The Atlas of Reality
Page 84
Next, it is convenient to define the relation of mereological overlap or ‘overlap’ for short. Two things overlap just in case they have a part in common. Overlap is obviously a reflexive and symmetric relation, where R is a symmetric relation if and only if, if a thing A stands in the R to a thing B, then B stands in R to A as well. Everything overlaps itself, and x overlaps y if and only if y overlaps x. Overlap is not transitive, however.
Def D23.2 Overlap. x overlaps y if and only if there exists a z such that z is part of x and z is part of y.
Figure 23.2 Mereological Overlap
There are other plausible principles of mereology included by Leśniewski, Leonard, and Goodman as axioms of mereology. First, suppose that one thing is a proper part of another. For example, my hand is a proper part of my body. This seems to entail that my body has some other proper part, one that does not itself overlap my hand. This principle is Weak Supplementation:
(MA4) Weak Supplementation. If x is a proper part of y, then y has some part z that does not overlap x.
Figure 23.3 Weak Supplementation
Second, suppose something isn't a part of RCK's body. Suppose RCK is standing in the middle of the room. The north half of the room is not a part of RCK's body, although it does overlap his body. In that case, the north half of the room must have some part that doesn't overlap RCK's body. Otherwise, his body must include the north half of the room. Thus, if x is not a part of y, then x must have some part that does not overlap y. This is Strong Supplementation:
(MA5) Strong Supplementation. If x is not a part of y, then x has some part z that does not overlap y.
Figure 23.4 Strong Supplementation
Third, and most controversially, Leśniewski, Leonard, and Goodman assume that if S is a non-empty set or collection of things (a non-empty set is one that has at least one member), then there exists something that every member of S is part of, and which has nothing more as parts. An object o is said to be a sum of the members of S just in case everything that overlaps a member of S overlaps o, and everything that overlaps o overlaps at least one of the members of S.
Def D23.3 Sum. x is a sum of set S if and only if S is a non-empty set, and for all z, z overlaps x if and only if z overlaps one of the members of S. (Equivalent to our definition D18.2 in Chapter 18.)
This controversial principle, that every set of things has a sum, is Arbitrary Sums:
(MA6) Arbitrary Sums. If S is a non-empty set, then there exists at least one sum of S.
Arbitrary Sums is controversial because it entails the existence of some very odd objects, including highly scattered objects. Consider the non-empty set where the only two members are your left big toe and the Eiffel Tower. Arbitrary Sums entails that there exists a scattered object having your left big toe and the Eiffel Tower as parts, but which overlaps nothing but things that overlap either your big toe or the Eiffel Tower. Such a scattered object has a part here and a part in Paris, but no parts in between the two objects. Arbitrary Sums also entails that there exists something we could call “Big Aluminum”, an entity that includes as a part everything made wholly of aluminum and nothing that isn't made of aluminum. This object includes every aluminum can as a part, but none of the air or packaging that separates one can from another. Even worse, it entails the existence of sums that are both scattered and randomly heterogeneous, such as the northern half of every molecule on Earth. A common sense inventory of the world's furniture wouldn't include these bizarre amalgams!
There is a somewhat weaker version of Arbitrary Sums to consider, namely, Arbitrary Parts.
(MA6′) Arbitrary Parts. If there is a sum of the members of set S, and S′ is a subset of S, then there is a sum of the members of S′.
Arbitrary Parts is weaker than Arbitrary Sums because it is restricted to those sets of things that are already parts of some other thing. Suppose that there is no such thing as the World, a single object containing everything as a part. Then it might be that President Obama's left hand and the Eiffel Tower are not parts of any one thing. If so, then the axiom Arbitrary Parts does not entail that there is a sum of those two things. However, if Mr. Obama's left hand and his right foot are both parts of his body, then the axiom Arbitrary Parts would entail that there is something that is exactly the sum of his left hand and his right foot.
Finally, we can define what it is for some things to compose something else.
Def D23.4 Composition. The x's compose y if and only if y is a sum of the set having the x's as its only members.
With this terminology and the theory comprised of axioms (MA1–MA6) on the table as a starting point, we can consider the metaphysics of composition.
What sort of relation is parthood? Clearly it is binary and non-symmetric. For A to be a part of B does not entail that B is a part of A. In the next section, we move beyond these initial, uncontroversial observations.
23.2 Three (or Four) Answers to the General Composition Question
Given our work on universals and relations in Chapters 7 and 10, we can identify a range of possible answers to the General Composition Question.
1. Fundamental or natural parthood. First, there is there is the view that the parthood relation is either fundamental or natural:
23.1T Compositional Realism. The parthood relation, or some closely related relation, like joint-composition, is fundamental or corresponds to a relational universal.
Compositional Realism comes in two varieties, depending on whether we take parthood to be absolutely fundamental and ungrounded, or whether we take it to be grounded in the instantiation of some universal (or grounded in a set of fundamental resemblance facts):
23.1T.1 Fundamental Parthood. The parthood relation, or some closely related relation, like joint-composition, is fundamental.
23.1T.2 Natural Parthood. The x is-a-part-of y relation, or some closely related relation, like the x's jointly compose y, corresponds to a relational universal or a natural relational resemblance class.
The idea behind these views is that parthood is like instantiation for UP-Realists (7.1T.1T) or resemblance for Resemblance Nominalists (8.1T.4), or else part-whole pairs instantiate some relational universal or belong to a perfect similarity class. For Constituent Ontologists (9.1T), parthood may be the only fundamental relation. For Ostrich Nominalists (7.1A.1A), the distinction between Fundamental Parthood and Natural Parthood collapses, since for them the natural properties just are the fundamental ones.
2. Compositional Anti-Realism. The second answer to the General Composition Question is Compositional Anti-Realism:
23.1A Compositional Anti-Realism. The parthood relation is neither fundamental nor natural.
If Compositional Anti-Realism is true, then no fundamental truth involves parthood. Compositional Anti-Realism comes in two versions, moderate and extreme.
23.1A.1T Moderate or Reductive Compositional Anti-Realism. There are truths about proper parthood, but all truths about proper parthood are grounded in more fundamental, non-mereological truths.
23.1A.1A Compositional Nihilism. Nothing is a proper part of anything else.
Compositional Nihilism implies that everything we ordinarily say and believe about wholes and parts is simply false. Moderate Compositional Anti-Realists instead might propose some sort of translation or paraphrase of talk using mereological vocabulary into a theory without parthood or composition relations. Alternatively, Moderate Compositional Anti-Realists can treat wholes as wholly derivative or grounded entities. On this view, fundamental reality consists only of simple things.1 When we refer to composite things, we are merely referring to something wholly grounded in the simple parts and their arrangement.
All Compositional Anti-Realists must assume that, if anything exists at all, the fundamental things are all simple or atomic (lacking proper parts). Either composite things don't exist at all (Compositional Nihilism) or else they can simply be identified with derived entities that are wholly grounded by pluralities of atoms.
A
mong Moderate Compositional Anti-Realists, there is a further distinction to make, between those Moderate Compositional Anti-Realists who want to make Arbitrary Sums come out true, and those who do not. The members of the first group are Arbitrary Compositional Reductionists, and those of the second are Non-Arbitrary Arbitrary Compositional Reductionists.
23.1A.1T.1T Arbitrary Compositional Reductionism. All truths about proper parthood are grounded in more fundamental, non-mereological truths in such a way that Arbitrary Sums is true.
23.1A.1T.1A Non-Arbitrary Compositional Reductionism. All truths about proper parthood are grounded in more fundamental, non-mereological truths in such a way that Arbitrary Sums is false.
All Compositional Anti-Realists reject the existence of a metaphysically fundamental composition relation, the relation that holds between some x's and y when the x's compose y. However, not all Compositional Reductionists need suppose that any set of atoms constitute a composite thing. Non-Arbitrary Compositional Reductionists can suppose that whether something is a part of another depends upon special facts about those things and their non-compositional relations. The Arbitrary Compositional Reductionists, in contrast, think that we can talk truthfully about composites formed from every set of simples, since there are no relevant facts to distinguish some sets of simples from others.
Arbitrary Compositional Reductionists can offer something like the following explications or definitions of the part-whole and composition relations:
(1) If S is a non-empty set of atoms, then the members of S compose x if and only if x is that unique entity whose existence and intrinsic character is wholly grounded in the existence, character, and arrangement of the members of S.
(2) If A is composed of the atoms in S, and B is composed of the atoms in S', then A is a part of B if and only if S is a subset of S'.
Thus, it seems that Arbitrary Compositional Reductionists should think of the grounding of facts of composition and parthood in non-mereological facts as a case of conceptual grounding (Section 3.4), with the implication that composite objects do not really exist. Non-Arbitrary Compositional Reductionists, in contrast, could instead think of the grounding relation as extra-conceptual, dependent on the real essences of the kinds of composite things there are.
A major drawback with Arbitrary Compositional Reductionism is that it rules out the possibility of metaphysically fundamental composite things, in opposition to F.H. Bradley's argument against the universal priority of parts (Section 11.2.3), and in opposition to evidence for emergent substances (Section 22.6).
3. Other Forms of Compositional Reductionism. There are several versions of Non-Arbitrary Compositional Reductionism that deserve consideration. One specific proposal to consider is that one thing x is a part of y if and only if the location of x is part of the location of y. This proposal suffers from two problems. First, it seems that two things could be located in the same place without sharing any parts. For example, physics allows multiple bosons (like photons) to co-exist in the same location without sharing any physical parts. We might also think of various fields (like magnetic and gravitational fields) as mereologically separate entities that share the same location. Second, we shouldn't rule out the possibility of non-located objects, like souls or mathematical objects. It might turn out that some of these have parts. Finally, and most importantly, the proposal doesn'treally define the parthood relation in a non-circular way, since it presupposes that we have an account of what it is for one place to be a part of another.
Another proposal for defining composition is found in Koons (2014b), Aristotelian Reductionism. On this view, composition and the part-whole relationship (for substances) is wholly grounded in facts about causal powers and real processes.
23.1A.1T.1A.1 Aristotelian Compositional Reductionism. Some things, the x's, compose y at time t if and only if the x's are more than one, and there is some process P involving exactly the x's as participants such that (i) the existence of y at time t is causally explained by the continuation of P during some I that is terminated by t, (ii) all of the power-conferring properties of each of the x's are at t wholly grounded in the power-conferring properties of y at t, and (iii) every property of y that confers an active or passive power on y at t ontologically depends on one or more of the x's.
We can use this model of composition to define the part-of relation:
Aristotelian Definition of Proper Part. A substance z is an immediate proper part of y at t if and only if z is among some x's that compose y at t.
This account provides a two-way dependency between parts and wholes, one that respects the principle of Redundancy that we discussed in Chapter 22. The whole depends causally and diachronically (across time) on the activities of its parts, and the parts depend synchronically (at each moment) upon the whole for their occurrent causal powers. In addition, every active or passive power of the whole essentially involves one or more of the parts: that is, the whole cannot act or be acted upon except through some of its parts (thanks to clause iii). Clauses (ii) and (iii) provide grounds for thinking that a whole is spatially located exactly where its parts are located. In addition, the definition explicitly permits mereological inconstancy: wholes that gain or lose parts over time.
4. Composition as identity. The fourth answer to the General Composition Question was suggested by Donald Baxter (1988) and has been endorsed, with some important qualifications, by David Lewis (1991).
23.1T.1.1 Composition as Identity (CAI). Every whole is identical to its parts: to be a proper part is to be one of the things that are (collectively) identical to the whole.
Put simply, Composition as Identity is the view that wholes are simply identical to their parts (taken collectively, not distributively). If an encyclopedia is composed of ten volumes, then the encyclopedia simply is the ten volumes, and the ten volumes are the encyclopedia. The cloud is identical to its constituent droplets, and the droplets are identical to the cloud. Thus, to be a part of a whole W is simply to be one of the things that are (taken collectively) identical to W. On this view, parthood is simply a form of identity, an unproblematic, merely logical notion. Thus, this view is a form of Fundamental Parthood, since the identity relation is plausibly fundamental.
Unlike Compositional Reductionists, CAI theorists are not obviously committed to the view that only atoms are fundamental, or even to the view that there are atoms at all. CAI theorists are not saying that parts are more fundamental than the wholes they compose. In fact, they are equally fundamental, since the parts just are the whole.
Since according to CAI, the whole is nothing “over and above” its parts, a theory that posits the existence of a whole, given any collection of parts, doesn't commit its endorsers to any addition to reality, over and above the parts themselves. One might then think that if the droplets exist, so does the cloud; if the volumes exist, so does the encyclopedia; and, in general, for any non-empty set S, the members of S jointly form a whole, the whole being nothing more than the parts themselves. However, as Kris McDaniel (2010) has pointed out, CAI is compatible with Extreme Compositional Nihilism because it does not entail that the members of any set ever compose anything at all. Nonetheless, we might conjecture that a modified or stronger version of CAI would entail the axiom Arbitrary Sums (MA6) by requiring that the members of any non-empty set are collectively identical to something. (We expand this point in Section 23.3.)
Peter van Inwagen (1994) has complained that Baxter's theory can't be formulated without violating the rules of grammar. The identity predicate of standard logic is a binary predicate that combines two singular terms: x is identical to y. Baxter's theory requires us to use identity when referring to both a single thing (the whole) and a plurality (the parts): x is identical to the y's. Moreover, it seems that linking singular and plural terms with identity violates one of the fundamental principles of logic, Leibniz's Law of the Indiscernibility of Identicals (not to be confused with the converse principle, the Identity of Indiscernibles 7.3T).
Leibniz's Law says that if A is identical to B, then A and B must exemplify all the same properties. But a whole is one thing, and its parts are many (more than one). If the whole were identical to the parts, then the parts would also be one thing, and not more than one. So the whole cannot be identical to the parts.
In response, the defender of CAI could reply that grammatical convention is a poor guide to metaphysical truth. The fact that ordinary language and standard logic requires a sharp distinction between singular and plural terms shouldn't constrain our metaphysical theorizing.
In response to the charge of inconsistency concerning one-ness and many-ness, we should say that composite wholes are both one (in a way) and more than one (in another). The encyclopedia would be one encyclopedia and more than one volume, and the cloud would be one cloud and many droplets. It's not one cloud and more than one cloud or one droplet and more than one droplet, so no contradiction is involved.
CAI bears some similarity to Compositional Nihilism, even if its proponents are committed to the denial of Nihilism. If the world were ultimately composed entirely of simples, then both CAI and Compositional Nihilism would deny that there is anything fundamental over and above the simples. However, CAI does not force us to say that wholes are less fundamental than their parts; instead, we can say that the composite things exist and are not wholly grounded in any of their proper parts, since they are simply identical to their proper parts (taken collectively). Thus, CAI theory is consistent with Compositional Equivalence (22.1A.1T.1), as we discussed in the last chapter. Wholes are not posterior to their parts, but they are nothing over and above their parts, either. In addition, CAI is consistent, and Compositional Anti-Realism is not, with the Existence of Mereological Gunk (22.4A).