Friday, February 18, 2005

Van Inwagen and Lewis on Composition and Identity

Modifying an example employed by Donald Baxter and David Lewis, suppose I own a parcel of land A consisting of exactly two adjoining lots B and C. It would be an insane boast were I to claim to own three parcels of land, B, C, and A. That would be ‘double-counting’: I count B, C, and A, and give three as the number of parcels I own. Lewis, rejecting ‘double-counting,’ will say that A = (B + C). Thus A is identical to what composes it.

Peter van Inwagen, who opposes composition as identity, argues against it as follows:

Suppose that there exists nothing but my big parcel of land and such parts as it may have. And suppose it has no proper parts but the six small parcels. . . . Suppose that we have a bunch of sentences containing quantifiers, and that we want to determine their truth-values: ‘ExEyEz(y is a part of x & z is a part of x & y is not the same size as z)’; that sort of thing. How many items in our domain of quantification? Seven, right? That is, there are seven objects, and not six objects or one object, that are possible values of our variables, and that we must take account of when we are determining the truth-value of our sentences. ("Composition as Identity," Philosophical Perspectives 8 (1994), p. 213)

In terms of my original example, Lewis is saying that A is identical to what composes it. Van Inwagen is denying this and saying that A is not identical to what composes it. His reason is that there must be at least three entities in the domain of quantification to make the relevant quantified sentences true. A is therefore a third entity in addition to B and C.

Van Inwagen’s argument strikes me as a non sequitur. Consider this sentence:

1. For any x, there is a y such that x = y.

(1) features two distinct bound variables, ‘x’ and ‘y.’ But it does not follow that there must be two entities in the domain of quantification for (1) to be true. It might be that the domain consists of exactly one individual a. Applying Existential Instantiation to (1), we get

2. a = a.

Relative to a domain consisting of a alone, (1) and (2) are logically equivalent. From the fact that there are two variables in (1), it does not follow that there are two entities in the domain relative to which (1) is evaluated. Now consider

3. There is an x, y and z such that x is a proper part of z & y is a proper part of z.

(3) contains three distinct variables, but it does not follow that the domain of quantification must contain three distinct entities for (3) to be true. Suppose that Lewis is right, and that A = (B + C). It will then be possible to existentially instantiate (3) using only two entities, thus:

4. B is a proper part of (B + C) & C is a proper part of (B + C).

If van Inwagen thinks that a quantified sentence in n variables can be evaluated only relative to a domain containing n entities (or values), then I refute him using (1) above. If van Inwagen holds that (3) requires three entities for its evaluation, then I say he has simply begged the question against Lewis by assuming that (B+C) is not identical to A.

It is important not to confuse the level of representation with the level of reality. That there are two different names for a thing does not imply that there are really two things. (‘Hesperus’ and ‘Phosphorus’ both name the same planet, Venus, to coin an example.) Likewise, the fact that there are two distinct bound variables at the level of linguistic representation does not entail that at the level of reality there are two distinct values. There might be or there might not be.

Here is the problem. ‘A = (B + C)’ is the logical contradictory of ‘~ (A = (B + C)).’ Thus one will be tempted to plump for one or the other limb of the contradiction. But there are reasons to reject both limbs. Surely A is more than the mere sum of B and C. This is because A involves a further ontological ingredient, namely, the connectedness of A and B. To put it another way, A is a unity of its parts, not a pure manifold. The Lewis approach leaves out unity.

The connectedness, however, which makes for the unity, is not a third entity, and this for two reasons. First, it is empirically undetectable, and second, if there were a third entity, we would be involved in a Bradleyan regress. To simple announce that A is a third entity in addition to B and C is to fail to see the problem. It is obvious that A is not wholly distinct from B and C inasmuch as A is composed of B and C as its sole proper parts. Analysis of A discloses nothing other than B and C. That’s all she wrote.

Therefore, if A is distinct from (B + C), then there must be something that makes them distinct. There must be something which, when added to (B + C), results in A. But what could that be? To simply announce that A is distinct from B + C is to help oneself to the very thing whose existence in reality is in question.

In short, both limbs of the contradiction are unacceptable. How then are we to avoid it? Or should we accept it? And then what?

FURTHER READING: Composition as Identity Resource Page