Universals in Lewis's Compositional Ontology — Epoche C1
Place three objects on a tatami mat: a tea bowl (茶碗, chawan ), a bamboo whisk (茶筅, chasen ), and a quantity of powdered green tea (抹茶, matcha ). By David Lewis's mereology — his theory of the part-whole relation, set out in On the Plurality of Worlds (1986) and developed in Parts of Classes (1991) — there are now not three objects on the mat but seven. The count is exact and the arithmetic is given below. Only one of those seven is the tea set that a practitioner of chanoyu would recognise, and nothing in the mereology distinguishes it from the six others. That is the observation from which the original version of this essay argued that Lewis's ontology leaves the unity of composites unexplained, and that universals — repeatable properties, capable of being wholly present in many things at once — must be brought back in to supply it. The observation is right and the argument built on it needs substantial repair. Lewis has an argument for refusing to add a unity condition, and it is a good one. He also has an elite class of properties doing very nearly the work the essay demands of universals, which the original text overlooked. But the essay's instinct survives all of that, provided it is aimed at the right target: there is something mereology demonstrably cannot represent, and it is not unity but structure. The arithmetic of unrestricted composition Classical extensional mereology takes parthood as primitive and adds three sorts of principle: that parthood is transitive, so that a part of a part is a part of the whole; a supplementation principle, ensuring that if one thing is a proper part of another there is something else making up the difference; and unrestricted composition , which says that for any objects whatever there exists a fusion — an object having all of them as parts and containing nothing foreign to them. Extensionality follows: objects with the same parts are the same object. Take a world, or a mat, containing $n$ mereological atoms — objects with no proper parts. Every non-empty set of atoms has exactly one fusion, and by extensionality distinct sets have distinct fusions. The number of objects is therefore the number of non-empty subsets of an $n$-element set: $$2^{n} - 1 .$$ The exponent is $n$ because each atom is independently either in or out of a given subset, which is two choices per atom; the subtraction of $1$ removes the empty set, which has no fusion, because classical mereology admits no null individual. With the three objects on the mat, $2^{3} - 1 = 7$: the three items themselves, the three pairs (bowl-and-whisk, bowl-and-tea, whisk-and-tea), and the triple. The tea set is the triple. The pair consisting of the whisk and the tea is exactly as much an object, on this theory, as the tea set is. Lewis holds that this profusion costs nothing, on the ground he calls the ontological innocence of mereology: the fusion is nothing over and above its parts, so in accepting it one has not accepted a further thing. It is worth being precise about the thesis, because the original essay attributed to Lewis a stronger one. In Parts of Classes Lewis endorses composition as identity only in an analogical form. Literal identity is one-one; composition is many-one, and Lewis explicitly declines the strong claim that the many parts simply are the one whole. What he claims is that the fusion is not an additional item in the inventory — that describing the parts and then describing the whole is describing the same portion of reality twice. Why Lewis refuses to add a unity condition The natural demand, and the one the original essay makes, is for a restriction: parts should compose a whole only when they are suitably connected, cohesive, functionally integrated. Peter van Inwagen named this the Special Composition Question in Material Beings (1990) — under what conditions do some things compose something? — and gave an answer of exactly the kind the essay wants. Composition occurs, van Inwagen says, when and only when the activity of the parts constitutes a life. The consequence is that there are simples and there are organisms, and there are no tables: what exists in a dining room is simples arranged tablewise. Whatever else one says of this, it is not a soft option, and it shows that a genuine unity condition costs something enormous. Lewis's reply is the argument from vagueness, given in On the Plurality of Worlds (§4.3) and formalised by Theodore Sider in Four-Dimensionalism (2001). It runs in four steps. Any restriction on composition of the kind wanted — contact, cohesion, causal integration, joint function — would have to be a vague one, since one can construct a continuous series of cases from a clear instance to a clear non-instance with no non-arbitrary line anywhere in it. If composition is vague in that way, then it is sometimes indeterminate whether some things compose something, and hence indeterminate how many objects there are. But how many objects there are is answered by a number, and a count admits no borderline cases: existential quantification and identity are not vague, so there is no coherent state of affairs in which the number of things is vaguely 5. Therefore composition is not restricted in that way. Either it never occurs, or it occurs among any things whatever. The turn to notice is at step 3. The argument does not claim that "table" is a precise term — Lewis is perfectly happy for our ordinary sortal predicates to be vague, and for it to be indeterminate which fusion a given use of "table" picks out. What he denies is that the vagueness can be located in the world's inventory rather than in our language. The compressed version of this essay assumed that Lewis simply had no answer to the question of what makes a heap of legs and screws into a table. He has one, and it is that this is not a question about composition at all: the fusion exists either way, and what changes when the table is assembled is which properties the fusion has and whether it me