23.3 Accounting for the Correct Principles of Mereology
Can any of the proposed answers to the General Composition Question provide a basis for a plausible theory about the correct principles of mereology? Can any account for the essential features of the parthood relation? Five of the most interesting candidate features are Anti-Symmetry, Transitivity, Weak Supplementation, Strong Supplementation, and Arbitrary Sums, repeated here for convenience:
(MA2) Mereological Anti-Symmetry. If x is a part of y, and y is a part of x, then x = y.
(MA3) Mereological Transitivity. If x is a part of y, and y is a part of z, then x is a part of z.
(MA4) Weak Supplementation. If x is a proper part of y, then y has some part z that does not overlap x.
(MA5) Strong Supplementation. If x is not a part of y, then x has some part z that does not overlap y.
(MA6) Arbitrary Sums. If S is a non-empty set, then there exists at least one sum of S.
These principles are widely, but not universally, accepted by metaphysical mereologists. Metaphysicians have three ways to go here. (1) They could simply deny that these principles hold without exception. (2) They could admit that they hold, but suppose that their holding is a brute, inexplicable fact. Or (3) they could explain their holding by appealing to other principles and assumptions that are already part of their theory. Obviously, (3) would be the best response, although a theory's being able to endorse response (3) would not necessarily be a decisive reason for preferring that theory to its competitors.
TRANSITIVITY Transitivity can be explained by Moderate Compositional Anti-Realists, whether they embrace the Arbitrary or the Non-Arbitrary version of Compositional Reductionism. If y and z are composite, then in speaking of them we are really speaking of their composite simples. Let's say that the composite simples that make up y are the u's, and the simples that make up z are the v's. Then, to say that x is part of y is to say either that x is one of the u's, or that x is composed of some of the u's. To say that y is part of z is simply to say that all of the u's are also v's. Therefore, x is one of the v's or is composed entirely of v's, and so x is also part of z.
Somewhat surprisingly, Composition as Identity cannot provide a similarly elegant, logical explanation of Transitivity. The problem is that CAI theory does not entail that composite things are wholly composed of simples. All we can say is the following:
(A) x is part of y if and only if there are some u's such that: y is identical to the u's, and x is one of the u's.
(B) y is part of z if and only if there are some v's such that: z is identical to the v's, and y is one of the v's.
(C) x is part of z if and only if there are some w's such that: z is identical to the w's, and x is one of the w's.
Let's suppose that x is part of y and y is part of z. So, y is among the v's such that z is the v's, and x is among the u's such that y is the u's. To prove that x is part of z, we would have to prove either that x is one of the v's or that there are some w's such that x is among the w's and z is identical to the w's. However, neither of these follows by logic alone. Thus, CAI theorists must accept Transitivity as an additional law of metaphysics, as do those who accept Fundamental or Natural Parthood.
To be precise, CAI theorists must adopt an axiom somewhat weaker than Transitivity itself in order to be able to derive Transitivity as a result. In particular, they must assume Plural Expansion:
Plural Expansion. If the members of set A are jointly (or plurally) identical to the members of set B, then the members of the union of A and C are jointly identical to the members of the union of B and C.
Plural Expansion is a plausible principle, but it is not in any sense a theorem of logic, not even of Boolos's plural logic. At first glance, this claim is wrong: here's how you might try to derive Plural Expansion by logic alone. If the members of A are identical to the members of B, then each member of A is a member of B, and vice versa. So, by the extensionality axiom for sets, A = B. But then the union of A and C is identical to the union of B and C, and so their members must also be identical. However, this proof is fallacious, since it confuses two different forms of identity that can link pluralities. There is a definable form of identity, call it member-by-member identity or = M, and it is true that whenever the x's = M the y's, then each of the x's is one of the y's, and vice versa. However, Composition as identity insists that primitive, indefinable identity holds, not only between individuals, but also between individuals and pluralities, and, by transitivity, between pluralities. Let's call this collective identity, whenever it holds between two pluralities. For CAI, it is possible for the members of a set A to be collectively identical to the members of set B, even though set A does not have the same members (considered individually) as B does. For example, suppose x has two proper parts, y and z. Then, according to CAI, the members of the set {x} are collectively identical to the members of the set {y, z}, even though {x} ≠ {y, z}.
If we assume Plural Expansion, then Transitivity follows. Assume that x is a part of y and y is a part of z. Then there is some set A such that x is a member of A and y is identical to the members of A (taken collectively), and there is some set B such that y is a member of B and z is identical to the members of B, taken collectively. Obviously, y is identical to the members of the set {y}, and so the members of A are identical to the members of {y}. By Plural Expansion, the members of the union of A and B are jointly identical to the members of the union of {y} and B, taken collectively. Since y is a member of B, the union of {y} and B is identical to B. So, the members of the union of A and B are jointly identical to the members of B. Thus, the members of the union of A and B (taken collectively) are jointly identical to z. Since x is a member of A, x is a part of z.
Alternatively, the CAI theorist could try to build transitivity into their definition of the part-whole relation. For example, suppose that CAI theorists defined a ‘completely plurality’ in the following way:
Definition of Completed plurality (1). The x's are a completed plurality if and only if for every y such that y is one of the x's, if y = the z's, then each of the z's are one of the x's.
CAI theorists can now define the part-whole and the composition relation in this way:
CAI 1.1 Composition. The x's compose y if and only if y = the z's, the z's are a completed plurality, each of the x's is one of the z's, and the z's are the smallest completed plurality that includes each of the x's.
CAI 1.2 Part-whole. x is a part of y if and only if y = the z's, the z's are a completed plurality, and x is one of the z's.
Now we can prove that if x is part of y, and y is part of z, then x is part of z. The definition of part-of now insures that z is identical to some w's such that the w's are a completed totality. This guarantees that if y is one of the w's, and x is part of y, then x is also one of the w's, and so also a part of z.
Turning finally to Aristotelian Compositional Reductionism, we see that such Aristotelians can explain the transitivity of parthood by appealing to the transitivity of the three components of their reduction: the grounding relation, the causal explanation relation, and the ontological dependence relation (see Chapter 3 on grounding, and especially 3.3 on ontological dependence).
ANTI-SYMMETRY Anti-Symmetry is also easy to explain for Compositional Reductionists, whether of the Arbitrary or Non-Arbitrary variety. Where we talk of two composite things, there are really just two pluralities of simples, the x's and the y's. One composite thing is part of another just in case the simples corresponding to the first are included in the simples corresponding to the second. If each composite is a part of the other, then we are really speaking about exactly the same simples in both cases. Hence, symmetrical parthood entails identity.
Advocates of Fundamental or Natural Parthood and of Composition as Identity can offer no interesting explanation for Anti-Symmetry. They must suppose it to be a metaphysically brute fact. It may be a little surprising that CAI cannot derive Anti-Symmetry from the laws of logic. If x is a part of y,
and y is a part of x, then x must be identical to some things, the z's, and y must be identical to some things, the w's, such that x is one of the z's and y is one of the w's. Still, we cannot use logic alone to prove that all of the w's are z's, or that all the z's are w's, and so we cannot prove that x and y are identical.
However, Strong CAI (option 3) can explain Anti-Symmetry if it includes Plural Expansion. Suppose that x is a part of y and y is a part of x. Then x is a member of A and y is identical to the members of A, taken together, and y is a member of B and x is identical to the members of B, taken together. Since x is a member of {x}, we know that the members of B are collectively identical to the members of {x}. Consequently, by Plural Expansion, the members of the union of A and B are collectively identical to the union of the members of A and {x}. Since x is a member of A, the union of A and {x} is simply A. So, the members of the union of A and B are collectively identical to the members of A, that is, to y. By a similar argument we can show that the members of the union of A and B are identical to x. By the transitivity of identity, x is identical to y.
Once again, however, we should note that Plural Expansion is a substantive assumption and not itself derivable by logic alone.
What if we use instead the definitions of composition and part-of in terms of completed totalities (CAI 1.1 and CAI 1.2)? If x is a part of y, and y is a part of x, then both x and y must be completed pluralities. So, let's say that x = the z's and y = the w's. To show that x = y, we must show that each of the z's is one of the w's, and each of the w's is one of the z's. But this follows immediately from transitivity.
For Aristotelian Compositional Reductionists, the anti-symmetry of the part-whole relation derives from the more fundamental asymmetry of the proper part relation, and that asymmetry in turn derives from the still more fundamental asymmetry of the grounding relation. It is impossible for all of the power-conferring properties of x to be wholly grounded in the power-conferring properties of y, while the power-conferring properties of y are also wholly grounded in the power-conferring properties of x, which is what would be required if x and y were each proper parts of the other.
ARBITRARY SUMS As we mentioned above, Arbitrary Compositional Reductionism is designed to validate Arbitrary Sums. In contrast, many mereologists who accept Fundamental or Natural Parthood simply deny Arbitrary Sums.
What about Composition as Identity? Some, like Lewis (1991) and Sider (2007), have assumed that CAI automatically validates Arbitrary Sums. It is true that if S has some members, then there is something (a whole) that is identical to those members, and of which each of those members is a part. However, is this whole a sum of that set in the precise, Leśniewski sense? Recall the definition of sums:
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.
Let W(S) be the whole that is identical to the members of S. According to CAI, there is such a whole. The members of S are all parts of W(S). Is W(S) a sum of S? Is it the case that, for all z, z overlaps W(S) if and only if z overlaps one of the members of S? Given Transitivity, the right to left direction is no problem. If z overlaps one of the members of S, namely u, then z and u have a common part, v. Since u is a member of S, u counts as a part of W(S), and so z has a part that is a part of one of the parts of W(S), and thus z overlaps W(S).
However, the other direction is problematic. Suppose z overlaps W(S). Is there any way to prove, by logic and CAI alone, that z overlaps some member of S? In a word, No. If z overlaps W(S), then they have some common part u. That is, u is part of z and a part of W(S). However, just because u is part of W(S), it doesn't follow that u is a member of the set S.
Could we solve this by re-introducing the notion of a completed plurality? Our first definition of completed plurality took care of Transitivity, but it won't help with the left-to-right direction of the proof that W(S) is a sum of S. We can, however, fix this by replacing W(S), the plurality of members of S, with the completion of S, C(S).
Definition of CAI completion. The completion C(S) = the z's if and only if the z's are the smallest completed plurality that includes all of the members of S.
We can now prove that the completion of any set S is a sum of the members of that set, so long as each member of S is either an atom or a completed plurality. Suppose that z overlaps C(S). Then they have some common part u. That is, u is part of z and a part of C(S). Since C(S) is the smallest completed plurality including the members of S, this means that u is either a member of S, or a member of a member of S, and so on. Since each member of S is either an atom or a completed totality, it follows that u will be a member of some member of S.
Therefore, CAI theorists can claim a restricted version of Arbitrary Sums: every set S of atoms and completed pluralities has a sum, namely, the CAI completion of S. However, as we shall see, CAI theory has difficulty validating the axioms of Weak and Strong Supplemenation, which calls into question whether these CAI “sums” are really sums at all.
Non-Arbitrary Compositional Reductionists, including Aristotelian Reductionists, will simply reject the axiom of Arbitrary Sums, opting for a much more restrictive principle of composition.
WEAK SUPPLEMENTATION To consider the grounds of Weak Supplementation (MA4) raises the question of why something cannot have only a single proper part. What is it about wholes that requires a multiplicity of disjoint proper parts? Arbitrary Compositional Reductionists have an obvious answer to this question, since they translate ‘x is a proper part of y’ into ‘The x's are some of but not all of the y's.’ For example, to say that the giraffe's neck is a proper part of the giraffe is to say that the particles arranged giraffe's-neckwise are some but not all of the particles arranged giraffe-wise. From this translation, Weak Supplementation follows as a matter of simple logic: these (the x's) can be some but not all of those (the y's) only if there are some other things (the z's) that are not among these (the x's) but are among those (the y's).
Non-Arbitrary Compositional Reductionists can make use of the same explanation, on the plausible assumption that if the y's cohere together, and the x's are among the y's, then the x's cohere together as well. In other words, in order to explain Weak Supplementation, Non-Arbitrary Compositional Reductionists might also want to accept Arbitrary Parts.
Aristotelian Compositional Reductionists can explain Weak Supplementation in a different way, however. Their definition of proper parthood requires multiple, metaphysically independent entities to cooperate in some process that generates and sustains the composite entity (see Koons 2014b for more detail).
Advocates of Fundamental or Natural Parthood and of Composition as Identity must appeal to some primitive and inexplicable fact about the parthood relation or the universal PARTHOOD or parthood tropes in order to ground Weak Supplementation, or else they must simply deny Weak Supplementation altogether. Some metaphysicians have in fact denied Weak Supplementation, supposing that in some cases a thing can have a proper part that includes (or at least) overlaps all of its other proper parts. For example, Alfred North Whitehead supposed that a solid sphere has its interior (the sphere minus its outer surface) as a proper part, but he denied that the sphere had any other proper parts that were not parts of the interior, since he refused to count the surface of the sphere as a real entity in its own right. Here's another example. Classical Bundle Theorists (9.1T.1T.1A) might suppose that each universal in a bundle is a proper part of a part of the bundle (a trope-like part) that contains only that universal as a part. The universal and its trope-like container would be different, the universal a part of the container, even though the universal is the container's only proper part.
What about Composition as Identity? Suppose that x is a part of y, and x ≠ y. Does it follow that y has a part that has no part in common with x? This doesn't follow from CAI plus plural logic alone. If x is part of y, then y = the z's, and x is one of the z's. If x is not identical to y, then (given Anti-Symmetr
y), it must be the case that y is not a part of x. So, either x is an atom, or x = some w's, and y is not one of the w's. This tells us nothing about other parts of y, much less parts of y that have no part in common with x.
Does it help if we assume Plural Expansion? We don't think so. What about our proposed definitions of composition and part-of in terms of completed pluralities? No, those won't help either. Thus, CAI theory fails to provide an independent account of the essential principles of mereology. We will consider one more attempted solution to this problem: a solution that, if it worked, would simultaneously support both Weak and Strong Supplementation.
STRONG SUPPLEMENTATION Strong Supplementation raises a different question: why can't two distinct wholes share the same proper parts? Strong Supplementation is equivalent to the General Principle of Constituent Identity, the most general version of the various forms of PCI that we considered in Chapter 9:
PMeta 4 General Principle of Constituent Identity. If x and y both have proper parts, and they have exactly the same proper parts, then they must be identical.
If we accept Strong Supplementation, then we don't have to worry about Weak Supplementation, since Strong Supplementation, Reflexivity, Anti-Symmetry, and Transitivity together entail Weak Supplementation.
Once again, Arbitrary Compositional Reductionsists have a ready answer. The x's are different from the y's only if something is among the x's but not among the y's or vice versa. The other options must either reject PCI for Composite Objects or accept it as a further, brute metaphysical fact.
Those who reject General PCI (and Strong Supplementation) face two serious problems, one metaphysical and one epistemological. These problems concern the number of wholes composed by a given set of constituents. If the members of set S compose at least one thing, how many do they compose? If we reject General PCI, we cannot say, in all cases: exactly one. In some cases, the members of a set might compose two or three or a billion or even an infinite number of distinct composite entities, each having exactly the same proper parts as all the others. It would be like supposing that the atoms in my favorite tie compose not just one tie but exactly 17 ties. The metaphysical question is this: what accounts for the fact that the things in set S compose exactly n composites and not n + 1 or n − 1?
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