Mereology (Stanford Encyclopedia of Philosophy) Cite this entry Search the SEP • Advanced Search • Tools • RSS FeedTable of Contents• What's New• Archives• Projected ContentsEditorial Information• About the SEP• Editorial Board• How to Cite the SEP• Special CharactersSupport the SEPContact the SEP ©Metaphysics Research Lab,CSLI,Stanford University  Open access to the SEP is made possible by a world-wide funding initiative. Please Read How You Can Help Keep the Encyclopedia FreeMereologyFirst published Tue May 13, 2003; substantive revision Thu Aug 21, 2003Mereology (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 awhole. Its roots can be traced back to the early days of philosophy,beginning with the Presocratic atomists and continuing throughout thewritings of Plato (especially the Parmenides and theThaetetus), Aristotle (especially the Metaphysics,but also the Physics, the Topics, and De partibusanimalium), and Boethius (especially In CiceronisTopica). Mereology has also occupied a prominent role in thewritings of medieval ontologists and scholastic philosophers such asGarland the Computist, Peter Abelard, Thomas Aquinas, Raymond Lull, andAlbert of Saxony, as well as in Jungius's Logica Hamburgensis(1638), Leibniz's Dissertatio de arte combinatoria (1666) andMonadology (1714), and Kant's early writings (theGedanken of 1747 and the Monadologia physica of1756). As a formal theory of parthood relations, however, mereologymade its way into modern philosophy mainly through the work of FranzBrentano and of his pupils, especially Husserl's third LogicalInvestigation (1901). The latter may rightly be considered thefirst attempt at a rigorous formulation of the theory, though in aformat that makes it difficult to disentagle the analysis ofmereological concepts from that of other ontologically relevant notions(such as the relation of ontological dependence). It is not untilLeśniewski's Foundations of a General Theory of Manifolds(1916, in Polish) that the pure theory of part-relations as we know ittoday was given an exact formulation. And because Leśniewski'swork was largely inaccessible to non-speakers of Polish, it is onlywith the publication of Leonard and Goodman's The Calculus ofIndividuals (1940) that this theory has become a chapter ofcentral interest for modern ontologists and metaphysicians.In the following we shall focus mostly on contemporary formulationsof mereology as they grew out of these recent theories --Leśniewski's and Leonard and Goodman's. Indeed, although suchtheories came in different logical guises, they are sufficientlysimilar to be recognized as a common basis for most subsequentdevelopments. To properly assess the relative strength and weaknesses,however, it will be convenient to proceed in steps. First we considersome core mereological notions and principles. Then we proceed to anexamination of the stronger theories that can be erected on thisbasis.1. ‘Part’ and Parthood2. Basic Principles 2.1. Parthood as a Partial Ordering2.2. Other Mereological Concepts3. Supplementation Principles 3.1. Parts and Remainders3.2. Identity and Extensionality4. Closure Principles 4.1. Finitary Operations4.2. Unrestricted Fusions4.3. Composition, Existence, and Identity5. Atomismistic and Atomless MereologiesBibliography Historical SurveysMonographsCited WorksOther Internet ResourcesRelated Entries1. ‘Part’ and ParthoodA preliminary caveat is in order. It concerns the very notion ofparthood that mereology is about. The word ‘part’ has manydifferent meanings in ordinary language, not all of which correspond tothe same relation. Broadly speaking, it can be used to indicate anyportion of a given entity, regardless of whether it is attached to theremainder, as in (1), or detached, as in (2); cognitively salient, asin (1)-(2), or arbitrarily demarcated, as in (3); self-connected, as in(1)-(3), or disconnected, as in (4); homogeneous, as in (1)-(4), orgerrymandered, as in (5); material, as in (1)-(5), or immaterial, as in(6); extended, as in (1)-(6), or unextended, as in (7); spatial, as in(1)-(7), or temporal, as in (8); and so on.(1) The handle is part of the cup.(2)This cap is part of my pen.(3)The left half is your part of the cake.(4)The US is part of North America.(5)The contents of this bag is only part of what I bought.(6)That corner is part of the living room.(7)The outermost points are part of the perimeter.(8)The first act was the best part of the play.All of these cases illustrate the notion of parthood that forms thefocus of mereology. Often, however, the word ‘part’ is usedin English in a restricted sense. For instance, it can be used todesignate only the cognitively salient relation of parthood illustratedin (1) and (2) as opposed to (3). In this sense, the parts of an objectx are just its “components”, i.e., those partsthat are available as individual units regardless of their interactionwith the other parts of x. (A component is a part ofan object, rather than just part of it; see Tversky 1989).Clearly, the properties of such restricted relations may not coincidewith those of parthood broadly understood, so the principles ofmereology should not be expected to carry over automatically.Also, the word ‘part’ is sometimes used in a broadersense, for instance to designate the relation of material constitution,as in (9), or the relation of mixture composition, as in (10), or evena relation of conceptual inclusion, as in (11):(9) The clay is part of the statue.(10)Gin is part of martini.(11)Writing detailed comments is part of being a good referee.The mereological status of these relations, however, iscontroversial. For instance, although the constitution relationexemplified in (9) was included by Aristotle in his threefold taxonomy(Metaphysics, Δ, 1023b), many contemporary authors wouldrather construe it as a sui generis, non-mereological relation(see e.g. Wiggins 1980, Rea 1995, and Thomson 1998). Similarly, theingredient-mixture relationship exemplified in (10) is subject tocontroversy, as the ingredients may involve significant structuralconnections besides spatial proximity and may therefore fail to retaincertain important chemical characteristics they have in isolation (seeSharvy 1983). As for cases like (11), it may simply be contended thatthe term ‘part’ appears only in the surface grammar anddisappears at the level of logical form, e.g., if (11) is paraphrasedas “Every good referee writes detailed comments.” (For moreexamples and tentative taxonomies, see Winston et al. 1987,Iris et al. 1988, and Gerstl and Pribbenow 1995.)Finally, it is worth stating explicitly that mereology assumes noontological restriction on the field of ‘part’. The relatacan be as different as material bodies, events, geometric entities, orgeographical regions, as in (1)-(8), as well as numbers, sets, types,or properties, as in the following examples:(12) 2 is part of 3.(13)The integers are part of the reals.(14)The first chapter is part of the novel.(15)Humanity is part of personhood.Thus, although both Leśniewski's and Leonard and Goodman'soriginal theories betray a nominalistic stand, resulting in aconception of mereology as an ontologically parsimonious alternative toset theory, there is no necessary link between the analysis of parthoodrelations and the philosophical position of nominalism.[1] As a formaltheory (in Husserl's sense of ‘formal’, i.e., as opposed to‘material’) mereology is simply an attempt to lay down thegeneral principles underlying the relationships between an entity andits constituent parts, whatever the nature of the entity, just as settheory is an attempt to lay down the principles underlying therelationships between a class and its constituent members. Unlike settheory, mereology is not committed to the existence ofabstracta: the whole can be as concrete as the parts. Butmereology carries no nominalistic commitment either: the parts can beas abstract as the whole. David Lewis's Parts of Classes(1991), which provides a mereological analysis of the set-theoreticuniverse, is a good illustration of this “ontologicalinnocence” of mereology.2. Basic PrinciplesWith these provisos, and barring for the moment the complicationsarising from the consideration of intensional factors (such as time andmodalities), let us now review some core mereological principles. Tosome extent, these may be thought of as lexical axioms fixing theintended meaning of the relational predicate ‘part’.However, the boundary of what is philosophically uncontroversial isdifficult to draw, so it will be convenient to proceed step by step,starting from the obvious and adding more substantive principles as wego on.2.1 Parthood as a Partial OrderingThe obvious is this: No matter how one feels about matters ofontology, if ‘part’ stands for the general relationexemplified by all of (1)-(8) above, then it stands for a partialordering -- a reflexive, antisymmetric, transitive relation:(16)Everything is part of itself.(17)Two distinct things cannot be part of each other.(18)Any part of any part of a thing is itself part of thatthing.To be sure, this characterization is not entirely uncontroversial.In particular, since Rescher (1955) several authors have had misgivingsabout the transitivity principle (18) (see e.g. Lyons 1977: 313, Cruse1979, and Moltman 1997). Rescher writes:In military usage, for example, persons can be parts ofsmall units, and small units parts of larger ones; but persons arenever parts of large units. Other examples are given by the varioushierarchical uses of ‘part’. A part (i.e., biologicalsubunit) of a cell is not said to be a part of the organ of which thatcell is a part. (1955: 10)Arguably, however, such misgivings stem from the aforementionedambiguity of ‘part’. What counts as a biological subunit ofa cell may not count as a subunit (a distinguished part) of the organ,but it is nonetheless part of the organ. The military example is moreto the point, yet it also trades on an ambiguity. If there is a senseof ‘part’ in which soldiers are not part of larger units,it is a restricted sense: a soldier is not directly part of abatallion -- the soldier does not report to the head of the batallion.Likewise, one can argue that a handle is a functional part of a door,the door is a functional part of the house, and yet the handle is not afunctional part of the house. But this involves a departure from thebroader notion of parthood that mereology is meant to capture. To putit differently, if the general intended interpretation of‘part’ is narrowed by additional conditions (e.g., byrequiring that parts make a direct contribution to the functioning ofthe whole), then obviously transitivity may fail. In general, ifx is a φ-part of y and y is a φ-partof z, x need not be a φ-part of z: thepredicate modifier ‘ φ ’ may not distribute overparthood. But that shows the non-transitivity of‘φ-part’ (e.g., of direct part, orfunctional part), not of ‘part’. And within asufficiently general framework this can easily be expressed with thehelp of explicit predicate modifiers.The other two properties -- reflexivity and antisymmetry -- are lesscontroversial, though also in this regard some qualifications are inorder. Concerning reflexivity (16), a familiar objection -- due againto Rescher -- is thatmany legitimate senses of ‘part’ arenonreflexive, and do not countenance saying that a whole is a part (inthe sense in question) of itself. The biologists' use of‘part’ for the functional subunits of an organism is a casein point. (1955: 10)This is of little import, though. Taking reflexivity (andantisymmetry) as constitutive of the meaning of ‘part’amounts to regarding identity as a limit (improper) case of parthood. Astronger relation, whereby nothing counts as part of itself, canobviously be defined in terms of the weaker one, hence there is no lossof generality (see section 2.2 below). Vice versa, one could frame amereological theory by taking proper parthood as a primitive instead.This is merely a question of choosing a suitable primitive. Formally,the issue therefore boils down to the previous point: a φ-part maynot quite behave as a part simpliciter, where φ is theadditional condition of being distinct from the whole.Finally, concerning the antisymmetry postulate (17), one may observethat this rules out “non-well-founded” mereologicalstructures. Sanford (1993: 222) refers to Borges's Aleph as a case inpoint:I saw the earth in the Aleph and in the earth the Alephonce more and the earth in the Aleph… (Borges 1949:151)In this case, a plausible reply (following van Inwagen 1993: 229) isthat fiction delivers no guidance to conceptual investigations.Conceivability may well be a guide to possibility, but literary fantasyis by itself no evidence of conceivability. However, the idea of anon-well-founded parthood relation is not pure fantasy. In view ofcertain developments in non-well-founded set theory (i.e., set theorytolerating cases of self-membership and, more generally, of membershipcircularities -- see Aczel 1988; Barwise and Moss 1996), one mightindeed suggest building mereology on the basis of an equally lessrestrictive notion of parthood that allows for closed loops. This isparticularly significant in view of the possibility of reformulatingset theory in mereological terms -- a possibility that is explored inthe works of Bunt (1985) and Lewis (1991, 1993). Thus, in this casethere is legitimate concern that one of the “obvious”meaning postulates for ‘part’ is in fact too restrictive.At present, however, no systematic study of non-well-founded mereologyhas been put forward in the literature, so in the following we shallconfine ourselves to theories that accept the antisymmetry postulatealong with relexivity and transitivity.2.2. Other Mereological ConceptsIt is convenient at this point to introduce some degree offormalization before we proceed further. This avoids ambiguities (suchas those involved in the above-mentioned objections) and facilitatescomparisons and developments. For definiteness, we shall work withinthe framework of a standard first-order language with identity,supplied with a distinguished binary predicate constant,‘P’, to be interpreted as the parthood relation. Taking theunderlying logic to be a standard predicate calculus withidentity,[2] the above minimal requisites on parthoodmay then be regarded as forming a first-order theory characterized bythe following proper axioms for ‘P’:(P.1)PxxReflexivity(P.2)(Pxy & Pyx) → x=yAntisymmetry(P.3)(Pxy & Pyz) → PxzTransitivity (Here and in the following we simplify notation by dropping allinitial universal quantifiers. All formulas are to be understood asuniversally closed.) We may call such a theory GroundMereology -- M for short[3] -- regarding it as thecommon basis of any comprehensive part-whole theory.Given (P.1)-(P.3), a number of additional mereological predicatescan be introduced by definition. For example:(19) Oxy =df  z(Pzx & Pzy)Overlap(20)Uxy =df z(Pxz & Pyz)Underlap(21)PPxy =df Pxy & ¬PyxProper Part(22)OXxy =df Oxy & ¬PxyOver-crossing(23)UXxy =df Uxy & ¬PyxUnder-crossing(24)POxy =df OXxy &OXyxProper Overlap(25)PUxy =df UXxy &UXyx.Proper UnderlapAn intuitive model for these relations, with ‘P’interpreted as spatial inclusion, is given in Figure 1. Figure 1. Basic patterns of mereological relations.In the leftmost pattern, the relations in parenthesis hold if there isa larger z including both x and y.It is immediately verified that overlap is reflexive and symmetric,though not transitive:(26) Oxx(27)Oxy → Oyx.Likewise for underlap. By contrast, it follows from (P.1)-(P.3) thatproper parthood is transitive but irreflexive and asymmetric -- astrict partial ordering:(28)¬ PPxx(29)PPxy → ¬ PPyx(30)(PPxy & PPyz) →PPxz.As mentioned, one could use proper parthood as an alternativestarting point (using (28)-(30) as axioms). This follows from the factthat the following equivalence is provable in M:(31) Pxy ↔ (PPxy  x=y)and one could therefore use the right-hand side of (31) to define‘P’ in terms of ‘PP’ and ‘=’. Onthe other hand, as with every partial ordering, it is worth observingthat identity could itself be introduced by definition, due to thefollowing immediate consequence of (P.2):(32) x=y ↔ (Pxy& Pyx).Accordingly, theory M could be formulated in a purefirst-order language by assuming (P.1) and (P.3) and replacing (P.2)with the following variant of the Leibniz axiom for identity (whereφ is any formula):(P.2′) (Pxy & Pyx)→ (φx ↔ φy).In the following, however, we shall continue to assume thatM is formulated in a language with both‘P’ and ‘=’ as primitives.3. Supplementation PrinciplesTheory M may be viewed as embodying the common coreof any mereological theory. Not just any partial ordering qualifies asa part-whole relation, though, and establishing what further principlesshould be added to (P.1)-(P.3) is precisely the question a goodmereological theory is meant to answer. These further principles aremore substantive, and they are to some extent stipulative. However,some main options can be identified.Generally speaking, a mereological theory may be viewed as theresult of extending M by means of principles assertingthe (conditional) existence of certain mereological items given theexistence of other items. Thus, one may consider the idea thatwhenever an object has a proper part, it has more than one --i.e., that there is always some mereological difference between a wholeand its proper parts. This need not be true in every model forM: a world with only two items, one of which isP-related to the other but not vice versa, would be a counterexample,though not one that could be illustrated with the sort of geometricdiagram used in Figure 1. Similarly, one may consider the idea thatthere is always a mereological sum of two or more parts --i.e., that for any number of objects there exists a whole that consistsexactly of those objects. Again, this need not be true in a model forM, and it is a matter of controversy whether the ideashould hold unrestrictedly. More generally, one may consider extendingM by requiring that the domain of discourse be closed-- on certain conditions -- under various mereological operations (sum,product, difference, and possibly others). Finally, one may considerthe question of whether there are any mereological atoms(objects with no proper parts), and also whether every object isultimately made up of atoms (or on what conditions an objectmay be assumed to be made up of atoms). Both of these options arecompatible with M, and the possibility of addingcorresponding axioms has interesting philosophical ramifications.3.1. Parts and RemaindersLet us begin with the first sort of extension. The underlying ideacan take at least two distinct forms. The simpler one consists instrengthening M by adding a fourth axiom to the effectthat every proper part must be supplemented by another,disjoint part -- a remainder:(P.4)PPxy → z(Pzy & ¬Ozx)Weak SupplementationCall this extension Minimal Mereology(MM). Some authors (most notably Peter Simons 1987,from whom the term ‘supplementation’ is borrowed) regard(P.4) as constitutive of the meaning of ‘part’ and wouldaccordingly list it along with the basic postulates of mereology.However, some theories in the literature violate this principle and itis therefore convenient to keep it separate from (P.1)-(P.3). A case inpoint would be Brentano's 1933 theory of accidents, according to whicha soul is a proper part of a thinking soul even though there is nothingto make up for the difference. (See Chisholm 1978; for an assessmentsee Baumgartner and Simons 1994.) Another example is provided byWhitehead's 1929 theory of extensive connection, where no boundaryelements are included in the domain of quantification: on this theory atopologically closed region includes its open interior as a proper partin spite of there being no boundary elements to distinguish them. (SeeClarke 1981 for a rigorous formulation.)The second way of expressing the supplementation intuition isstronger. It corresponds to the following axiom, which differs from(P.4) in the antecedent:(P.5)¬Pyx → z(Pzy & ¬Ozx)Strong SupplementationThis says that if an object fails to include another amongits parts, then there must be a remainder. It is easily seen that (P.5)implies (P.4), so any theory rejecting (P.4) will a fortiorireject (P.5). (For instance, on Whitehead's boundary-free theory ofextensive connection, a closed region is not part of its interiorthough they have exactly the same extended parts.) However, theconverse does not hold. Consider a model with four distinct objects,a, b, c, d, such that cand d are P-related to both a and b. Thenthe corresponding instance of (P.4) is true, since each proper partcounts as a supplement of the other; yet (P.5) is false, since bothparts of a are part of (and therefore overlap) b, andboth parts of b are part of (and overlap) a.Admittedly, it is difficult to imagine such objects; it isdifficult to draw a picture illustrating two distinct objects with thesame parts, because drawing an object is drawing its parts.Once the parts are drawn, there is nothing left to be done to get adrawing of the whole object. But this only proves that pictures arebiased towards (P.5). In the non-spatial domain, for instance, theenvisaged countermodel to (P.5) can be set up by identifying aand b with the ordered pairs <c, d>and <d, c>, respectively, interpreting‘P’ as the relation of membership for ordered sets.The theory obtained by adding (P.5) to (P.1)-(P.3) is thus a properextension of the theory of Minimal Mereology obtained by adding (P.4).We label this stronger theory Extensional Mereology(EM). The attribute ‘extensional’ isjustified precisely by the exclusion of countermodels that, like theones just mentioned, contain distinct objects with the same properparts. In fact, the following is a theorem of EM:(33) zPPzx → ( z(PPzx → PPzy) →Pxy).from which it follows that non-atomic objects with the same properparts are identical:(34) (zPPzx zPPzy) →(x=y ↔ z(PPzx ↔PPzy)).(The analogue for ‘P’ is already provable inM, since P is reflexive and antisymmetric.) This isthe mereological counterpart of the familiar set-theoreticextensionality principle, as it reflects the view that an object isexhaustively defined by its constituent parts, just as a set isexhaustively defined by its constituent elements. Nelson Goodmanappropriately termed this mereological principle“hyper-extensionalism” (1958: 66), relating it to theontological parsimony of nominalism:A class (e.g., that of the counties of Utah) is differentneither from the single individual (the whole state of Utah) thatexactly contains its members nor from any other class (e.g., that ofacres of Utah) whose members exactly exhaust this same whole. Theplatonist may distinguish these entities by venturing into a newdimension of Pure Form, but the nominalist recognizes no distinction ofentities without a distinction of content. (Goodman 1951:26)3.2. Identity and ExtensionalityIs EM a plausible theory? Apart from thecounterexamples to (P.5) mentioned above, several objections have beenraised against (34), in spite of its intuitive plausibility in thecontext of Goodman's geographic example. On the one hand, it issometimes argued that sameness of parts is not sufficient foridentity, as some entities may differ exclusively with respect to thearrangement of their parts. Two sentences made up of the same words --‘John loves Mary’ and ‘Mary loves John’ --would be a case in point (Hempel 1953: 110; Rescher 1955: 10).Likewise, the identity of a bunch of flowers may depend crucially onthe arrangements of the individual flowers (Eberle 1970: §2.10). Asecond familiar objection is familiar from the literature on materialconstitution, where the principle of mereological extensionality issometimes taken to contradict the possibility that an object may bedistinct from the matter constituting it. A cat can survive theannihilation of its tail, it is argued. But the amount of feline tissueconsisting of the cat's tail and the rest of the cat's body cannotsurvive the annihilation of the tail. Thus, a cat and the correspondingamount of feline tissue have different (tensed or modal) properties andshould not be identified in spite of their sharing exactly the sameactual parts. (See e.g. Wiggins 1968, Doepke 1982, Lowe 1989, Johnston1992, and Baker 1999, Sanford 2003 for this line of objection.)Conversely, if the identity relation is taken to extend over times orover possible worlds, as in standard tensed and modal talk, then thepossibility of mereological change implies that sameness of parts isnot necessary for identity. If a cat survives the annihilationof its tail, then the cat with tail (before the accident) andthe cat without tail (after the accident) are numerically thesame in spite of their having different proper parts (Wiggins 1980). Ifany of these arguments is accepted, then clearly (34) is too strong aprinciple to be imposed on the parthood relation. And since (34)follows from (P.5), it might be concluded that EMshould be rejected in favor of the weaker mereological theoryMM.A thorough discussion of these issues is beyond the scope of thisentry. (See the entries on Identity and Persistence).Some remarks, however, are in order. Concerning the sufficiency ofmereological extensionality, i.e., the right-to-left conditional in theconsequent of (34):(35) z(PPzx ↔ PPzy) →x=y,it should be noted that the first sort of objection mentioned abovemay be dispensed with easily. Sentences made up of the same words, itcan be argued, are best described as different sentencetokensmade up of distinct tokens of the same wordtypes. There is,accordingly, no violation of (35) in the opposition between ‘Johnloves Mary’ and ‘Mary loves John’ (for instance),hence no reason to reject (P.5) on these grounds. Besides, even withrespect to types it could be pointed out that the sentences ‘Johnloves Mary’ and ‘Mary loves John’ do not shareall their proper parts. The string ‘John loves’,for instance, is only included in the first sentence. As for such moreconcrete examples as a bunch of flowers, a planetary system, or a fleetformation, it should be noted that these violate extensionality onlyinsofar as we engage in tensed or counterfactual talk. It might beplausible to hold that a bunch of flowers would not (or no longer) bewhat it is if the flowers were arranged differently, or if they werescattered all over the floor. So, if the variables in (35) are taken torange over entities existing at different times, or at differentpossible worlds, then indeed (35) would appear to be too strong. Itdoes not follow, however, that we have found a counterexample toextensionality if we confine ourselves to issues of syncronic identityin the actual world. (In essence, this amounts to treating sentences ofEM as present tensed. So the interesting question is:Does perfectly general mereology, if it makes this move, require tenseand modal logic?)This leads to the second objection to the sufficiency ofextensionality, which is more delicate. As a sufficient condition forindividual identity (35) is indeed very strict. At the same time,abandoning it may lead to massive ontological multiplication: if thecat is different from the mereological aggregate tail+remainder, itmust also be different from the aggregate head+remainder, and from theaggregate nose+remainder, and so on. How many entities then occupy theregion occupied by the cat? To what principled criterion can we appealto avoid this slippery slope? (Similarly, if a bunch of flowers isdistinguished from the mere aggregate of the individual flowersconstituting it on account of the fact that they have different modalproperties -- the latter could while the former could not surviverearrangement of the parts -- then these must be distinguished alsofrom many other mereological aggregates: the one consisting ofrose#1+remainder, the one consisting of tulip#2+ remainder, and soon.)On behalf of EM, and to resist such ontologicalexhuberance, it should be noted that the appeal to Leibniz's law inthis context is to be carefully evaluated. Let ‘Tibbles’name our cat and ‘Tail’ its tail, and let us grant thetruth of(36) Tibbles can survive the annihilation of Tail.There is, indeed, an intuitive sense in which the following is alsotrue:(37) The amount of feline tissue consisting of Tail and the rest ofTibbles's body cannot survive the annihilation of Tail.However, this intuitive sense corresponds to a de dictoreading of the modality, where the description in (37) has narrowscope:(38) In every possible world, the amount of feline tissue consisting ofTail and the rest of Tibbles's body has Tail as a proper part.On this reading (37) is hardly negotiable (in fact, logically true).Yet this is irrelevant in the present context, for (38) does not amountto an ascription of a modal property and cannot be used in connectionwith Leibniz's law. (Compare the following fallacious argument: GeorgeW. Bush might not have been a US President; the 43rd USPresident is necessarily a US President; hence George W. Bush is notthe 43rd US President.) On the other hand, consider a dere reading of (37), where the description has wide scope:(39) The amount of feline tissue consisting of Tail and the rest ofTibbles's body has Tail as a proper part in every possible world.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 (36) and (37) (i.e., (39)) to conclude thatTibbles is distinct from the relevant amount of feline tissue. However,there is no obvious reason why (37) should be regarded as true on thisreading. That is, there is no obvious reason to suppose that the amountof feline tissue that in the actual world consists of Tail and the restof Tibbles's body -- that amount of feline tissue that is nowresting on the carpet -- cannot survive the annihilation of Tail.Indeed, it would appear that any reason in favor of this claimvis-à-vis the truth of (36) would have topresuppose the distinctness of the entities in question, so noappeal to Leibniz's law would be legitimate to establish thedistinctess (on pain of circularity). This is not to say that theputative counterexample to (35) is wrong-headed. But it requiresgenuine metaphysical work and it makes the rejection of the strongsupplementation principle (P.5) a matter of genuine philosophicalcontroversy. (Similar remarks would apply to any argument intended toreject extensionality on the basis of competing modal intuitionsregarding the possibility of mereological rearrangement,rather than mereological change, as with the flowers example.On a de re reading, the claim that a bunch of flowers couldnot survive rearrangement of the parts -- while the aggregate of theindividual flowers composing it could -- must be backed up by a genuinemetaphysical theory about these entities.)Finally, consider the objection against (P.5) based on the intuitionthat sameness of parts is not necessary for identity, contraryto the left-to-right conditional in the consequent of (34):(40) x=y → z(PPzx ↔PPzy).This objection proceeds from the consideration that ordinaryentities such as cats and other living organisms (and possibly otherentities as well, such as statues and ships) survive all sorts ofgradual mereological changes. Clearly this a serious objection, unlessthose entities are construed as fictional entia succcessiva(Chisholm 1976). However, the difficulty is not peculiar to extensionalmereology. For (40) is just a corollary of the identity axiom(ID) x=y →(φx ↔ φy).And it is well known that this axiom calls for revisions when‘=’ is given a diachronic reading. Arguably, any suchrevisions will affect the case at issue as well, and in this sense theabove-mentioned objection to (40) can be disregarded. For example, ifthe basic parthood predicate were reinterpreted as a time-indexedrelation (Thomson 1983), then the problem would disappear as the tensedversion of (P.5) would only warrant the following variant of (40):(41) x=y → tz(PPtzx ↔PPtzy).Similarly, the problem would disappear if the variables in (40) weretaken to range over four-dimensional entities whose parts may extend intime as well as in space (Heller 1984, Sider 1997), or if identityitself were construed as a contingent relation that may hold at sometimes but not at others (Gibbard 1975, Myro 1985, Gallois 1998). Suchrevisions may be regarded as an indicator of the limited ontologicalneutrality of extensional mereology. But their independent motivationalso bears witness to the fact that the controversies aboutextensionality, and particularly about (40), stem from genuine andfundamental philosophical conundrums and cannot be assessed byappealing to our intuitions about the meaning of‘part’.4. Closure PrinciplesLet us now consider the second way of extending M,corresponding to the idea that a mereological domain must be closedunder various operations.4.1. Finitary OperationsTake first the operations of sum and product. (Mereological sum issometimes called “fusion”.) If two things underlap, then wemay assume that there is a smallest thing of which they are part -- athing that exactly and completely exhausts both. For instance, yourleft thumb and index finger underlap, since they are both parts of you.There are other things of which they are part -- e.g., your left hand.And we may assume that there is a smallest such thing: that part ofyour left hand which consists exactly of your left thumb and indexfinger. Likewise, if two things overlap (e.g., two intersecting roads),then we may assume that there is a largest thing that is part of both(the common part at their junction). These two assumptions can beexpressed by means of the following axioms, respectively:(P.6)Uxy → zw(Owz ↔ (Owx Owy))Sum (P.7)Oxy → zw(Pwz ↔ (Pwx &Pwy))Product Call the extension of M obtained by adding (P.6)and (P.7) Closure Mereology (CM). The resultof adding these axioms to MM or EMinstead yields corresponding Minimal or ExtensionalClosure Mereologies (CMM andCEM), respectively.The intuitive idea behind these two axioms is best appreciated inthe presence of extensionality, for in that case the entities whoseconditional existence is asserted by (P.6) and (P.7) must be unique.Thus, if the language has a description operator‘ι’,[4] CEM supports the followingdefinitions:(42)x+y =dfιzw(Owz ↔ (Owx Owy))(43)x×y =dfιzw(Pwz ↔ (Pwx &Pwy))and (P.6) and (P.7) can be rephrased more perspicuously as(P.6′)Uxy → z(z = x+y)(P.7′)Oxy → z(z = x ×y).In other words, any two underlapping things have a uniquemerelogical sum, and any two overlapping things have a unique product.Actually the connection with extensionality is more subtle. In thepresence of the Weak Supplementation Principle (P.4), the productclosure (P.7) implies the Strong Supplementation Principle (P.5). Thus,CMM turns out to be the same theory asCEM.One could consider adding further closure postulates. For instance,it may be reasonable to require that a mereological domain be closedunder the operations of mereological difference and mereologicalcomplement. In the presence of extensionality these notions can bedefined as follows:(44)x−y =dfιzw(Pwz ↔ (Pwx &¬Owy))(45)~x =df ιzw(Pwz ↔¬Owx)The corresponding closure principles can therefore be statedthus:(P.8)¬Pyx → z(z =y−x)Remainder (P.9)z¬Pzx → z(z = ~x)Complementation The first of these is equivalent to (P.5), but the second isindependent of any of the principles considered so far. In manyversions, a closure theory also involves a postulate to the effect thatthe domain has an upper bound -- that is, there is something of whicheverything is part:(P.10)zxPxz TopAgain, in the presence of extensionality such a “universalindividual” is unique and easily defined:(46) U =dfιzxPxzThe existence of U makes the algebraic structure ofCEM even neater, since it guarantees that any twoentities underlap and, hence, have a sum. Thus, in the presence of(P.10) the antecedent in (P.6) may be dropped. On the other hand, fewauthors have gone so far as to postulate the existence of a “nullentity” that is part of everything:(P.11)zxPzxBottom(Two exceptions are Martin 1965 and Bunt 1985; see also Bunge 1966for a theory with several null individuals.) Without such anentity, which one could hardly countenance except for good algebraicreasons, the existence of a mereological product is not alwaysguaranteed. Hence (P.7) must remain in conditional form. Likewise,differences and complements may not be defined -- e.g., relative to theuniverse U. Hence, the corresponding closure principles (P.8)and (P.9) must also remain in conditional form.4.2. Unrestricted FusionsIn the literature, closure merelogies are just as controversial asextensional mereologies, though for quite independent reasons. We shallattend to these reasons shortly. First, however, let us note thepossibility of adding infinitary closure conditions. One mayallow for sums of arbitrary non-empty sets of objects, and consequentlyalso for products of arbitrary sets of overlapping objects (the productof all members of a class A is just the sum of all thosethings that are part of every member of A). It is notimmediately obvious how this can be done if one wants to avoidcommitment to classes and stick to an ordinary first-order theory --e.g., without resorting to the machinery of plural quantification ofBoolos (1984). As a matter of fact, in some classical theories, such asthose of Tarski (1929) and Leonard and Goodman (1940), the formulationof these conditions involves explicit reference to classes. (Goodmanproduced a class-free version of the calculus of individuals in 1951.)We can, however, avoid such reference by relying on an axiomschema that involves only predicates or open formulas.Specifically, we can say that for every satisfied property or conditionφ there is an entity consisting of all those things that satisfyφ. Since an ordinary first-order language has a denumerable supplyof open formulas, at most denumerably many classes (in any givendomain) can be specified in this way. But this limitation is, in a way,negligible, especially if one is inclined to deny that classes existexcept as nomina. We thus arrive at what has come to be knownas Classical or General Mereology(GM), which is obtained from Mbyadding the axiom schema(P.12)xφ → zy(Oyz ↔ x(φ & Oyx))Unrestricted Fusion(where, again, φ is any formula in the language). The result ofadding this schema to EM or MM yieldscorrespondingly stronger mereological theories. In fact, bothMM and EM extend to the sameextensional strengthening of GM -- the theory ofGeneral Extensional Mereology, or GEM --since (P.12) implies (P.7) and (P.7)+(P.4) imply (P.5) (Simons 1987:31). It is also clear that both GM andGEM are extensions of CM andCEM, since (P.6) also follows from (P.12). The logicalspace of all these theories can thus be represented schematically as inFigure 2. Figure 2. Hasse diagram of mereological theories(from weaker to stronger, going uphill).It is worth observing that if the extensionality principle issatisfied, then again at most one entity can satisfy the consequent of(P.12). Accordingly, in GEM we can define theoperations of general sum (σ) and product (π):(47)σxφ =df ιzy(Oyz ↔ x(φ &Oyx))(48)πxφ =df σzx(φ →Pzx).(P.12) then becomes(P.12′) xφ → z(z =σxφ),which implies(49) (xφ & yx(φ → Pyx)) → z(z =πxφ),and we have the following definitional identities whenever therelevant existential presuppositions are satisfied:(50) x+y = σz(Pzx Pzy)(51)x × y = σz(Pzx& Pzy)(52)x−y = σz(Pzx &¬Ozy)(53)~x = σz¬Ozx (54)U = σzPzz (It may be instructive to compare these identities with thedefinitions of the corresponding set-theoretic notions, with setabstraction in place of the fusion operator.) This gives us the fullstrength of GEM, which is in fact known to have a richalgebraic structure: Tarski (1935) proved that the parthood relationaxiomatized by GEMhas the same properties as theset-inclusion relation -- more precisely, as the inclusion relationrestricted to the set of all non-empty subsets of a given set, which isto say a complete Boolean algebra with the zero element removed.(Compare Clay 1974 for a corresponding result in relation toLeśniewski's mereology, which is not based on classicallogic.)Various other equivalent formulations of GEM arealso available, using different primitives or different sets of axioms.For instance, it is a theorem of every extensional mereology thatparthood amounts to inclusion of overlappers:(55) Pxy ↔ z(Ozx →Ozy).It follows that in an extensional mereology ‘O’ could beused as a primitive and ‘P’ defined accordingly. In fact,the theory defined by postulating (55) together with the Fusion Axiom(P.12′) and the Antisymmetry Axiom (P.2) is equivalent toGEM, but more elegant. Another elegant axiomatizationof GEM, due to Tarski (1929), is obtained by taking asonly postulates the Transitivity Axiom (P.3) and the Unique FusionAxiom (P.12′).4.3. Composition, Existence, and IdentityThe algebraic strength of GEM, and of its weakerfinitary variants, reflects substantive mereological postulates thatsome may find unattractive. Indeed, as anticipated above, closuremerelogies are just as controversial -- philosophically -- asextensional mereologies. Two objections, in particular, have been givenserious consideration in the literature. The first is that suchtheories are ontologically exuberant -- they involve asignificant increase in the number of entities to be included in aninventory of the world, contrary to the thought that mereology shouldbe “ontologically innocent”. The second objection is thatthey are ontologically extravagant -- they involve acommitment to a wealth of entities that are utterly counterintuitiveand for which we have no place in our conceptual scheme, contrary tothe thought that mereology should be “ontologicallyneutral”.To begin with the first objection, there is no question that atypical C(E)M model(not to mention aG(E)M model) is moredensely populated than a corresponding(E)M (or MM) model.If the ontological commitment of a theory is measured exclusively inQuinean terms -- via the dictum “to be is to be a valueof a bound variable” -- then clearly a mereological theoryaccepting a closure principle such as (P.6), (P.7), or (P.12) willinvolve greater ontological commitments than a theory rejecting suchprinciples, and one might find this unpalatable. This is particularlytrue for theories that accept the sum principles (P.7) or (P.12) butnot the Strong Supplementation postulate (P.5) -- hence theextensionality principle (34) -- for then the ontological exhuberanceof such theories may yield massive multiplication, as seen in section3.2. There is, accordingly, no question that the acceptance of aclosure principle requires substantive philosophical defense and canhardly be motivated exclusively in terms of the meaning of‘part’. Nonetheless, two sorts of remarks could be offeredon behalf of GEM and of its weaker finitaryvariants.First, it could be observed that the ontological exhuberanceassociated with the relevant closure principles is not substantive --that the increase of entities in the domain of quantification of aclosure mereology involves no substantive additional commitmentsbesides those already involved before the closure. This isperhaps best appreciated in the case of a closure principle such as(P.7), to the effect that any two overlapping entities have amereological product. After all, a product adds nothing. Butthe same could be said with respect to such principles as (P.6) and(P.12), which assert the existence of finitary or infinitarymereological sums. At least, this seems reasonable in the presence ofextensionality. For in that case it can be argued that even a sum is,in a sense, nothing “over and above” its constituent parts.As David Lewis put it:Given a prior commitment to cats, say, a commitment tocat-fusions is not a further commitment. The fusion is nothingover and above the cats that compose it. It just is them. Theyjust are it. Take them together or take them separately, thecats are the same portion of Reality either way. (1991:81)Thus, the ‘are’ of mereological composition -- themany-one relation of the parts to a whole -- is for Lewis a sort ofplural form of the ‘is’ of identity. For some authors(e.g., Baxter 1988) mereological composition is more than analogous toordinary identity. It is identity. The fusion is just theparts counted loosely; it is strictly a multitude and loosely a singlething. This is the thesis known in the literature as “compositionis identity”. And if this view is accepted, then one can speak ofa powerful mereological theory such as GEM as being“ontologically innocent” after all -- not only insofar asit is topic-neutral and domain-independent, but also insofar as it doesnot carry any additional ontological commitments besides those thatcome already with the choice of any model for a weaker theory such asEM. (For more discussion on this issue see van Inwagen1994, Yi 1999, Merricks 2000, Varzi 2000.)Secondly, it could be observed that the objection in question doesnot bite at the right level. If, given two objectsx1 and x2, the countenance of asum x1 + x2 is regarded as acase of further ontological commitment, then given a mereologicallycomposite object y1 + y2 thecountenance of its proper parts y1 andy2 could also be regarded as a case of furtherontological commitment. After all, every object is distinct from itsproper parts. So the objection in question would apply in the lattercase as well -- there would be ontological exhuberance in countenancingy1 and y2 along withy1 + y2. Yet this has nothingto do with the Sum axiom; it is, rather, a question of whether there isany point in countenancing a whole along with its parts. And if theanswer is negative, then there seems to be little use for mereologytout court. From the point of view of the present objection,it would appear that the only thoroughly parsimonious account would beone that rejects, not only some logically admissible sums, butany such sum. The only existing entities would be mereologicalatoms, entities with no proper parts. And such an account, thoughperfectly defensible, would be mereologically uninteresting: nothingwould be part of anything else and parthood would collapse to identity.(This account is sometimes referred to as mereologicalnihilism -- in contrast to the mereologicaluniversalism represented by adherence to the principle ofUnrestricted Composition. The terminology is from van Inwagen 1990:72ff.[5] For a detailed defense of nihilism see Rosenand Dorr 2002.)The second line of objection to closure mereologies -- to the effectthat they are ontologically extravagant -- might well becalled the objection from counter-intuitiveness and applies especiallyto theories accepting the principle of unrestricted fusion (P.12).According to this objection, it is all right to countenance certainmereological sums as bona fide entities -- for example whenthe summands make up an ordinary object or event. Even when thesummands are spatially scattered material objects (for instance) it maysometimes be reasonable to speak of them jointly as formingone thing, as when we speak of Mary's new bikini, of my copyof Proust's Recherche, of the solar system, or of some printedinscription consisting of separate letter tokens (see Cartwright 1975).However -- the objection goes -- a principle such as (P.12) would forceus to countenance all sorts of scattered objects, all sorts ofqueer entities consisting of scattered or otherwise ill-assortedsummands, such as you and I, my cat and your umbrella, or Chisholm'sleft foot and the top of the Empire State Building -- not to mentioncategorially distinct summands such as Chisholm's left foot andSebastian's stroll, your life and my favorite Chinese restaurant, orthe color red and the number 2. Such “sums” fail to displayany degree of integrity whatsoever and there appears to be no groundsfor treating them as unified wholes. There appears to be no reasonwhatsoever to postulate them on top of their constituent parts and,indeed, common sense disregards them altogether. (This objection goesback to the early debate on the calculus of individuals: see Lowe 1953and, again, Rescher 1955, with replies in Goodman 1956, 1958; for morerecent formulations see, e.g., Wiggins 1980, Chisholm 1987, and vanInwagen 1987, 1990.)A sympathiser of this objection need not adhere to a nihilistposition concerning mereological composition. More simply, theobjection reflects the intuitive view that only some mereologicalcomposites exist -- not all. And no doubt common sense supports thissort of intuition. In spite of this, two sorts of replies have beenoffered on behalf of (P.12), both of which are rather popular in theliterature. The first reply is that the question of whichfusions exist (what van Inwagen 1990 calls the “generalcomposition question”) cannot be successfully answered in arestricted way. Of course, it may well be that whenever some entitiescompose a bigger one, it is just a brute fact that they do so(Markosian 1998b). But if we are unhappy with brute facts, then thechallenge is to come up with a specification of the circumstances underwhich the facts obtain, so as to replace (P.12) with a restrictedversion. And according to the reply in question this is not a feasibleoption. Any attempt to do away with queer fusions by restrictingcomposition would have to do away with too much else besides the queerentities; for queerness comes in degrees whereas parthood and existencecannot be a matter of degree. In David Lewis's words:The question whether composition takes place in a givencase, whether a given class does or does not have a mereological sum,can be stated in a part of language where nothing is vague. Thereforeit cannot have a vague answer. … No restriction on compositioncan be vague. But unless it is vague, it cannot fit the intuitivedesiderata. So no restriction on composition can serve theintuitions that motivate it. So restriction would be gratuitous. (1986:213)(This line of argument, or some modified version of it, isparticularly congenial to authors adhering to a four-dimensionalontology of material objects; see e.g. Heller 1990: 49f, Jubien 1993:83ff; Sider 2001: 121ff, and Hudson 2001: 99ff.)The second reply on behalf of (P.12) is that the objection rests onpsychological biases that should have no bearing on ontological issues.Granted, we may feel uneasy about treating unheard-of fusions asbona fide entities, but this is no ground for doing away withthem altogether. We may ignore such things when we tally up the thingswe care about in ordinary contexts, but this is not to say they do notexist. We seldom speak with our quantifiers wide open; we normallyquantify subject to restrictions, as when we say “There is nobeer” meaning that there is no beer in the refrigerator. So inthat sense we may want to say that there are no cat-umbrellas andstroll-feet -- there truly aren't any such things among the things wecare about. But they are all there nonetheless, like the warm beer inthe garage. As James Van Cleve put it:Even if one came up with a formula that jibed with allordinary judgments about what counts as a unit and what does not, whatwould that show? Not … that there exist in nature such objects(and such only) as answer the formula. The factors that guide ourjudgments of unity simply do not have that sort of ontologicalsignificance. (1986: 145)From this perspective, the endorsment of (P.12) is certainly notneutral with respect to the question of what there is. But it loses itsflavor of counterintuitiveness, especially if combined with the“composition as identity” account mentioned in relation tothe first objection above.In recent years, further objections have been raised against closuremereologies -- especially against the the full strength ofGEM. These include objections to the effect thatunrestricted composition does not sit well with certain fundamentalintuitions about persistence through time (van Inwagen 1990, 75ff), orthat it implies that an entity must necessarily have the parts it has(Merricks 1999), or that it is incompatible with certain models ofspace (Forrest 1996b), or that it -- or the weaker closure principle(P.10) -- leads to paradoxes similar to the ones afflicting naïveset theory (Bigelow 1996). Such objections are still the subject ofon-going controversy and a detailed examination is beyond the scope ofthis entry. Some discussion of the first point, however, is alreadyavailable in the literature: see especially Rea 1998, McGrath 1998,2001, and Hudson 2001: 93ff. Hudson 2001: 95ff also contains adiscussion of the last point.5. Atomistic and Atomless MereologiesWe conclude this review of mereology by briefly considering 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:(56) Ax =df ¬yPPyx.Are there any such entities? And if there are, is everythingentirely made up of atoms? Does everything comprise at least someatoms? Or is everything made up of atomless gunk? These are deep anddifficult questions, which have been the focus of philosophicalinvestigation since the early days of philosophy and have been centerstage also in many recent disputes in mereology (see, for instance, vanInwagen 1990, Sider 1993, Zimmerman 1996, Markosian 1998a, Mason 2000.)Here we shall confine ourselves to pointing out that all options arelogically compatible with the mereological principles examined so farand can therefore be treated on independent grounds.The two main options, to the effect that there are no atoms at all,or that everything is ultimately made up of atoms, correspond to thefollowing postulates, respectively:(P.13)¬AxAtomlessness(P.14)y(Ay &Pyx).AtomicityThese postulates are mutually incompatible, but taken in isolationthey can consistently be added to any mereological theoryX considered in the previous sections. Adding (P.14)yields a corresponding Atomistic version, AX.By contrast, adding (P.13) yields an Atomless version,AX, in which the existence of a bottom level ofmereological entities is rejected -- everything is made up of“atomless gunk”. Since finitude together with theantisymmetry of parthood (P.2) jointly imply that decomposition intoparts must eventually come to an end, it is clear that any finite modelof M (and a fortiori of any extension ofM) must be atomistic. Accordingly, an atomlessmereology AX admits only models of infinitecardinality. (A world containing such wonders as Borges's Aleph, whereparthood is not antisymmetric, might by contrast be finite and yetatomless.) An example of such a model, establishing the consistency ofany atomless theory up to AGEM, is provided bythe regular open sets of a Euclidean space, with ‘P’interpreted as set-inclusion (Tarski 1935). On the other hand, theconsistency of any atomistic theory is guaranteed by the trivialone-element model (with ‘P’ interpreted as identity),though the full strength of AGEM is best appreciatedby considering that it is isomorphic with an atomic Boolean algebrawith the zero element removed.It bears emphasis that atomistic mereologies admit of significantsimplifications in the axioms. For instance, AEM canbe simplified by replacing (P.5) and (P.14) with(P.5′) ¬Pxy → z(Az &Pzx & ¬Pzy),which in turns implies the following atomistic variant of theextensionality thesis (34):(57) x=y ↔ z(Az →(Pzx ↔ Pzy))Thus, any atomistic extensional mereology is truly hyperextensionalin Goodman's sense: things built up from exactly the same atoms areidentical. Similarly, AGEM could be simplified byreplacing the Unrestricted Fusion postulate (P.12) with(P.12″) xφ → zy(Ay → (Pyz ↔ x(φ &Pyx))).An interesting question, discussed at some length in the late 1960's(Yoes 1967, Eberle 1968, Schuldenfrei 1969) and taken up more recentlyby Simons (1987: 44f), is whether there is any atomless analogue of(57). Is there any predicate that can play the role of ‘A’in an atomless mereology? Such a predicate would identify the“base” (in the topological sense) of the system and wouldtherefore enable mereology to cash out Goodman's hyperextensionalintuitions even in the absence of atoms. This question is particularlysignificant from a nominalist perspective, but it has deepramifications also in other fields (e.g., in connection with theWhiteheadian conception of space mentioned in section 3.1, according towhich space contains no parts of lower dimensions such as points orboundary elements; see Forrest 1996a and Roeper 1997). In special casesthere is no difficulty in providing a positive answer. For example, inthe AGEM model consisting of the open regularsubsets of the real line, the open intervals with rational end pointsform a base in the relevant sense. It is unclear, however, whether ageneral answer can be given that applies to any sort of domain,regardless of its specific composition. If not, then the only optionwould appear to be an account where the notion of a “base”is relativized to entities of a given sort. In Simons's terminology, wecould say that the G-ers form a base for the F-ersiff the following variants of (P.14) and (P.5′) aresatisfied:(P.14*) Fx → y(Gy & Pyx))(P.5*)(Fx & Fy) → (¬Pxy → z(Gz &Pzx & ¬Pzy)).An atomistic mereology would then correspond to the limit case where‘G’ is identified with ‘A’ for every choice of‘F’. In an atomless mereology, by contrast, the choice ofthe base would depend each time on the level of granularity set by therelevant specification of ‘F’.Between the two main options corresponding to Atomicity andAtomlessness, there is of course room for intermediate positions. Forinstance, it can be held that there are atoms, though not everythingneed have a complete atomic decomposition, or it can be held that thereis atomless gunk, though not everything need be gunky. (The latterposition is defended e.g. by Zimmerman 1996.) It is not difficult toprovide a formal statement of these views:(P.15)xAxWeak Atomicity(P.16)xy(Pyx →¬Ay)Weak AtomlessnessHowever, at present no thorough investigation of the resultingsystems has been entertained.Let us also mention, in closing, the option corresponding to thenihilist position mentioned in the previous section. 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