Complementation

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

(P.6)

Complementation[15]
¬Pyx → ∃z∀w(Pwz ↔ (Pwy ∧ ¬ Owx)).

This says that if y is not part of x, there exists something that comprises exactly those parts of ythat are disjoint from x—something we may call the difference or relative complement between yand x. It is easily checked that this principle implies (P.5). On the other hand, the diagram in Figure 5 shows that the converse does not hold: there are two parts of y in this diagram that do not overlap x, namely z and w, but there is nothing that consists exactly of such parts, so we have a model of (P.5) in which (P.6) fails.

اقتباس :: Figure 5. A strongly supplemented model violating Complementation.

Any misgivings about (P.5) may of course be raised against (P.6). But what if we agree with the above arguments in support of (P.5)? Do they also give us reasons to accept the stronger principle (P.6)? The answer is in the negative. Plausible as it may initially sound, (P.6) has consequences that even an extensionalist may not be willing to accept. For example, it may be argued that although the base and the stem of this wine glass jointly compose a larger part of the glass itself, and similarly for the stem and the bowl, there is nothing composed just of the base and the bowl (= the difference between the glass and the stem), since these two pieces are standing apart. More generally, it appears that (P.6) would force one to accept the existence of a wealth of “scattered” entities, such as the aggregate consisting of your nose and your thumbs, or the aggregate of all mountains higher than Mont Blanc. And since V. Lowe (1953), many authors have expressed discomfort with such entities regardless of extensionality. (One philosopher who explicitly accepts extensionality but feels uneasy about scattered entities is Chisholm 1987.) As it turns out, the extra strength of (P.6) is therefore best appreciated in terms of the sort of mereological aggregates that this principle would force us to accept, aggregates that are composed of two or more parts of a given whole. This suggests that any additional misgivings about (P.6), besides its extensional implications, are truly misgivings about matters of composition. We shall accordingly postpone their discussion to Section 4, where we shall attend to these matters more fully. For the moment, let us simply say that (P.6) is, on the face of it, not a principle that can be added to M without further argument.

3.4 Atomism, Gunk, and Other Options

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

(32)	Atom Ax =df ¬∃yPPyx.

By definition of ‘PP’, all atoms are pairwise disjoint and can only overlap things of which they are part. Are there any such entities? And, if there are, is everything entirely made up of atoms? Is everything comprised of at least some atoms? Or is everything made up of atomless “gunk”—as Lewis (1991: 20) calls it—that divides forever into smaller and smaller parts? These are deep and difficult questions, which have been the focus of philosophical investigation since the early days of philosophy and throughout the medieval and modern debate on anti-divisibilism, up to Kant's antinomies in the Critique of Pure Reason (see the entries on ancient atomism and atomism from the 17th to the 20th century). Along with nuclear physics, they made their way into contemporary mereology mainly through Nicod's (1924) “geometry of the sensible world”, Tarski's (1929) “geometry of solids”, and Whitehead's (1929) theory of “extensive connection” mentioned in Section 3.1, and are now center stage in many mereological disputes at the intersection between metaphysics and 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, and the papers collected in Hudson 2004; see also Sobociński 1971 and Eberle 1967 for some early treatments of these questions in the spirit of Leśniewski's Mereology and of Leonard and Goodman's Calculus of Individuals, respectively). Here we shall confine ourselves to a brief examination.
The two main options, to the effect that everything is ultimately made up of atoms, or that there are no atoms at all, are typically expressed by the following postulates, respectively:

(P.7)	Atomicity ∃y(Ay ∧ Pyx)
(P.8)	Atomlessness ∃yPPyx.

(See e.g. Simons 1987: 42.) These postulates are mutually incompatible, but taken in isolation they can consistently be added to any standard mereological theory X considered here. Adding (P.7) yields a corresponding Atomistic version, AX; adding (P.8) yields an Atomless version, ÃX. Since finitude together with the antisymmetry of parthood (P.3) jointly imply that mereological decomposition must eventually come to an end, it is clear that any finite model of M—and a fortiori of any extension of M—must be atomistic. Accordingly, an atomless mereology ÃXadmits only models of infinite cardinality. An example of such a model, establishing the consistency of the atomless versions of most standard mereologies considered in this survey, is provided by the regular open sets of a Euclidean space, with ‘P’ interpreted as set-inclusion (Tarski 1935). On the other hand, the consistency of an atomistic theory is typically guaranteed by the trivial one-element model (with ‘P’ interpreted as identity), though one can also have models of atomistic theories that allow for infinite domains. A case in point is provided by the closed intervals on the real line, or the closed sets of a Euclidean space (Eberle 1970). In fact, it turns out that even when X is as strong as the full calculus of individuals, corresponding to the theory GEM of Section 4.4, there is no purely mereological formula that says whether there are finitely or infinitely many atoms, i.e., that is true in every finite model of AX but in no infinite model (Hodges and Lewis 1968).
Concerning Atomicity, it is also worth noting that (P.7) does not quite say that everything is ultimately made up of atoms; it merely says that everything has atomic parts.[16] As such it rules out gunky worlds, but one may wonder whether it fully captures the atomistic intuition. In a way, the answer is in the affirmative. For, assuming Reflexivity and Transitivity, (P.7) is equivalent to the following

(33)	Pzx → ∃y(Ay ∧ Pyx ∧ Oyz),

which is logically equivalent to

(34)	((Ay ∧ Pyx) → Pyx) ∧ (Pzx → ∃y(Ay ∧ Pyx ∧ Oyz))

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

(35)	(φy → Pyx) ∧ (Pzx → ∃y(φy ∧ Oyz)).

And (35) is the closest we can get to saying that x is composed of the φs, i.e., all and only those entities that satisfy the given condition φ (in the present case: being an atomic part of x): every φ is part of x, and any part of x overlaps some φ. Indeed, provided the φs are pairwise disjoint, this is the standard definition of what it means for something x to be composed of the φs (van Inwagen 1990: 29), and surely enough, if the φs are all atomic, then they are pairwise disjoint. Thus, although (P.7) does not say that everything is ultimately composed of atoms, it implies it—at least in the presence of (P.1) and (P.2). (Of course, non-standard mereologies in which either postulates is rejected may not warrant the initial equivalence, so in such theories (33) would perhaps be a better 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 atomistic doctrine. A simple example is a descending chain of decomposition that never “bottoms out”, as in Figure 6: here x is ultimately composed of atoms, but the pattern of decomposition that goes down the right branch “looks” awfully similar to a gunky precipice. For a concrete example (from Eberle 1970: 75), consider the set of all subsets of the natural numbers, with parthood modeled 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 is ultimately composed of atoms. However, they violate the idea that everything can bedecomposed into its ultimate constituents. And this may be found problematic if atomism is meant to carry the weight of metaphysical grounding: as J. Schaffer puts it, the atomist's ontology seems to drain away “down a bottomless pit” (2007: 184); being is “infinitely deferred, never achieved” (2010: 62). Are there any ways available to the atomist to avoid this charge? One option would simply be to require that every model be finite, or that it involve only a finite set of atoms. Yet such requirements, besides being philosophically harsh and controversial even among atomists, cannot be formally implemented in first-order mereology, the former for well-known model-theoretic reasons and the latter in view of the above-mentioned result by Hodges and Lewis (1968). The only reasonable option would seem to be a genuine strengthening of Atomicity in the spirit of what Cotnoir (2013c) calls “superatomism”. Given any object x, (P.7) guarantees the existence of some parthood chain that bottoms out at an atom. Superatomicity would require that every parthood chain of x bottoms out—a property that fails in the model of Figure 6. At the moment, such ways of strengthening (P.7) have not been explored. However, in view of the connection between classical mereology and Boolean algebras (see below, Section 4.4), mathematical models for superatomistic mereologies may be recovered from the work on superatomic Boolean algebras initiated by Mostowski and Tarski (1939) and eventually systematized in Day (1960). (A Boolean algebra is superatomic if and only if every subalgebra is atomic, as with the algebra generated by the finite subsets of a given set; see Day 1967 for an overview.) See also Shiver (2015) for ways of strengthening (P.7) in the context of stronger mereologies such as GEM (Section 4.4), or within theories formulated in languages enriched with set variables or plural quantification.

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

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

(P.5′)

Atomistic Supplementation
¬Pxy → ∃z(Az ∧ Pzx ∧ ¬Pzy),

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

(27′)

x=y ↔ ∀z(Az → (Pzx ↔ Pzy)).

Thus, any atomistic extensional mereology is truly “hyperextensional” in Goodman's (1958) sense: things built up from exactly the same atoms are identical. In particular, if the domain 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 would identify the “base” (in the topological sense) of the system and would therefore enable mereology to cash out Goodman's hyperextensional intuitions even in the absence of atoms. The question is therefore significant especially from a nominalistic perspective, but it has deep ramifications also in other fields (e.g., in connection with a Whiteheadian conception of space according to which space itself contains no parts of lower dimensions such as points or boundary elements; see Forrest 1996a, Roeper 1997, and Cohn and Varzi 2003). In special cases there is no difficulty in providing a positive answer. For example, in the ÃEM model consisting of the open regular subsets of the real line, the open intervals with rational end points form a base in the relevant sense. It is unclear, however, whether a general answer can be given that applies to any sort of domain. If not, then the only option would appear to be an account where the notion of a “base” is relativized to entities of a given sort. In Simons's terminology, we could say that the ψ-ers form a base for the φ-ers if and only if the following variants of (P.5′) and (P.7) are satisfied:

(P.5φ/ψ)	Relative Supplementation (φx ∧ φy) → (¬Pxy → ∃z(ψz ∧ Pzx ∧ ¬Pzy))
(P.7φ/ψ)	Relative Atomicity φx → ∃y(ψy ∧ Pyx).

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, by contrast, the choice of the base would depend each time on the level of “granularity” set by the relevant specification of ‘φ’.
Concerning atomless mereologies, one more remark is in order. For just as (P.7) is too weak to rule out unpleasant atomistic models, so too the formulation of (P.8) may be found too weak to capture the intended idea of a gunky world. For one thing, as it stands (P.8) presupposes Antisymmetry. Absent (P.3), the symmetric two-element pattern in Figure 3, left, would qualify as atomless. To rule out such models independently of (P.3), one should understand (P.8) in terms of the stronger notion of ‘PP’ given in (20′), i.e.,

(P.8′)

Proper Atomlessness
∃y(Pyx ∧ ¬Pxy).

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

(P.4′′)

Atomless Supplementation

(P.4′′)

Atomless Supplementation
Pxy → ∃z(PPzy ∧ (Ozx → x=y)).

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 denumerably many, possibly continuum-many parts; but can it have more? Is there an upper bound on the cardinality on the number of pieces of gunk? Should it be allowed that for every cardinal number there may be more than that many pieces of gunk? (P.8) is silent on these questions. Yet these are certainly aspects of atomless mereology that deserve scrutiny. It may even be thought that the world is not mere gunk but “hypergunk”, as Nolan (2004: 305) calls it—gunk such that, for any set of its parts, there is a set of strictly greater cardinality containing only its parts. It is not known whether such a theory is consistent (though Nolan conjectured that a model can be constructed using the resources of standard set theory with Choice and urelements together with some inaccessible cardinal axioms), and even if it were, some philosophers would presumably be inclined to regard hypergunk as a mere logical possibility (Hazen 2004). Nonetheless the question is indicative of the sort of leeway that (P.8) leaves, and that one might want to regiment.
So much for the two main options, corresponding to atomicity and atomlessness. What about theories that lie somewhere between these two extremes? Surely it may be held that there are atoms, though not everything need be made up of atoms; or it may be held that there is atomless gunk, though not everything need be gunky. (The latter position is defended e.g. by Zimmerman 1996.) Formally, these possibilities can be put again in terms of suitable restrictions on (P.7) and (P.8), by requiring that the relevant conditions hold exclusively of certain entities:

(P.7φ)	φ-Atomicity φx → ∀y(Pyx → ∃z(Az ∧ Pzy))
(P.8φ)	φ-Atomlessness φx → ∀y(Pyx → ∃zPPzy).

And the options in question would correspond to endorsing (P.7φ) or (P.8φ) for specific values of ‘φ’. At present, no thorough formal investigation has been pursued in this spirit (though see Masolo and Vieu 1999 and Hudson 2007b). Yet the issue is particularly pressing when it comes to the mereology of the spatio-temporal world. For example, it is a plausible thought that while the question of atomism may be left open with regard to the mereological structure of material objects (pending empirical 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.7φ) or (P.8φ) in which ‘φ’ is understood as a condition that is satisfied exclusively by regions of space-time. Some may find it hard to conceive of a world in which an atomistic space-time is inhabited by entities that can be decomposed indefinitely (pace McDaniel 2006), in which case accepting (P.7φ) for regions would entail the stronger principle (P.7). However, (P.8φ) would be genuinely independent of (P.8) unless it is assumed that every mereologically atomic entity should be spatially unextended, an assumption that is not part of definition (32) and that has been challenged by van Inwagen (1981) and Lewis (1991: 32) (and extensively discussed in recent literature; see e.g. MacBride 1998, Markosian 1998a, Scala 2002, J. Parsons 2004, Simons 2004, Tognazzini 2006, Braddon-Mitchell and Miller 2006, Hudson 2006a, McDaniel 2007, Sider 2007, Spencer 2010). More generally, such issues depend on the broader question of whether the mereological structure of a thing should always “mirror” or be in perfect “harmony” with that of its spatial or spatio-temporal receptacle, a question addressed in J. Parsons (2007) and Varzi (2007: §3.3) and further discussed in Schaffer (2009), Uzquiano (2011) and Saucedo (2011). (For more on this, see the entry location and mereology.)
Similar considerations apply to other decomposition principles that may come to mind at this point. For example, one may consider a requirement to the effect that ‘PP’ forms a dense ordering, as already Whitehead (1919) had it:

(P.9)

Density
PPxy → ∃z(PPxz ∧ PPzy).

As a general decomposition principle, (P.9) might be deemed too strong, especially in an atomistic setting. (Whitehead's own theory assumes Atomlessness.) However, it is plausible to suppose that (P.9) should hold at least with respect to the domain of spatio-temporal regions, regardless of whether these are construed as atomless gunk or as aggregates of spatio-temporal atoms. For more on this, see Eschenbach 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 to the postulate

(P.10)

Bottom
∃x∀yPxy,

then such an entity would perforce be an atom. Accordingly, no atomless mereology is compatible with this assumption. But it bears emphasis that (P.10) is at odds with a host of other theories as well. For, given (P.10), the Antisymmetry axiom (P.3) will immediately entail that the atom in question is unique, while the Reflexivity axiom (P.1) will entail that it overlaps everything, hence that everything overlaps everything. This means that under such axioms the Supplementation principle (P.4) cannot be satisfied except in models whose domain includes a single element. Indeed, this is also true of the weaker Quasi-supplementation principle, (P.4d). It follows, therefore, that the result of adding (P.10) to any theory at least as strong as (P.1) + (P.3) + (P.4d), and a fortiori to MM and any extension thereof, will immediately collapse to triviality in view of the following corollary:

(36)	∃x∀y x=y.

‘Triviality’ may strike one as the wrong word here. After all, there have been and continue to be philosophers who hold radically monistic ontologies—from the Eleatics (Rea 2001) to Spinoza (J. Bennett 1984) all the way to contemporary authors such as Horgan and Potrč (2000), whose comparative ontological parsimony results in the thesis that the whole cosmos is but one huge extended atom, an enormously complex but partless “blobject”. For all we know, it may even be that the best ontology for quantum mechanics, if not for Newtonian mechanics, consists in a lonely atom speeding through configuration-space (Albert 1996). None of this is trivial. However, none of this corresponds to fully endorsing (36), either. For such philosophical theories do not, strictly speaking, assert the existence of one single entity—which is what (36) says—but only the existence of a single material substance along with entities of other kinds, such as properties or spatio-temporal regions. In other words, they only endorse a sortally restricted version of (36). In its full generality, (36) is much stronger and harder to swallow, and most mereologists would rather avoid it. The bottom line, therefore, is that theories endorsing (P.10) are likely to be highly non-standard, pace Carnap's persuasion that the null item would be a “natural and convenient choice” for certain purposes (such as providing a referent for all defective descriptions; see 1947: 37). A few authors have indeed gone that way, beginning with Martin (1943, 1965), who rejects unrestricted Reflexivity and characterizes the null item as “that which is not part of itself”. Other notable exceptions include Bunt (1985) and Meixner (1997) and, more recently, Hudson (2006) and Segal (2014), both of whom express sympathy for the null 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 paraconsistent recasting of the underlying logic. Still another option would be to treat the null item as a mere algebraic “fiction” and to amend the entire mereological machinery accordingly, carefully distinguishing between trivial cases of parthood and overlap (those that involve the infectious null item) and genuine, non-trivial ones:

(37)	Genuine Parthood GPxy =df Pxy ∧ ∃z¬Pxz
(38)	Genuine Overlap GOxy =df ∃z(GPzx ∧ GPzy).

The basic M-axioms need not be affected by this distinction. But stronger principles such as Supplementation could give way to their “genuine” counterparts, as in

(P.4G)

Genuine Supplementation
PPxy → ∃z(GPzy ∧ ¬GOzx),

and this would suffice to block the inference to (36) while keeping with the spirit of standard mereology. This strategy is not uncommon, especially in the mathematically oriented literature (see e.g. Mormann 2000, Forrest 2002, Pontow and Schubert 2006), and we shall briefly return to it in Section 4.4 below. In general, however, mereologists tend to side with traditional wisdom and steer clear of (P.10) altogether.

Complementation

3.4 Atomism, Gunk, and Other Options

Complementation :: تعاليق

Complementation