (P.6) is, on the face of it, not a principle that canbe added toM without further argument.

3.4 Atomism, Gunk, and Other Options

One last important family of decomposition principles concerns thequestion of atomism. Mereologically, an atom (or “simple”)is an entity with no proper parts, regardless of whether it ispoint-like or has spatial (and/or temporal) extension:

By definition of ‘PP’, all atoms are pairwise disjoint andcan only overlap things of which they are part. Are there any suchentities? And, if there are, is everything entirely made up of atoms?Is everything comprised of at least some atoms? Or is everything madeup of atomless “gunk”—as Lewis (1991: 20) callsit—that divides forever into smaller and smaller parts? Theseare deep and difficult questions, which have been the focus ofphilosophical investigation since the early days of philosophy andthroughout the medieval and modern debate on anti-divisibilism, up toKant's antinomies in theCritique of Pure Reason (see theentries onancient atomism andatomism from the 17th to the 20th century). Along with nuclear physics, they made their way into contemporarymereology mainly through Nicod's (1924) “geometry of thesensible world”, Tarski's (1929) “geometry ofsolids”, and Whitehead's (1929) theory of “extensiveconnection” mentioned in Section 3.1, and are now center stagein many mereological disputes at the intersection between metaphysicsand the philosophy of space and time (see, for example, Sider 1993,Forrest 1996a, Zimmerman 1996, Markosian 1998a, Schaffer 2003,McDaniel 2006, Hudson 2007a, Arntzenius 2008, and J. Russell 2008, andthe papers collected in Hudson 2004; see also Sobociński 1971 andEberle 1967 for some early treatments of these questions in the spiritof Leśniewski'sMereology and of Leonard and Goodman'sCalculus of Individuals, respectively). Here we shall confineourselves to a brief examination.

The two main options, to the effect that everything is ultimately madeup of atoms, or that there are no atoms at all, are typicallyexpressed by the following postulates, respectively:

(See e.g. Simons 1987: 42.) These postulates are mutuallyincompatible, but taken in isolation they can consistently be added toany standard mereological theoryX considered here.Adding (P.7) yields a correspondingAtomistic version,AX; adding (P.8) yields anAtomless version,ÃX. Since finitude together with theantisymmetry of parthood (P.3) jointly imply that mereologicaldecomposition must eventually come to an end, it is clear that anyfinite model ofM—anda fortiori ofany extension ofM—must be atomistic.Accordingly, an atomless mereologyÃX admitsonly models of infinite cardinality. An example of such a model,establishing the consistency of the atomless versions of most standardmereologies considered in this survey, is provided by the regular opensets of a Euclidean space, with ‘P’ interpreted asset-inclusion (Tarski 1935). On the other hand, the consistency of anatomistic theory is typically guaranteed by the trivial one-elementmodel (with ‘P’ interpreted as identity), though one canalso have models of atomistic theories that allow for infinitedomains. A case in point is provided by the closed intervals on thereal line, or the closed sets of a Euclidean space (Eberle 1970). Infact, it turns out that even whenX is as strong asthe full calculus of individuals, corresponding to the theoryGEM of Section 4.4, there isno purelymereological formula that says whether there are finitely orinfinitely many atoms, i.e., that is true in every finite model ofAX but in no infinite model (Hodges and Lewis1968).

Concerning Atomicity, it is also worth noting that (P.7) does notquite say that everything is ultimatelymade up of atoms; itmerely says that everythinghas atomic parts.^[16] As such it rules out gunky worlds, but one may wonder whether itfully captures the atomistic intuition. In a way, the answer is in theaffirmative. For, assuming Reflexivity and Transitivity, (P.7) isequivalent to the following

which is logically equivalent to

(adding a tautological conjunct), which is an instance of the generalschema

And (35) is the closest we can get to saying thatx iscomposed of the φs, i.e., all and only those entities that satisfythe given condition φ (in the present case: being an atomic partofx): every φ is part ofx, and any part ofx overlaps some φ. Indeed, provided the φs arepairwise disjoint, this is the standard definition of what it meansfor somethingx to be composed of the φs (van Inwagen1990: 29), and surely enough, if the φs are all atomic, then theyare pairwise disjoint. Thus, although (P.7) does notsay thateverything is ultimately composed of atoms, it implies it—atleast in the presence of (P.1) and (P.2). (Of course, non-standardmereologies in which either postulates is rejected may not warrant theinitial equivalence, so in such theories (33) would perhaps be abetter way to express the assumption of atomism.) In another way,however, (34) may still not be enough. For if the domain is infinite,(P.7) admits of models that seem to run afoul of the atomisticdoctrine. A simple example is a descending chain of decomposition thatnever “bottoms out”, as in Figure 6: herex isultimately composed of atoms, but the pattern of decomposition thatgoes down the right branch “looks” awfully similar to agunky precipice. For a concrete example (from Eberle 1970: 75),consider the set of all subsets of the natural numbers, with parthoodmodeled by the subset relation. In such a universe, each singleton{n} will count as an atom and each infinite set {m:m >n} will be “made up” of atoms.Yet the set of all such infinite sets will be infinitely descending.Models of this sort do not violate the idea that everything isultimately composed of atoms. However, they violate the idea thateverything can bedecomposed into its ultimate constituents.And this may be found problematic if atomism is meant to carry theweight of metaphysical grounding: as J. Schaffer puts it, theatomist's ontology seems to drain away “down a bottomlesspit” (2007: 184); being is “infinitely deferred, neverachieved” (2010: 62). Are there any ways available to theatomist to avoid this charge? One option would simply be to requirethat every model be finite, or that it involve only a finite set ofatoms. Yet such requirements, besides being philosophically harsh andcontroversial even among atomists, cannot be formally implemented infirst-order mereology, the former for well-known model-theoreticreasons and the latter in view of the above-mentioned result by Hodgesand Lewis (1968). The only reasonable option would seem to be agenuine strengthening of Atomicity in the spirit of what Cotnoir(2013c) calls “superatomism”. Given any objectx,(P.7) guarantees the existence ofsome parthood chain thatbottoms out at an atom. Superatomicity would require thatevery parthood chain ofx bottoms out—aproperty that fails in the model of Figure 6. At the moment, such waysof strengthening (P.7) have not been explored. However, in view of theconnection between classical mereology and Boolean algebras (seebelow, Section 4.4), mathematical models for superatomisticmereologies may be recovered from the work on superatomic Booleanalgebras initiated by Mostowski and Tarski (1939) and eventuallysystematized in Day (1960). (A Boolean algebra is superatomic if andonly if every subalgebra is atomic, as with the algebra generated bythe finite subsets of a given set; see Day 1967 for an overview.) Seealso Shiver (2015) for ways of strengthening (P.7) in the context ofstronger mereologies such asGEM (Section 4.4), orwithin theories formulated in languages enriched with set variables orplural quantification.

Figure 6. An infinitely descending atomistic model.(The ellipsis indicates repetition of the branching pattern.)

Another thing to notice is that, independently of their philosophicalmotivations and formal limitations, atomistic mereologies admit ofsignificant simplifications in the axioms. For instance,AEM can be simplified by replacing (P.5) and (P.7)with

which in turns implies the following atomistic variant of theextensionality thesis (27):

Thus, any atomistic extensional mereology is truly“hyperextensional” in Goodman's (1958) sense: things builtup from exactly the same atoms are identical. In particular, if thedomain of anAEM-model has only finitely many atoms,the domain itself is bound to be finite. An interesting question,discussed at some length in the late 1960's (Yoes 1967, Eberle 1968,Schuldenfrei 1969) and taken up more recently by Simons (1987: 44f)and Engel and Yoes (1996), is whether there are atomless analogues of(27′). Is there any predicate that can play the role of‘A’ in an atomless mereology? Such a predicate wouldidentify the “base” (in the topological sense) of thesystem and would therefore enable mereology to cash out Goodman'shyperextensional intuitions even in the absence of atoms. The questionis therefore significant especially from a nominalistic perspective,but it has deep ramifications also in other fields (e.g., inconnection with a Whiteheadian conception of space according to whichspace itself contains no parts of lower dimensions such as points orboundary elements; see Forrest 1996a, Roeper 1997, and Cohn and Varzi2003). In special cases there is no difficulty in providing a positiveanswer. For example, in theÃEM modelconsisting of the open regular subsets of the real line, the openintervals with rational end points form a base in the relevant sense.It is unclear, however, whether a general answer can be given thatapplies to any sort of domain. If not, then the only option wouldappear to be an account where the notion of a “base” isrelativized to entities of a given sort. In Simons's terminology, wecould say that the ψ-ers form a base for the φ-ers if and onlyif the following variants of (P.5′) and (P.7) are satisfied:

An atomistic mereology would then correspond to the limit case where‘ψ’ is identified with the predicate ‘A’for every choice of ‘φ’. In an atomless mereology, bycontrast, the choice of the base would depend each time on the levelof “granularity” set by the relevant specification of‘φ’.

Concerning atomless mereologies, one more remark is in order. For justas (P.7) is too weak to rule out unpleasant atomistic models, so toothe formulation of (P.8) may be found too weak to capture the intendedidea of a gunky world. For one thing, as it stands (P.8) presupposesAntisymmetry. Absent (P.3), the symmetric two-element pattern inFigure 3, left, would qualify as atomless. To rule out such modelsindependently of (P.3), one should understand (P.8) in terms of thestronger notion of ‘PP’ given in (20′), i.e.,

Likewise, note that the pattern in Figure 2, middle, will qualify as amodel of (P.8) unless Supplementation is assumed, though again such apattern does not quite correspond to what philosophers ordinarily havein mind when they talk about gunk. It is indeed an interestingquestion whether Supplementation (or perhaps Quasi-supplementation, assuggested by Gilmore 2016) is in some sense presupposed by theordinary concept of gunk. To the extent that it is, however, thenagain one may want to be explicit, in which case the relevantaxiomatization may be simplified. For instance,ÃMM can be simplified by merging (P.4) and(P.8) into a single axiom:

There is, in addition, another, more important sense in which (P.8)may seem too week. After all, infinite divisibility is loose talk.Given (P.8) (and also given (P.8′)), gunk may have denumerablymany, possibly continuum-many parts; but can it have more? Is there anupper bound on the cardinality on the number of pieces of gunk? Shouldit be allowed that forevery cardinal number there may bemore than that many pieces of gunk? (P.8) is silent on thesequestions. Yet these are certainly aspects of atomless mereology thatdeserve scrutiny. It may even be thought that the world is not meregunk but “hypergunk”, as Nolan (2004: 305) callsit—gunk such that, for any set of its parts, there is a set ofstrictly greater cardinality containing only its parts. It is notknown whether such a theory is consistent (though Nolan conjecturedthat a model can be constructed using the resources of standard settheory with Choice and urelements together with some inaccessiblecardinal axioms), and even if it were, some philosophers wouldpresumably be inclined to regard hypergunk as a mere logicalpossibility (Hazen 2004). Nonetheless the question is indicative ofthe sort of leeway that (P.8) leaves, and that one might want toregiment.

So much for the two main options, corresponding to atomicity andatomlessness. What about theories that lie somewhere between these twoextremes? Surely it may be held that there are atoms, though noteverything need be made up of atoms; or it may be held that there isatomless gunk, though not everything need be gunky. (The latterposition is defended e.g. by Zimmerman 1996.) Formally, thesepossibilities can be put again in terms of suitable restrictions on(P.7) and (P.8), by requiring that the relevant conditions holdexclusively of certain entities:

And the options in question would correspond to endorsing(P.7_φ) or (P.8_φ) for specific values of‘φ’. At present, no thorough formal investigation hasbeen pursued in this spirit (though see Masolo and Vieu 1999 andHudson 2007b). Yet the issue is particularly pressing when it comes tothe mereology of the spatio-temporal world. For example, it is aplausible thought that while the question of atomism may be left openwith regard to the mereological structure of material objects (pendingempirical findings from physics), one might be able to settle it(independently) with regard to the structure of space-time itself.This would amount to endorsing a version of either(P.7_φ) or (P.8_φ) in which‘φ’ is understood as a condition that is satisfiedexclusively by regions of space-time. Some may find it hard toconceive of a world in which an atomistic space-time is inhabited byentities that can be decomposed indefinitely (pace McDaniel2006), in which case accepting (P.7_φ) for regions wouldentail the stronger principle (P.7). However, (P.8_φ)would be genuinely independent of (P.8) unless it is assumed thatevery mereologically atomic entity should be spatially unextended, anassumption that is not part of definition (32) and that has beenchallenged by van Inwagen (1981) and Lewis (1991: 32) (and extensivelydiscussed in recent literature; see e.g. MacBride 1998, Markosian1998a, Scala 2002, J. Parsons 2004, Simons 2004, Tognazzini 2006,Braddon-Mitchell and Miller 2006, Hudson 2006a, McDaniel 2007, Sider2007, Spencer 2010). More generally, such issues depend on the broaderquestion of whether the mereological structure of a thing shouldalways “mirror” or be in perfect “harmony”with that of its spatial or spatio-temporal receptacle, a questionaddressed in J. Parsons (2007) and Varzi (2007: §3.3) and furtherdiscussed in Schaffer (2009), Uzquiano (2011) and Saucedo (2011). (Formore on this, see the entrylocation and mereology.)

Similar considerations apply to other decomposition principles thatmay come to mind at this point. For example, one may consider arequirement to the effect that ‘PP’ forms a denseordering, as already Whitehead (1919) had it:

As a general decomposition principle, (P.9) might be deemed toostrong, especially in an atomistic setting. (Whitehead's own theoryassumes Atomlessness.) However, it is plausible to suppose that (P.9)should hold at least with respect to the domain of spatio-temporalregions, regardless of whether these are construed as atomless gunk oras aggregates of spatio-temporal atoms. For more on this, seeEschenbach and Heydrich (1995) and Varzi (2007: §3.2).

Finally, it is worth noting that if one assumed the existence of a“null item” that is part of everything, corresponding tothe postulate

then such an entity would perforce be an atom. Accordingly, noatomless mereology is compatible with this assumption. But it bearsemphasis that (P.10) is at odds with a host of other theories as well.For, given (P.10), the Antisymmetry axiom (P.3) will immediatelyentail that the atom in question is unique, while the Reflexivityaxiom (P.1) will entail that it overlaps everything, hence thateverything overlaps everything. This means that under such axioms theSupplementation principle (P.4) cannot be satisfied except in modelswhose domain includes a single element. Indeed, this is also true ofthe weaker Quasi-supplementation principle, (P.4_d). Itfollows, therefore, that the result of adding (P.10) to any theory atleast as strong as (P.1) + (P.3) + (P.4_d), and a fortioritoMM and any extension thereof, will immediatelycollapse to triviality in view of the following corollary:

‘Triviality’ may strike one as the wrong word here. Afterall, there have been and continue to be philosophers who holdradically monistic ontologies—from the Eleatics (Rea 2001) toSpinoza (J. Bennett 1984) all the way to contemporary authors such asHorgan and Potrč (2000), whose comparative ontological parsimonyresults in the thesis that the whole cosmos is but one huge extendedatom, an enormously complex but partless “blobject”. Forall we know, it may even be that the best ontology for quantummechanics, if not for Newtonian mechanics, consists in a lonely atomspeeding through configuration-space (Albert 1996). None of this istrivial. However, none of this corresponds to fully endorsing (36),either. For such philosophical theories do not, strictly speaking,assert the existence of one singleentity—which is what(36) says—but only the existence of a single material substancealong with entities of other kinds, such as properties orspatio-temporal regions. In other words, they only endorse a sortallyrestricted version of (36). In its full generality, (36) is muchstronger and harder to swallow, and most mereologists would ratheravoid it. The bottom line, therefore, is that theories endorsing(P.10) are likely to be highly non-standard,pace Carnap'spersuasion that the null item would be a “natural and convenientchoice” for certain purposes (such as providing a referent forall defective descriptions; see 1947: 37). A few authors have indeedgone that way, beginning with Martin (1943, 1965), who rejectsunrestricted Reflexivity and characterizes the null item as“that which is not part of itself”. Other notableexceptions include Bunt (1985) and Meixner (1997) and, more recently,Hudson (2006) and Segal (2014), both of whom express sympathy for thenull individual at the cost of foregoing unrestricted(Quasi-)Supplementation. See also Priest (2014a and 2014b: §6.13)and Cotnoir and Weber (2014), who avoid (36) through a paraconsistentrecasting of the underlying logic. Still another option would be totreat the null item as a mere algebraic “fiction” and toamend the entire mereological machinery accordingly, carefullydistinguishing between trivial cases of parthood and overlap (thosethat involve the infectious null item) and genuine, non-trivialones:

The basicM-axioms need not be affected by thisdistinction. But stronger principles such as Supplementation couldgive way to their “genuine” counterparts, as in

and this would suffice to block the inference to (36) while keepingwith the spirit of standard mereology. This strategy is not uncommon,especially in the mathematically oriented literature (see e.g. Mormann2000, Forrest 2002, Pontow and Schubert 2006), and we shall brieflyreturn to it in Section 4.4 below. In general, however, mereologiststend to side with traditional wisdom and steer clear of (P.10)altogether.

4. Composition Principles

Let us now consider the second way of extendingMmentioned at the beginning of Section 3. Just as we may want toregiment the behavior of P by means of decomposition principles thattake us from a whole to its parts, we may look at compositionprinciples that go in the opposite direction—from the parts tothe whole. More generally, we may consider the idea that the domain ofthe theory ought to be closed under mereological operations of varioussorts: not only mereological sums, but also products, differences, andmore.

4.1 Upper Bounds

Conditions on composition are many. Beginning with the weakest, onemay consider a principle to the effect that any pair of suitablyrelated entities must underlap, i.e., have an upper bound:

Exactly how ‘ξ’ should be construed is, of course, animportant question by itself—a version of what van Inwagen(1987, 1990) calls the “Special Composition Question”. Anatural choice would be to identify ξ with mereological overlap,the rationale being that such a relation establishes an important tiebetween what may count as two distinct parts of a larger whole. As weshall see (Section 4.5), with ξ so construed (P.11_ξ)is indeed rather uncontroversial. By contrast, the most liberal choicewould be to identify ξ with the universal relation, in which case(P.11_ξ) would reduce to its consequent and assert theexistence of an upper bound forany pair of entitiesx andy. An axiom of this sort was used, forinstance, in Whitehead's (1919, 1920) mereology of events.^[17] In any case, and regardless of any specific choice, it is apparentthat (P.11_ξ) does not express a strong condition oncomposition, as the consequent is trivially satisfied in any domainthat includes a universal entity of which everything is part, or anyentity sufficiently large to include bothx andy asparts regardless of how they are related.

4.2 Sums

A stronger condition would be to require that any pair of suitablyrelated entities must have aminimalunderlapper—something composed exactly of their parts andnothing else. This requirement is sometimes stated by saying that anysuitable pair must have a mereological “sum”, or “fusion”,^[18] though it is not immediately obvious how this requirement should beformulated. Consider the following definitions:

(‘S_izxy’ may be read:‘z is a sum_i ofx andy’. The first notion is found e.g. in Eberle 1967,Bostock 1979, and van Benthem 1983; the second in Tarski 1935 andLewis 1991; the third in Needham 1981, Simons 1987, and Casati andVarzi 1999.) Then, for eachi ∈ {1, 2, 3}, one couldextendM by adding a corresponding axiom as follows,where again ξ specifies a suitable binary condition:

In a way, (P.12_ξ,1) would seem the obvious choice,corresponding to the idea that a sum of two objects is just a minimalupper bound of those objects relative to P (a partial ordering).However, this condition may be regarded as too weak to capture theintended notion of a mereological sum. For example, with ξconstrued as overlap, (P.12_ξ,1) is satisfied by themodel of Figure 7, left: herez is a minimal upper bound ofx andy, yetz hardly qualifies as a sum“made up” ofx andy, since its partsinclude also a third, disjoint itemw. Indeed, it is a simplefact about partial orderings that among finite models(P.12_ξ,1) is equivalent to (P.11_ξ), hencejust as weak.

By contrast, (P.12_ξ,2) corresponds to a notion of sumthat may seem too strong. In a way, it says—literally—thatany pair of suitably ξ-related entitiesx andycompose something, in the sense already discussed in connection with(35): they have an upper bound all parts of which overlap eitherx ory. Thus, it rules out the model on the left ofFigure 7, precisely becausew is disjoint from bothx andy. However, it also rules out the model on theright, which depicts a situation in whichz may be viewed asan entity truly made up ofx andy insofar as it isultimately composed of atoms to be found either inx or iny. Of course, such a situation violates the StrongSupplementation principle (P.5), but that's precisely the sense inwhich (P.12_ξ,2) may seem too strong: ananti-extensionalist might want to have a notion of sum that does notpresuppose Strong Supplementation.

The formulation in (P.12_ξ,3) is the natural compromise.Informally, it says that for any pair of suitably ξ-relatedentitiesx andy there is something that overlapsexactly those things that overlap eitherx ory.This is strong enough to rule out the model on the left, but weakenough to be compatible with the model on the right. Note, however,that if the Strong Supplementation axiom (P.5) holds, then(P.12_ξ,3) is equivalent to (P.12_ξ,2).Moreover, it turns out that if the stronger Complementation axiom(P.6) holds, then all of these principles are trivially satisfied inany domain in which there is a universal entity: in that case,regardless of ξ, the sum of any two entities is just the complementof the difference between the complement of one minus the other. (Suchis the strength of (P.6), a genuine cross between decomposition andcomposition principles.)

Figure 7. A sum₁ that is not asum₃, and a sum₃ that is not a sum₂.

The intuitive idea behind these principles is in fact best appreciatedin the presence of (P.5), hence extensionality, for in that case therelevant sums must be unique. Thus, consider the following definition,wherei ∈ {1, 2, 3} and ‘℩’ is thedefinite descriptor):

In the context ofEM, each(P.12_ξ,i) would then imply that thecorresponding sum operator has all the “Boolean”properties one might expect (Breitkopf 1978). For example, as long asthe arguments satisfy the relevant condition ξ,^[20] each +_i is idempotent, commutative, andassociative,

and well-behaved with respect to parthood:

(Note that (47) would warrant defining ‘P’ in terms of‘+_i’, treated as a primitive. Fori=3, this was actually the option endorsed in Leonard 1930:187ff.)

Indeed, here there is room for further developments. For example, justas the principles in (P.12_ξ,i) assert theexistence of a minimal underlapper for any pair of suitably relatedentities, one may at this point want to assert the existence of amaximal overlapper, i.e., not a “sum” but a“product” of those entities. In the present context, suchan additional claim can be expressed by the following principle:

where

and ‘ξ’ is at least as strong as ‘O’(unless one assumes the Bottom principle (P.10)). InEM one could then introduce the corresponding binaryoperator,

and it turns out that, again, such an operator would have theproperties one might expect. For example, as long as the argumentssatisfy the relevant condition ξ, × is idempotent,commutative, and associative, and it interacts with each+_i in conformity with the usual distributionlaws:

Now, obviously (P.13_ξ) does not qualify as a compositionprinciple in the main sense that we have been considering here, i.e.,as a principle that yields a whole out of suitably ξ-related parts.Still, in a derivative sense it does. It asserts the existence of awhole composed of parts that areshared by suitably relatedentities. Be that as it may, it should be noted that such anadditional principle is not innocuous unless ‘ξ’expresses a condition stronger than mere overlap. For instance, wehave said that overlap may be a natural option if one is unwilling tocountenance arbitrary scattered sums. It would not, however, be enoughto avoid embracing scattered products. Think of two C-shaped objectsoverlapping at both extremities; their sum would be a one-pieceO-shaped object, but their product would consist of two disjoint,separate parts (Bostock 1979: 125). Moreover, and independently, ifξ were just overlap, then (P.13_ξ) would beunacceptable for anyone unwilling to embrace mereologicalextensionality. For it turns out that the Strong Supplementationprinciple (P.5) would then be derivable from the weakerSupplementation principle (P.4) using only the partial ordering axiomsfor ‘P’ (in fact, using only Reflexivity and Transitivity;see Simons 1987: 30f). In other words, unless ‘ξ’expresses a condition stronger than overlap,MMcum (P.13_ξ) wouldautomaticallyincludeEM. This is perhaps even more remarkable, foron first thought the existence of products would seem to have nothingto do with matters ofdecomposition, let alone adecomposition principle that is committed to extensionality. On secondthought, however, mereological extensionality is really adouble-barreled thesis: it says that two wholes cannot be decomposedinto the same proper parts but also, by the same token, that twowholes cannot be composed out of the same proper parts. So it is notentirely surprising that as long as proper parthood is well behaved,as per (P.4), extensionality might pop up like this in the presence ofsubstantive composition principles. (It is, however, noteworthy thatit already pops up as soon as (P.4) is combined with a seeminglyinnocent thesis such as the existence of products, so theanti-extensionalist should keep that in mind.)

4.3 Infinitary Bounds and Sums

One can get even stronger composition principles by consideringinfinitary bounds and sums. For example, (P.11_ξ) can begeneralized to a principle to the effect that any non-emptyset of (two or more) entities satisfying a suitable conditionψ has an upper bound. Strictly speaking, there is a difficulty inexpressing such a principle in a standard first-order language. Someearly theories, such as those of Tarski (1929) and Leonard and Goodman(1940), require explicit quantification over sets (see Niebergall2009a, 2009b; Goodman produced a set-free version of the calculus ofindividuals in 1951). Others, such as Lewis's (1991), resort to themachinery of plural quantification of Boolos (1984). One can, however,avoid all this and achieve a sufficient degree of generality byrelying on an axiom schema where sets are identified by predicates oropen formulas. Since an ordinary first-order language has adenumerable supply of open formulas, at most denumerably many sets (inany given domain) can be specified in this way. But for most purposesthis limitation is negligible, as normally we are only interested inthose sets of objects that we are able to specify. Thus, for mostpurposes the following axiom schema will do, where ‘φ’is any formula in the language and ‘ψ’ expresses thecondition in question:

(The first conjunct in the antecedent is simply to guarantee that‘φ’ picks out anon-empty set, while in theconsequent the variable ‘z’ is assumed not tooccur free in ‘ψ’.) The three binary sum axiomscorresponding to the schema in (P.12_ξ,i) can bestrengthened in a similar fashion as follows:

where

(Here, ‘S_izφw’ may beread: ‘z is a sum_i of everyw such that φw’ and, again,‘z’ and ‘v’ are assumed notto occur free in φ; similar restrictions will apply below.) Thus,each (P.15_ψ,i) says that if there are someφ-ers, and if every φ-er satisfies condition ψ, then theφ-ers have a sum of the relevant type. It can be checked that eachvariant of (P.15_ψ,i) includes thecorresponding finitely principle (P.12_ψ,i) asa special case, taking ‘φw’ to be the formula‘w=x ∨w=y’ and‘ψw’ the condition‘(w=x → ξwy) ∧(w=y → ξxw)’. And, again, itturns out that in the presence of Strong Supplementation,(P.15_ψ,2) and (P.15_ψ,3) are equivalent.

One could also consider here a generalized version of the Productprinciple (P.13_ξ), asserting the conditional existenceof a maximal common overlapper—a common “nucleus”,in the terminology of Leonard and Goodman (1940)—for anynon-empty set of entities satisfying a suitable condition. Adaptingfrom Goodman (1951: 37), such a principle could be stated asfollows:

where

and ‘ψw’ expresses a condition at least asstrong as ‘∀x(φx →Owx)’ (again, unless one assumes the Bottom principle(P.10)). This principle includes the finitary version(P.13_ξ) as a special case, taking‘φw’ and ‘ψw’ asabove, so the remarks we made in connection with the latter applyhere. An additional remark, however, is in order. For there is a sensein which (P.16_ψ) might be thought to be redundant inthe presence of the infinitary sum principles in(P.15_ψ,i). Intuitively, a maximal commonoverlapper (i.e., a product) of a set of overlapping entities issimply a minimal underlapper (a sum) of their common parts; that isprecisely the sense in which a product principle qualifies as acomposition principle. Thus, intuitively, each of the infinitary sumprinciples above should have a substitution instance that yields(P.16_ψ) as atheorem, at least when‘ψw’ is as strong as indicated. However, itturns out that this is not generally the case unless one assumesextensionality. In particular, it is easy to see that(P.15_ψ,3) does not generally imply(P.16_ψ), for it may not even imply the binary version(P.13_ξ). This can be verified by taking‘ξxy’ and ‘ψw’ toexpressjust the requirement of overlap, i.e., the conditions‘Oxy’ and‘∀x(φx → Owx)’,respectively, and considering again the non-extensional modeldiagrammed in Figure 4. In that model,x andy donot have a product, since neither is part of the other and neitherz norw includes the other as a part. Thus,(P.13_ξ) fails, which is to say that(P.16_ψ) fails when ‘φ’ picks out theset {x, y}; yet (P.15_ψ,3) holds, for bothz andw are things that overlap exactly those thingsthat overlap some common part of the φ-ers, i.e., ofxandy.

In the literature, this fact has been neglected until recently (Pontow2004). It is, nonetheless, of major significance for a fullunderstanding of (the limits of) non-extensional mereologies. As weshall see in the next section, it is also important when it comes tothe axiomatic structure of mereology, including the axiomatics of themost classical theories.

4.4 Unrestricted Composition

The strongest versions of all these composition principles areobtained by asserting them as axiom schemas holding foreverycondition ψ, i.e., effectively, by foregoing any reference toψ altogether. Formally this amounts in each case to dropping thesecond conjunct of the antecedent, i.e., to asserting the schemaexpressed by the relevant consequent with the only proviso that thereare some φ-ers. In particular, the following schema is theunrestricted version of (P.15_ψ,i), to theeffect that every specifiable non-empty set of entities has asum_i:

Fori=3, the extension ofEM obtained byadding every instance of this schema has a distinguished pedigree andis known in the literature asGeneral Extensional Mereology,orGEM. It corresponds to the classical systems ofLeśniewski and of Leonard and Goodman, modulo the underlyinglogic and choice of primitives. The same theory can be obtained byextendingEM with (P.15₂) instead, for inthe presence of extensionality the two schemas are equivalent. Indeed,it turns out that the latter axiomatization is somewhat redundant:given just Transitivity and Supplementation, UnrestrictedSum₂ entails all the other axioms, i.e.,GEM is the same theory as (P.2) + (P.4) +(P.15₂). By contrast, extendingEM with(P.15₁) would result in a weaker theory (Figure 8), thoughone can still get the full strength ofGEM with thehelp of additional axioms. For example, Hovda (2009) shows that thefollowing will do:

(in which case, again, Transitivity and Supplementation would suffice,i.e.,GEM = (P.2) + (P.4) + (P.15₁) +(P.17)). For other ways of axiomatizatizing ofGEMusing (P.15₁), see e.g. Link (1983) and Landman (1991)(and, again, Hovda 2009). See also Sharvy (1980, 1983), where theextension ofM obtained by adding (P.15₁)is called a “quasi-mereology”.

Figure 8. A model ofEM +(P.15₂) but not ofGEM.

GEM is a powerful theory, and it was meant to be soby its nominalistic forerunners, who were thinking of mereology as agood alternative to set theory. It is also decidable (Tsai 2013a),whereas for example,M,MM, andEM, and many extensions thereof turn out to beundecidable. (For a comprehensive picture of decidability inmereology, see also Tsai 2009, 2011, 2013b.) Just how powerful isGEM? To answer this question, let us focus on theclassical formulation based on (P.15₃) and consider thefollowing generalized sum operator:

Then (P.15₃) and (P.5) can be simplified to a single axiomschema:

and we can introduce the following definitions:

Note that (55) and (56) yield the binary operators defined in(40₃) and (49) as special cases. Moreover, inGEM the General ψ-Product principle(P.16_ψ) is also derivable as a theorem, with‘ψ’ as weak as the requirement of mutual overlap, andwe can introduce a corresponding functor as follows:

The full strength of the theory can then be appreciated by consideringthat its models are closed under each of these functors, modulo thesatisfiability of the relevant conditions. To be explicit: thecondition ‘DzU’ is unsatisfiable, soUcannot have a complement. Likewise products are defined only foroverlappers and differences only for pairs that leave a remainder.Otherwise, however, (55)–(60) yield perfectly well-behavedfunctors. Since such functors are the natural mereological analoguesof the familiar set-theoretic operators, with ‘σ’ inplace of set abstraction, it follows that the parthood relationaxiomatized byGEM has essentially the sameproperties as the inclusion relation in standard set theory. Moreprecisely, it is isomorphic to the inclusion relation restricted tothe set of all non-empty subsets of a given set, which is to say acomplete Boolean algebra with the zero element removed—a resultthat can be traced back to Tarski (1935: n. 4) and first proved inGrzegorczyk (1955: §4).^[22]

There are other equivalent formulations ofGEM thatare noteworthy. For instance, it is a theorem of every extensionalmereology that parthood amounts to inclusion of overlappers:

This means that in an extensional mereology ‘O’ could beused as a primitive and ‘P’ defined accordingly, as inGoodman (1951), and it can be checked that the theory defined bypostulating (61) together with the Unrestricted Sum principle(P.15₃) and the Antisymmetry axiom (P.3) is equivalent toGEM (Eberle 1967). Another elegant axiomatization ofGEM, due to an earlier work of Tarski (1929),^[23] is obtained by taking just the Transitivity axiom (P.2) together withthe Sum₂-analogue of the Unique Unrestricted Sum axiom(P.18). By contrast, it bears emphasis that the result of adding(P.15₃) toMM isnot equivalenttoGEM, contrary to the “standard”characterization given by Simons (1987: 37) and inherited by muchliterature that followed, including Casati and Varzi (1999) and thefirst edition of this entry.^[24] This follows immediately from Pontow's (2004) counterexamplementioned at the end of Section 4.3, since the non-extensional modelin Figure 4 satisfies (P.15₃), and was first noted inPietruszczak (2000, n. 12). More generally, in Section 4.2 we havementioned that in the presence of the binary Product postulate(P.13_ξ), with ξ construed as overlap, the StrongSupplementation axiom (P.5) follows from the weaker Supplementationaxiom (P.4). However, the model shows that the postulate is notimplied by (P.15₃) any more than it is implied by itsrestricted variants (P.15_ψ,3). Apart from its relevanceto the proper characterization ofGEM, this result isworth stressing also philosophically, for it means that(P.15₃) is by itself too weak to generate a sum out of anyspecifiable set of objects. In other words, fully unrestrictedcomposition calls for extensionality, on pain of giving upboth supplementation principles. The anti-extensionalistshould therefore keep that in mind. (On the other hand, a friend ofextensionality may welcome this result as an argument in favor ofadopting (P.15₂) instead of (P.15₃), for we havealready noted that such a way of sanctioning unrestricted compositionturns out to be enough, inMM, to entail StrongSupplementation along with the existence of all products and, withthem, of all sums; see Varzi 2009, with discussion in Rea 2010 andCotnoir 2016 . In this sense, the standard way of characterizingcomposition given in (35), on which (P.15₂) is based, isnot as neutral as it might seem. On this and related matters,indicating that the axiomatic path to “classical extensionalmereology” is no straightforward business, see also Hovda 2009and Gruszczyński and Pietruszczak 2014.)

Would we get afull Boolean algebra by supplementingGEM with the Bottom axiom (P.10), i.e., by positingthe mereological equivalent of the empty set? One immediate way toanswer this question is in the affirmative, but only in a trivialsense: we have already seen in Section 3.4 that, under the axioms ofMM, (P.10) only admits of degenerate one-elementmodels. Such is the might of the null item. On the other hand, supposewe rely on the “non-trivial” notions of genuine parthoodand genuine overlap defined in (37)–(38). And suppose weintroduce a corresponding family of “non-trivial”operators for sum, product, etc. Then it can be shown that the theoryobtained fromGEM by adding (P.10) and replacing(P.5) and (P.15₃) with the following non-trivialvariants:

is indeed a full Boolean algebra under the new operators (Pontow andSchubert 2006). This shows that, mathematically, mereology does indeedhave all the resources to stand as a robust and yet nominalisticallyacceptable alternative to set theory, the real source of differencebeing the attitude towards the nature of singletons (as alreadyemphasized by Leśniewski 1916 and eventually clarified in Lewis1991). As already mentioned, however, from a philosophical perspectivethe Bottom axiom is by no means a favorite option. The null item wouldhave to exist “nowhere and nowhen” (as Geach 1949: 522 putit), or perhaps “everywhere and everywhen” (as in Efirdand Stoneham 2005), and that is hard to swallow. One may try tojustify the gulp in various ways, perhaps by construing the null itemas a non-existing individual (Bunge 1966), as a Meinongian objectlacking all nuclear properties (Giraud 2013), as an Heideggeriannothing that nothings (Priest 2014a and 2014b: §6.13), or as theultimate incarnation of divine omnipresent simplicity (Hudson 2006b,2009). But few philosophers would be willing to go ahead and swallowfor the sole purpose of neatening up the algebra.

Finally, it is worth recalling that the assumption of atomismgenerally allows for significant simplifications in the axiomatics ofmereology. For instance, we have already seen thatAEM can be simplified by subsuming (P.5) and (P.7)under a single Atomistic Supplementation principle, (P.5′).Likewise, it is easy to see thatGEM is compatiblewith the assumption of Atomicity (just consider the one-elementmodel), and the resulting theory has some attractive features. Inparticular, it turns out thatAGEM can be simplifiedby replacing any of the Unrestricted Sum postulates in(P.15_i) with the more perspicuous

which asserts, for any non-empty set of entities, the existence of asum_i composed exactly of all the atoms thatcompose those entities. Indeed,GEM also provides theresources to overcome the limits of the Atomicity axiom (P.7)discussed in Section 3.4. For, on the one hand, the infinitelydescending chain depicted in Figure 6 is not a model ofAGEM, since it is missing all sorts of sums. On theother, inGEM one can actually strenghten (P.7) insuch a way as to require explicitly that everything be made entirelyatoms, as in

(See Shiver 2015.) It should be noted, however, that such advantagescome at a cost. For regardless of the number of atoms one begins with,the axioms ofAGEM impose a fixed relationshipbetween that number, κ, and the overall number of things, whichis going to be 2^κ–1. As Simons (1987: 17)pointed out, this means that the possible cardinality of anAGEM-model is restricted. There are models with 1, 3,7, 15, 2^ℵ₀, and many more cardinalities,but no models with, say, cardinality 2, 4, 6, orℵ₀. Obviously, this is not a consequence of(P.15_i) alone but also of the other axioms ofGEM (the unsupplemented pattern in Figure 2, left,satisfies (P.15_i) for eachi and has 2elements, and can be expanded at will to get models of any finitecardinality, or indefinitely to get a model with ℵ₀elements, as in Figure 2, center; see also Figure 8 for a supplementednon-filtrated model of (P.15₁) with 4 elements and Figure7, right, for a supplemented non-extensional model of(P.15₃) with 6 elements). Still, it is a fact that in thepresence of such axioms each (P.15_i) rules out alarge number of possibilities. In particular, every finite model ofAGEM—hence ofGEM—isbound to involve massive violations of what Comesaña (2008)calls “primitive cardinality”, namely, the intuive thesisto the effect that, for any integern, there could be exactlyn things. And since the size of any atomistic domain canalways be reached from below by taking powers, it also follows thatAGEM cannot have infinite models of stronglyinaccessible cardinality. Such is, as Uzquiano (2006) calls it, the“price of universality” in the context of Atomicity.

What aboutÃGEM, the result of adding theAtomlessness axiom (P.8)? Obviously the above limitation does notapply, and the Tarski model mentioned in Section 3.4 will suffice toestablish consistency. However, note that everyGEMmodel—hence everyÃGEM model—isnecessarily bound at the top, owing to the existence of the universalentityU. This is not by itself problematic: while theexistence ofU is the dual the Bottom axiom, a top jumbo ofwhich everything is part has none of the formal and philosophicaloddities of a bottom atom that is part of everything (though seeSection 4.5 for qualifications). Yet a philosopher who believes ininfinite divisibility, or at least in its possibility, might feel thesame about infinite composability. Just as everything could be made ofatomless gunk that divides forever into smaller and smaller parts,everything might be mereological “junk”—as Schaffer(2010: 64) calls it—that composes forever into greater andgreater wholes. (One philosopher who held such a view is, again,Whitehead, whose mereology of events includes both the Atomlessnessprinciple and its upward dual, i.e.:

See Whitehead 1919: 101; 1920: 76).GEM is compatiblewith the former possibility, andÃGEM makes itinto a universal necessity. But neither has room for the latter.Indeed, the possibility of junk might be attractive also from anatomist perspective. After all, already Theophilus thought that eventhough everything is composed of monads, “there is never aninfinite whole in the world, though there are always wholes greaterthan othersad infinitum” (Leibniz,NewEssays, I-xiii-21). Is this a serious limitation ofGEM? More generally, is this a serious limitation ofany theory in which the existence ofU is atheorem—effectively, any theory endorsing at least theunrestricted version of (P.14_ψ)? (In the absence ofAntisymmetry, one may want to consider this question by understandingthe predicate ‘PP’ in (P.19) in terms of the strongerdefinition given in (20′); see above,ad (P.8′).)Some authors have argued that it is (Bohn 2009a, 2009b, 2010), giventhat junk is at least conceivable (see also Tallant 2013) and admitsof plausible cosmological and mathematical models (Morganti 2009,Mormann 2014). Others have argued that it isn't, because junk ismetaphysically impossible (Schaffer 2010, Watson 2010). Others stillare openly dismissive about the question (Simons 1987: 83). One mayalso take the issue to be symptomatic of the sorts of trouble thataffect any theory that involves quantification over absolutelyeverything, as the Unrestricted Sum principles in(P.15_i) obviously do (see Spencer 2012, though hisremarks focus on mereological theories formulated in terms of pluralquantification). One way or the other, from a formal perspective theincompatibility with Ascent may be viewed as an unpleasant consequenceof (P.15_i), and a reason to go for weakertheories. In particular, it may be viewed as a reason to endorse onlyfinitary sums, which is to say only instances of(P.12_ξ,i), or perhaps its unrestrictedversion:

(See Contessa 2012 and Bohn 2012: 216 for explicit suggestions in thisspirit.) This would be consistent with the existence of junky worldsas it is consistent with the existence of gunky worlds. Yet it shouldbe noted that even this move has its costs. For example, it turns outthat in a world that is both gunky and junky (what Bohn calls“hunk”) (P.12_i) is in tension with theComplementation principle (P.6) for eachi (Cotnoir 2014).Moreover, while (P.12_i) is compatible with junkyworlds, i.e., models that fully satisfy the Ascent axiom(P.19), it is in tension with the possibility of worlds containingjunky structures along with other, disjoint elements (Giberman 2015).

4.5 Composition, Existence, and Identity

The algebraic strength ofGEM, and of its weakerfinitary and infinitary variants, is worth emphasizing, but it alsoreflects substantive mereological postulates whose philosophicalunderpinnings leave room for considerable controversy well beyond thegunk/junk dispute. Indeed,all composition principles turnout to be controversial, just as the decomposition principles examinedin Section 3. For, on the one hand, it appears that the weaker,restricted formulations, from (P.11_ξ) to(P.15_ψ,i), are just not doing enough work: notonly do they depend on the specification of the relevant limitingconditions, as expressed by the predicates ‘ξ’ and‘ψ’; they also treat such conditions as merelysufficient for the existence of bounds and sums, whereas ideally weare interested in an account of conditions that are both sufficientand necessary. On the other hand, the stronger, unrestrictedformulations appear to go too far, for while they rule out thepossibility of junky worlds, they also commit the theory to theexistence of a large variety ofprima facie implausible,unheard-of mereological composites—a large variety of“junk” in the good old sense of the word.

Concerning the first sort of worry, one could of course construe everyrestricted composition principle as a biconditional expressing both asufficientand a necessary condition for the existence of anupper bound, or a sum, of a given pair or set of entities. But thenthe question of how such conditions should be construed becomescrucial, on pain of turning a weak sufficient condition into anexceedingly strong requirement. For example, with regard to(P.11_ξ) we have mentioned the idea of construing‘ξ’ as ‘O’, the rationale being thatmereological overlap establishes an important connection between whatmay count as two distinct parts of a larger (integral) whole. However,as a necessary condition overlap is obviously too stringent. The tophalf of my body and the bottom half do not overlap, yet they do forman integral whole. The topological relation of contact, i.e., overlapor abut, might be a better candidate. Yet even that would betoo stringent. We may have misgivings about the existence of scatteredentities consisting of totally disconnected parts, such as my umbrellaand your left shoe or, worse, the head of this trout and the body ofthat turkey (Lewis 1991: 7–8). Yet in other cases it appearsperfectly natural to countenance wholes that are composed of two ormore disconnected entities: a bikini, a token of the lowercase letter‘i’, my copy ofThe Encyclopedia of Philosophy(R. Cartwright 1975; Chisholm 1987)—indeed any garden-varietymaterial object, insofar as it turns out to be a swarm of spatiallyisolated elementary particles (van Inwagen 1990). Similarly for someevents, such as Dante's writing ofInferno versus the sum ofSebastian's stroll in Bologna and Caesar's crossing of Rubicon (seeThomson 1977: 53f). More generally, intuition and common sense suggestthatsome mereological composites exist, not all; yet thequestion ofwhich composites exist seems to be up for grabs.Consider a series of almost identical mereological aggregates thatbegins with a case where composition appears to obtain (e.g., the sumof all body cells that currently make up my body, the relativedistance among any two neighboring ones being less than 1 nanometer)and ends in a case where composition would seem not to obtain (e.g.,the sum of all body cells that currently make up my body, after theirrelative distance has been increased to 1 kilometer). Where should wedraw the line? In other words—and to limit ourselves to(P.15_ψ,i)—what value ofn wouldmark a change of truth-value in the soritical sequence generated bythe schema

when ‘φ’ picks out my body cells and‘ψ’ expresses the condition ‘less thann+1 nanometers apart from another φ-er’? It maywell be that whenever some entities compose a bigger one, it is just abrute fact that they do so (Markosian 1998b), perhaps a matter ofcontingent fact (Nolan 2005: 36, Cameron 2007). But if we are unhappywith brute facts, if we are looking for a principled way of drawingthe line so as to specify the circumstances under which the factsobtain, then the question is truly challenging. That is, it is achallenging question short of treating it as a mere verbal dispute, ifnot denying that it makes any sense to raise it in the first place(see Hirsch 2005 and Putnam 1987: 16ff, respectively; see also Dorr2005 and McGrath 2008 for relevant discussion). This is, effectively,van Inwagen's “Special Composition Question” mentioned inSection 4.1, an early formulation of which may be found in Hestevold(1981). For the most part, the literature that followed has focused onthe conditions of composition for material objects, as in Sanford(1993), Horgan (1993), Hoffman and Rosenkrantz (1997), Merricks(2001), Hawley (2006), Markosian (2008), Vander Laan (2010), and Silva(2013). Occasionally the question has been discussed in relation tothe ontology of actions, as in Chant (2006). In its most general form,however, the Special Composition Question may be asked with respect toany domain of entities whatsoever.

Concerning the second worry, to the effect that theunrestricted sum principles in (P.15_i)would go too far, its earliest formulations are almost as old as theprinciples themselves (see e.g. V. Lowe 1953 and Rescher 1955 on thecalculus of individuals, with replies in Goodman 1956, 1958). Here onepopular line of response, inspired by Quine (1981: 10), is simply toinsist that the pattern in (P.15_i) is the onlyplausible option, disturbing as this might sound. Granted, commonsense and intuition dictate that some and only some mereologicalcomposites exist, but we have just seen that it is hard to draw aprincipled line. On pain of accepting brute facts, it would appearthat any attempt to do away with queer sums by restricting compositionwould have to do away with too much else besides the queer entities;for queerness comes in degrees whereas parthood and existence cannotbe a matter of degree (though we shall return to this issue in Section5). As Lewis (1986b: 213) puts it, no restriction on composition canbe vague, but unless it is vague, it cannot fit the intuitivedesiderata. Thus, no restriction on composition could servethe intuitions that motivate it; any restriction would be arbitrary,hence gratuitous. And if that is the case, then either mereologicalcomposition never obtains or else the only non-arbitrary, non-brutalanswer to the question, Under what conditions does a set have asum_i?, would be the radical one afforded by(P.15_i): Under any condition whatsoever. (Thisline of reasoning is further elaborated in Lewis 1991: 79ff as well asin Heller 1990: 49f, Jubien 1993: 83ff; Sider 2001: 121ff, Hudson2001: 99ff, and Van Cleve 2008: §3; for reservations and criticaldiscussion, see Merricks 2005, D. Smith 2006, Nolan 2006, Korman 2008,2010, Wake 2011, Carmichael 2011, and Effingham 2009, 2011a, 2011c.)Besides, it might be observed that any complaints about thecounterintuitiveness of unrestricted composition rest on psychologicalbiases that should have no bearing on the question of how the world isactually structured. Granted, we may feel uneasy about treatingshoe-umbrellas and trout-turkeys asbona fide entities, butthat is no ground for doing away with them altogether. We may ignoresuch entities when we tally up the things we care about in ordinarycontexts, but that is not to say they do notexist. Even ifone came up “with a formula that jibed with all ordinaryjudgments about what counts as a unit and what does not” (VanCleve 1986: 145), what would that show? The psychological factors thatguide our judgments of unity simply do not have the sort ofontological significance that should be guiding our construction of agood mereological theory, short of thinking that composition itself ismerely a secondary quality (as in Kriegel 2008). In the words ofThomson (1998: 167): reality is like “an over-crowdedattic”, with some interesting contents and a lot of junk, in theordinary sense of the term. We can ignore the junk and leave it togather dust; but it is there and it won't go away. (One residualproblem, that such observations do not quite address, concerns thestatus ofcross-categorial sums. Absent any restriction, apluralist ontology might involve trout-turkeys and shoe-umbrellasalong with trout-promenades, shoe-virtues, color-numbers, and whatnot. It is certainly possible to conceive ofsome suchthings, as in the theory of structured propositions mentioned inSection 2.1, or in certain neo-Aristotelian metaphysics that construeobjects as mereological sums of a “material” and a“formal” part; see e.g. Fine 1999, 2010, Koslicki 2007,2008, and Toner 2012. There are also theories that allow for compositeobjects consisting of both “positive” and“negative” parts, e.g., a donut, as in Hoffman andRichards 1985. At the limit, however, the universal entityUwould involve parts ofall ontological kinds. And there wouldseem to be nothing arbitrary, let alone any psychological biases, inthe thought that at leastsuch monsters should be banned. Fora statement of this view, see Simons 2003, 2006; for a reply, seeVarzi 2006b.)

A third worry, which applies to all (restricted or unrestricted)composition principles, is this. Mereology is supposed to beontologically “neutral”. But it is a fact that the modelsof a theorycum composition principles tend to be moredensely populated than those of the corresponding composition-freetheories. If the ontological commitment of a theory is measured inQuinean terms—via the dictum “to be is to be avalue of a bound variable” (1939: 708)—it follows thatsuch theories involve greater ontological commitments than theircomposition-free counterparts. This is particularly worrying in theabsence of the Strong Supplementation postulate (P.5)—hence theextensionality principle (27)—for then the ontologicalexuberance of such theories may yield massive multiplication. But theworry is a general one: composition, whether restricted orunrestricted, is not an ontologically “innocent”operation.

There are two lines of response to this worry (whose earliestformulations go as far back as V. Lowe 1953). First, it could beobserved that the ontological exuberance associated with the relevantcomposition principles is not substantive—that the increase ofentities in the domain of a mereological theorycumcomposition principles involves no substantive additional commitmentsbesides those already involved in the underlying theorywithout composition. This is obvious in the case of modestprinciples in the spirit of (P.11_ξ) and(P.14_ψ), to the effect that all suitably relatedentities must have an upper bound. After all, there are small thingsand there are large things, and to say that we can always find a largething encompassing any given small things of the right sort is not tosay much. But the same could be said with respect to those strongerprinciples that require the large thing to be composedexactly of the small things—to be their mereologicalsum in some sense or other. At least, this seems reasonable in thepresence of extensionality. For in that case it can be argued thateven a sum is, in an important sense, nothingover and aboveits constituent parts. The sum is just the parts “takentogether” (Baxter 1998a: 193); it is the parts “countedloosely” (Baxter 1988b: 580); it is, effectively, “thesame portion of Reality” (Lewis 1991: 81), which is strictly amultitude and loosely a single thing. That's why, if you proceed witha six-pack of beer to the six-items-or-fewer checkout line at thegrocery store, the cashier is not supposed to protest your use of theline on the ground that you have seven items: either s/he'll count thesix bottles, or s/he'll count the one pack. This thesis, known in theliterature as “composition as identity”, is by no meansuncontroversial and admits of different formulations (see van Inwagen1994, Yi 1999, Merricks 1999, McDaniel 2008, Berto and Carrara 2009,Carrara and Martino 2011, Cameron 2012, Wallace 2013, Cotnoir 2013a,Hawley 2013, and the essays in Baxter and Cotnoir 2014). To the extentthat the thesis is accepted, however, the charge of ontologicalexuberance loses its force. The additional entities postulated by thesum axioms would not be a genuine addition to being; they would be, inArmstrong's phrase, an “ontological free lunch” (1997:13). In fact, if composition is in some sense a form of identity, thenthe charge of ontological extravagance discussed in connection withunrestricted composition loses its force, too. For if a sum is nothingover and above its constituent proper parts, whatever they are, and ifthe latter are all right, then there is nothing extravagant incountenancing the former: it just is them, whatever they are. (This isnot to say that unrestricted composition isentailed by thethesis that composition is identity; indeed, see McDaniel 2010 for anargument to the effect that it isn't.)

Secondly, it could be observed that the objection in question bites atthe wrong level. If, given some entities, positing their sum were tocount as further ontological commitment, then, given a mereologicallycomposite entity, positing its proper parts should also count asfurther commitment. After all, every entity is distinct from itsproper parts. But then the worry has nothing to do with thecomposition axioms; it is, rather, a question of whether there is anypoint in countenancing a whole along with its proper partsorvice versa (see Varzi 2000, 2014 and Smid 2015). And if the answer isin the negative, then there seems to be little use for mereologytout court. From the point of view of the present worry, itwould appear that the only thoroughly parsimonious account would beone that rejectsany mereological complex whatsoever.Philosophically such an account is defensible (Rosen and Dorr 2002;Grupp 2006; Liggins 2008; Cameron 2010; Sider 2013; Contessa 2014) andthe corresponding axiom,

is certainly compatible withM (up toEM and more). But the immediate corollary

says it all: nothing would be part of anything else and parthood wouldcollapse to identity. (This account is sometimes referred to asmereologicalnihilism, in contrast to the mereologicaluniversalism expressed by (P.15_i); seevan Inwagen 1990: 72ff.^[25] Van Inwagen himself endorses a restricted version of nihilism, whichleaves room for composite living things. So does Merricks 2000, 2001,whose restricted nihilism leaves room for composite consciousthings.)

In recent years, further worries have been raised concerningmereological theories with substantive compositionprinciples—especially concerning the full strength ofGEM. Among other things, it has been argued that theprinciple of unrestricted composition does not sit well with certainfundamental intuitions about persistence through time (van Inwagen1990, 75ff), that it is incompatible with certain plausible theoriesof space (Forrest 1996b), or that it leads to paradoxes similar to theones afflicting naïve set theory (Bigelow 1996). A detailedexamination of such arguments is beyond the scope of this entry. Forsome discussion of the first issue, however, see Rea (1998), McGrath(1998, 2001), Hudson (2001: 93ff) and Eklund (2002: §7). On thesecond, see Oppy (1997) and Mormann (1999). Hudson (2001: 95ff) alsocontains some discussion of the last point.

5. Indeterminacy and Fuzziness

We conclude with some remarks on a question that was briefly mentionedabove in connection with the Special Composition Question but thatpertains more generally to the underlying notion of parthood thatmereology seeks to systematize. All the theories examined so far, fromM toGEM and its variants, appear toassume that parthood is a perfectly determinate relation: given anytwo entitiesx andy, there is always an objective,determinate fact of the matter as to whether or notx is partofy. However, in some cases this seems problematic. Perhapsthere is no room for indeterminacy in the idealized mereology of spaceand time as such; but when it comes to the mereology of ordinaryspatio-temporal particulars (for instance) the picture looksdifferent. Think of objects such as clouds, forests, heaps of sand.What exactly are their constitutive parts? What are the mereologicalboundaries of a desert, a river, a mountain? Some stuff is positivelypart of Mount Everest and some stuff is positively not part of it, butthere is borderline stuff whose mereological relationship to Everestseems indeterminate. Even living organisms may, on closer look, giverise to indeterminacy issues. Surely Tibbles's body comprises his tailand surely it does not comprise Pluto's. But what about the whiskerthat is coming loose? It used to be a firm part of Tibbles and soon itwill drop off for good, yet meanwhile its mereological relation to thecat is dubious. And what goes for material bodies goes for everything.What are the mereological boundaries of a neighborhood, a college, asocial organization? What about the boundaries of events such aspromenades, concerts, wars? What about the extensions of such ordinaryconcepts as baldness, wisdom, personhood?

These worries are of no little import, and it might be thought thatsome of the principles discussed above would have to be revisitedaccordingly—not because of their ontological import but becauseof their classical, bivalent presuppositions. For example, theextensionality theorem ofEM, (27), says thatcomposite things with the same proper parts are identical, but in thepresence of indeterminacy this may call for qualifications. The modelin Figure 9, left, depictsx andy as non-identicalby virtue of their having distinct determinate parts; yet one mightprefer to describe a situation of this sort as one in which theidentity betweenx andy is itself indeterminate,owing to the partly indeterminate status of the two outer atoms.Conversely, in the model on the rightx andy havethe same determinate proper parts, yet again one might prefer tosuspend judgment concerning their identity, owing to the indeterminatestatus of the middle atom.

Figure 9. Objects with indeterminate parts (dashedlines).

Now, it is clear that a lot here depends on how exactly oneunderstands the relevant notion of indeterminacy. There are, in fact,two ways of understanding a claim of the form

depending on whether the phrase ‘it is indeterminatewhether’ is assigned wide scope, as in (64a), or narrow scope,as in (64b):

On the first understanding, the indeterminacy is merelydedicto: perhaps ‘a’ or‘b’ are vague terms, or perhaps‘part’ is a vague predicate, but there is no reason tosuppose that such vagueness is due to objective deficiencies in theunderlying reality. If so, then there is no reason to think that itshould affect the apparatus of mereology either, at least insofar asthe theory is meant to capture some structural features of the worldregardless of how we talk about it. For example, the statement

may owe its indeterminacy to the semantic indeterminacy of‘Tibbles’: our linguistic practices do not, on closerlook, specify exactly which portion of reality is currently picked outby that name. In particular, they do not specify whether the namepicks out something whose current parts include the whisker that iscoming loose and, as a consequence, the truth conditions of (65) arenot fully determined. But this is not to say that the stuff out thereis mereologically indeterminate. Each one of a large variety ofslightly distinct chunks of reality has an equal claim to being thereferent of the vaguely introduced name ‘Tibbles’, andeach such thing has a perfectly precise mereological structure: someof them currently include the lose whisker among their parts, othersdo not. (Proponents of this view, which also affords a way of dealingwith the so-called “problem of the many” of Unger 1980 andGeach 1980, include Hughes 1986, Heller 1990, Lewis 1993a, McGee 1997,and Varzi 2001.) Alternatively, one could hold that the indeterminacyof (65) is due, not to the semantic indeterminacy of‘Tibbles’, but to that of ‘part’ (as inDonnelly 2014): there is no one parthood relation; rather, severalslightly different relations are equally eligible as extension of theparthood predicate, and while some such relations connect the loosewisker to Tibbles, others do not. In this sense, the dashed lines inFigure 9 would be “defects” in the models, not in thereality that they are meant to represent. Either way, it is apparentthat, on ade dicto understanding, mereological indeterminacyneed not be due to the way the world is (or isn't): it may just be aninstance of a more general and widespread phenomenon of indeterminacythat affects our language and our conceptual apparatus at large. Assuch, it can be accounted for in terms of whatevertheory—semantic, pragmatic, or even epistemic—one findsbest suited for dealing with the phenomenon in its generality. (Seethe entry onvagueness.) The principles of mereology, understood as a theory of the parthoodrelation, or of all the relations that qualify as admissibleinterpretations of the parthood predicate, would hold regardless.^[26]

By contrast, on the second way of understanding claims of the form(64), corresponding to (64b), the relevant indeterminacy is genuinelyde re: there is no objective fact of the matter as to whethera is part ofb, regardless of the words we use todescribe the situation. For example, on this view (65) would beindeterminate, not because of the vagueness of ‘Tibbles’,but because of the vagueness of Tibbles itself: there simply would beno fact of the matter as to whether the whisker that is coming looseis part of the cat. Similarly, the dashed lines in Figure 9 would notreflect a “defect” in the models but a genuine, objectivedeficiency in the mereological organization of the underlying reality.As it turns out, this is not a popular view: already Russell (1923)argued that the very idea of worldly indeterminacy betrays a“fallacy of verbalism”, and some have gone as far assaying thatde re indeterminacy is simply not“intelligible” (Dummett 1975: 314; Lewis 1986b: 212) orruled outa priori (Jackson 2001: 657). Nonetheless, severalphilosophers feel otherwise and the idea that the world may includevague entities relative to which the parthood relation is not fullydetermined has received considerable attention in recent literature,from Johnsen (1989), Tye (1990), and van Inwagen (1990: ch. 17) toMorreau (2002), McKinnon (2003), Akiba (2004), N. Smith (2005), Hyde(2008: §5.3), Carmichael (2011), and Sattig (2013, 2014),inter alia. Even those who do not find that thoughtattractive might wonder whether ana priori ban on it mightbe unwarranted—a deep-seated metaphysical prejudice, as Burgess(1990: 263) puts it. (Dummett himself withdrew his earlier remark andspoke of a “prejudice” in his 1981: 440.) It is thereforeworth asking: How would such a thought impact on the mereologicaltheses considered in the preceding sections?

There is, unfortunately, no straightforward way of answering thisquestion. Broadly speaking, two main sorts of answer may beconsidered, depending on whether (i) one simply takes theindeterminacy of the parthood relation to be thereason whycertain statements involving the parthood predicate lack a definitetruth-value, or (ii) one understands the indeterminacy so thatparthood becomes a genuine matter ofdegree. Both options,however, may be articulated in a variety of ways.

On option (i) (initially favored by such authors as Johnsen and Tye),it could once again be argued that no modification of the basicmereological machinery is strictly necessary, as long as eachpostulate is taken to characterize the parthood relation insofar as itbehaves in a determinate fashion. Thus, on this approach, (P.1) shouldbe understood as asserting that everything is definitely part ofitself, (P.2) that any definite part of any definite part of a thingis itself a definite part of that thing, (P.3) that things that aredefinitely part of each other are identical, and so on, and the truthof such principles is not affected by the consideration that parthoodneed not be fully determinate. There is, however, some leewayas to how such basic postulates could be integrated with furtherprinciples concerning explicitly the indeterminate cases. For example,do objects with indeterminate parts have indeterminate identity?Following Evans (1978), many philosophers have taken the answer to beobviously in the affirmative. Others, such as Cook (1986), Sainsbury(1989), or Tye (2000), hold the opposite view: vague objects aremereologically elusive, but they have the same precise identityconditions as any other object. Still others maintain that the answerdepends on the strength of the underlying mereology. For instance, T.Parsons (2000: §5.6.1) argues that on a theory such asEMcum unrestricted binary sums,^[27] thede re indeterminacy of (65) would be inherited by

A related question is: Does countenancing objects with indeterminateparts entail that composition be vague, i.e., that there is sometimesno matter of fact whether some things make up a whole? A popular view,much influenced by Lewis (1986b: 212), says that it does. Others, suchas Morreau (2002: 338), argue instead that the link between vagueparthood and vague composition is unwarranted: perhaps thedere indeterminacy of (65) is inherited by some instances of

(for example,x could be something that is just like Tibblesexcept that the whisker is determinately not part of it); yet thiswould not amount to saying that composition is vague, for thefollowing might nonetheless be true:

Finally, there is of course the general question of how one shouldhandle logically complex statements concerning, at least in part,mereologically indeterminate objects. A natural choice is to rely on athree-valued semantics of some sort, the third value being, strictlyspeaking, not a truth value but rather a truth-value gap. In thisspirit, both Johnsen and Tye endorse the truth-tables of Kleene (1938)while Hyde those of Łukasiewicz (1920). However, it is worthstressing that other choices are available, includingnon-truth-functional accounts. For example, Akiba (2000) and Morreau(2002) recommend a form of “supervaluationism”. This wasoriginally put forward by Fine (1975) as a theory for dealing withde dicto indeterminacy, the idea being that a statementinvolving vague expressions should count as true (false) if and onlyit is true (false) on every “precisification” of thoseexpressions. Still, a friend ofde re indeterminacy mayexploit the same idea by speaking instead of precisifications of theunderlying reality—what Sainsbury (1989) calls“approximants”, Cohn and Gotts (1996)“crispings”, and T. Parsons (2000)“resolutions” of vague objects. As a result, one would beable to explain why, for example, (69) appears to be true and (70)false (assuming that Tibbles's head is definitely part of Tibbles),whereas both conditionals would be equally indeterminate on Kleene'ssemantics and equally true on Łukasiewicz's:

As for option (ii)—to the effect thatde remereological indeterminacy is a matter of degree—the picture isdifferent. Here the main motivation is that whether or not somethingis part of something else is really not an all-or-nothing affair. IfTibbles has two whiskers that are coming loose, then we may want tosay that neither is a definite part of Tibbles. But if one whisker islooser than the other, then it would seem plausible to say that thefirst is part of Tibbles to alesser degree than the second,and one may want the postulates of mereology to be sensitive to suchdistinctions. This is, for example, van Inwagen's (1990) view of thematter, which results in a fuzzification of parthood that parallels inmany ways to the fuzzification of membership in Zadeh's (1965) settheory, and it is this sort of intuition that also led to thedevelopment of such formal theories as Polkowsky and Skowron's (1994)“rough mereology” or N. Smith's (2005) theory of“concrete parts”. Again, there is room for some leewayconcerning matters of detail, but in this case the main features ofthe approach are fairly clear and uniform across the literature. Forlet π be the characteristic function associated with the parthoodrelation denoted by the basic mereological primitive, ‘P’.Then, if classically this function is bivalent, which can be expressedby saying that π(x,y) always takes, say, thevalue 1 or the value 0 according to whether or notx is partofy, to say that parthood may be indeterminate is to saythat π need not be fully bivalent. And whereas option (i) simplytakes this to mean that π may sometimes be undefined, option (ii)can be characterized by saying that the range of π may includevalues intermediate between 0 and 1, i.e., effectively, values fromthe closed real interval [0, 1]. In other words, on this latterapproach π is still a perfectly standard, total function, and theonly serious question that needs to be addressed is the genuinelymereological question of what conditions should be assumed tocharacterize its behavior—a question not different from the onethat we have considered for the bivalent case throughout the precedingsections.

This is not to say that the question is an easy one. As it turns out,the “fuzzification” of the core theoryMis rather straightforward, but its extensions give rise to variousissues. Thus, consider the partial ordering axioms (P.1)–(P.3).Classically, these correspond to the following conditions on π:

and one could argue that the very same conditions may be taken to fixthe basic properties of parthood regardless of whether π isbivalent. Perhaps one may consider weakening (P.2_π) asfollows (Polkowsky and Skowron 1994):

Or one may consider strengthening (P.3_π) as follows (N.Smith 2005):^[28]

But that is about it: there is little room for further adjustments.Things immediately get complicated, though, as soon as we move beyondM. Take, for instance, the Supplementation principle(P.4) ofMM. One natural way of expressing it interms of π is as follows:

There are, however, fifteen other ways of expressing (P.4) in terms ofπ, obtained by re-writing one or both occurrences of ‘=1’ as ‘> 0’ and one or both occurrences of‘= 0’ as ‘< 1’. In the presence ofbivalence, these would all be equivalent ways of saying the samething. However, such alternative formulations would not coincide ifπ is allowed to take non-integral values, and the question of whichversion(s) best reflect the supplementation intuition would have to becarefully examined. (See e.g. the discussion in N. Smith 2005: 397.)And this is just the beginning: it is clear that similar issues arisewith most other principles discussed in the previous sections, such asComplementation, Density, or the various composition principles. (Seee.g. Polkowsky and Skowron 1994: 86 for a formulation of theUnrestricted Sum axiom (P.15₂).)

On the other hand, it is worth noting that precisely because thedifficulty is mainly technical—the framework itself being fairlyfirm—now some of the questions raised in connection with option(i) tend to be less open to controversy. For example, the question ofwhether mereological indeterminacy implies vague identity is generallyanswered in the negative, especially if one adheres to the spirit ofextensionality. For then it is natural to say that non-atomic objectsare identical if and only if they have exactly the same parts to thesame degree—and that is not a vague matter (a point already madein Williamson 1994: 255). In other words, given that classically theextensionality principle (27) corresponds to the followingcondition:

it seems perfectly natural to stick to this condition even if therange of π is extended from {0, 1} to [0, 1]. Likewise, thequestion of whether mereological indeterminacy implies vaguecomposition or vague existence is generally answered in theaffirmative (though not always; see e.g. Donnelly 2009 and Barnes andWilliams 2009). Van Inwagen (1990: 228) takes this to be a ratherobvious consequence of the approach, but N. Smith (2005: 399ff) goesfurther and provides a detailed analysis of how one can calculate thedegree to which a given non-empty set of things has a sum, i.e., thedegree of existence of the sum. (Roughly, the idea is to begin withthe sum as it would exist if every element of the set were a definitepart of it, and then calculate the actual degree of existence of thesum as a function of the degree to which each element of the set isactually part of it).

The one question that remains widely open is how all of this should bereflected in the semantics of our language, specifically the semanticsof logically complex statements. As a matter of fact, there is atendency to regard this question as part and parcel of the moregeneral problem of choosing the appropriate semantics for fuzzy logic,which typically amounts to an infinitary generalization of sometruth-functional three-valued semantics. The range of possibilities,however, is broader, and even here there is room fornon-truth-functional approaches—including degree-theoreticvariants of supervaluationism (as recommended e.g. in Sanford 1993:225).

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Monographs and Collections

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Acknowledgments

The author would like to thank Aaron Cotnoir, Maureen Donnelly, CodyGilmore, Paolo Maffezioli, Anthony Shiver, and an anonymous refereefor helpful comments and suggestions.