In general, the plausibility of a "ontological denial," i.e. a metaphysical thesis stating that there are no things of such and such a sort (properties, sets, material objects, etc.), rests on the possibility of giving a paraphrase of sentences that apparently quantify over entities of that sort. (The other option, is, of course, to deny that such sentences are ever true; but, depending on the sort of entity in question, this option will often put the metaphysical thesis outside of the range of plausibility.) Mereological nihilism, the thesis that no composite objects exist, and Van Inwagen's thesis, that the only composite material objects that exist are living organisms, which is "almost mereological nihilism," are both ontological denials. As such, their plausibility rests on the possibility of giving paraphrases of sentences such as `There are two chairs over there' and `Some chairs are heavier than some tables'.
The mereological nihilist and Van Inwagen can do a good job with paraphrasis of these two sentences, utilizing only plural quantification and variably multigrade primitives such as `are arranged chairwise', `are disjoint from', and `are collectively heavier than'. `There are (at least) two chairs over there' can be paraphrased as `There are xs and ys such that the xs are disjoint from the ys, the xs are arranged chairwise, the ys are arranged chairwise, and the xs are over there, and the ys are over there'. `Some chairs are heavier than some tables' becomes `There are xs and ys such that the xs are arranged chairwise, the ys are arranged tablewise, and the xs are collectively heavier than the ys'.
There are other sentences that the nihilist and Van Inwagen cannot so easily paraphrase. Take, for example, the sentence `There are exactly as many chairs as there are tables'. I know of three ways to render this sentence into "the language of logic": two of which are unavailable to the nihilist and Van Inwagen, and the last of which involves quantifying over sets, which many philosophers, including me, find much more ontologically suspicious than tables and chairs.
The first way to render this sentence into "the language of logic" is by straightforward use of plural quantification over tables and chairs, along with a primitive predicate `are equinumerous': `There are some xs and ys such that all the xs are chairs, all chairs are one of the xs, all the ys are tables, all tables are one of the ys, and the xs and ys are equinumerous'. But this rendering of the sentence is clearly unavailable to the nihilist or to Van Inwagen, since it involves quantifying over chairs and tables, which "ontologically commits" one to the existence of chairs and tables.
The sentence can also be rendered into the language of first-order logic, without plural quantification, if one helps oneself to the full-blown calculus of individuals, to the predicate `is maximally connected', and to some sort of predicate that is mereologically additive, such as `has mass n' or `has volume n'. 1 (This is the Quine-Goodman trick from "Steps Toward a Constructive Nominalism," Journal of Symbolic Logic 12, 109-10.) We may render our sentence into first-order logic without plural quantification as follows: `There exists an x and a y such that x is part of the sum of the chairs and overlaps each chair, y is part of the sum of the tables and overlaps each table, and each maximally connected part of the sum of x and y is the same mass as every maximally connected part of the sum of x and y, and x is the same mass as y'. 2 But this rendering of the sentence is not available to the nihilist or to Van Inwagen, as it quantifies over tables and chairs. Worse still, from Van Inwagen's point of view, it makes use of the "Doctrine of Arbitrary Undetached Parts", a doctrine I suspect that Van Inwagen would continue to deny even if he were persuaded to give up his almost-nihilism and admit chairs and tables into his ontology.
The only way I know of that the nihilist or Van Inwagen can paraphrase our sentence without quantifying over tables and chairs is the following: `There is a set S1 and a set S2, every member of S1 is a set of simples arranged chairwise, there is no set of simples arranged chairwise that is not a member of S1, every member of S2 is a set of simples arranged tablewise, there is no set of simples arranged tablewise that is not a member of S2, and S1 and S2 are equivalent.' (Equivalence of sets is rigorously definable in terms of set membership; we must also assume that `arranged chairwise' and `arranged tablewise' are maximal predicates, i.e. that if the xs are arranged chairwise, then there are no ys properly among the xs such that the ys are arranged chairwise.) But this rendering of our sentence into logic would commit Van Inwagen or the nihilist to the existence of sets.
Van Inwagen would probably not be disturbed by this commitment to sets, as sets do not have the problems of persistence through change and counterfactual identity which motivate Van Inwagen's denial of composite objects other than living things, for sets have their members essentially. But many philosophers do find sets to be ontologically suspicious, something we should do without in our metaphysics if at all possible. The moral of the story, then, is this: if you don't believe in sets, you better believe in composite material objects. At the very least, I have shown that Van Inwagen is wrong when he says that "we can achieve all the powers of plural or collective reference we shall need for our discussion of composition without using singular terms that purport to refer to pluralities or aggregates or sets. We shall only need plural referring expressions" (Material Beings, 23).
1. A predicate P is mereologically additive just in case necessarily, if P(x,n) and P(y,m) (where n and m are real numbers) and x is disjoint from y, then P(the sum of x and y, n+m).
2. If the chairs and tables are composed of simples, then a necessary, but not sufficient, condition of this paraphrase working is that the simples are of commensurable masses. If there is a table or composed of simples having 1 g mass, and a table or chair composed of simples having square root of 2 g mass, then no maximally connected part of x+y will have the same mass as every maximally connected part of x+y.
Van Inwagen, Peter. Material Beings. Cornell, 1990.
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