There is a dilemma that faces the proponent of an ontological denial, i. e. any thesis to the effect that there are no entities of a certain kind, e. g. properties, numbers, or physical objects. Either the proponent must say that sentences that apparently quantify over entities of that sort are always, strictly speaking, false (although perhaps they are "useful fictions"); or she must propose some paraphrase for each of these sentences (or better, a systematic way of paraphrasing these sentences), a paraphrase that clearly does not quantify over entities of that sort. 1 Sometimes it will be relatively unproblematic to seize the first horn of the dilemma: one who denies the existence of unicorns may plausibly say that sentences that apparently quantify over unicorns, such as `Some unicorns are white', are always false. And sometimes the second horn of the dilemma will be unproblematic: one may deny that there is any such thing as "the average family," and offer the following paraphrase of `The average family has 2.5 children': `The total number of children had by families divided by the total number of families is 2.5'. Other ontological denials have more trouble with this dilemma. Phenomenalism, the doctrine that there are no entities other than sensations, for all its epistemological merits, could not escape the dilemma. It is miserably implausible to deny that sentences apparently quantifying over physical objects, such as `There is a wineglass on the table', are always false; but no one could issue anything other than a promissory note concerning paraphrase of these sentences into a form clearly quantifying over only sensations.

The purpose of this paper is to look at one particular ontological denial, mereological nihilism, the thesis that there are no composite objects, i. e. objects that have proper parts. Admittedly, there may be no actual mereological nihilists, but there is at least one philosopher who is almost a mereological nihilist, Peter van Inwagen. In his book Material Beings, van Inwagen advances the thesis that there are no composite objects other than living organisms. 2 There are human beings, dogs, cats, plants, there are single cells that compose these organisms, and there are simples, according to van Inwagen; but there are no chairs, tables, houses, or stones. And when it comes to sentences apparently quantifying over chairs, tables, houses, and stones, van Inwagen seizes the second horn of the dilemma, offering paraphrases of sentences such as `Some tables are heavier than some chairs'. I shall argue that there are certain sentences that the nihilist and van Inwagen can give adequate paraphrases of only by assuming an ontology that many philosophers, including me, find much more problematic than an ontology of composite physical objects, viz. an ontology of sets. 3 These of sentences will, as it turns out, be just those sentences that give trouble to the nominalist who is trying to give formal interpretations of sentences into first-order logic without making use of plural quantification: that is, those sentences that cannot be rendered into first-order logic without use of set theory. 4

Now van Inwagen does not explicitly repudiate sets in Material Beings; indeed, in chapter 17, dealing with the problem of the many, he makes explicit use of sets, although only in response to his imaginary interlocutor, who has herself assumed an ontology of sets in order to state the problem of the many without assuming composite entities. But this, I suspect, should be considered a move in the dialectic, since van Inwagen states elsewhere 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" (Material Beings, 23). 5 At the very least, then, I hope to show that if you don't believe in sets, you'd better believe in composites, including chairs, tables, houses, and stones. For many philosophers, including me, this will be reason enough to reject mereological nihilism as well as van Inwagen's limited nihilism.

I should first say a bit about van Inwagen's method of paraphrase (as well as that of my imaginary nihilist). Van Inwagen's method of paraphrase makes crucial use of irreducibly plural referring expressions and quantifiers, and the primitive relation `is one of'. `The Beatles' is a plural referring expression, referring to the same things we refer to when we say `John, Paul, George, and Ringo'. 6 John is one of the Beatles, and Paul is one of the Beatles, George is one of the Beatles, and Ringo is one of the Beatles. Furthermore, our use of `The Beatles' is ontologically innocent: we do not commit ourselves to a set containing John, Paul, George, and Ringo, nor do we commit ourselves to a mereological fusion of John, Paul, George, and Ringo. 7 Nor do I commit myself to any class or fusion of John, Paul, George, and Ringo when I make the following inference: `The Beatles recorded St. Pepper's Lonely Heart Club Band in 1967; therefore, there are some musicians who recorded St. Pepper's Lonely Heart Club Band in
1967'. The predicate `recorded St. Pepper's Lonely Heart Club Band in 1967' is a distinctly plural predicate, 8 for the following inference is not valid: `John, Paul, George, and Ringo recorded St. Pepper's Lonely Heart Club Band in 1967; therefore, John recorded St. Pepper's Lonely Heart Club Band in 1967'. (Compare this with the following valid inference: `John, Paul, George, and Ringo are musicians; therefore, John is a musician'.) Furthermore, many plural predicates will be "variably multigrade," that is, the plural referring expressions that combine with them to form a complete sentence may refer to any number of entities. For example, the plural predicate `carried a coffin' is variably multigrade: it may combine with a plural referring expression referring to four people, or six, or eight. In addition to plural referring expressions and plural existential quantifiers, `there are some xs such that. . .', we may make use of plural universal quantifiers, `for all xs. . .', which may be defined in terms of the plural existential quantifier: `For all xs, F' =df. `It is not the case that there are some xs such that it is not the case that F'.

Van Inwagen's method of paraphrase also makes use of certain variably multigrade plural predicates, which I shall call "predicates of arrangement." An example of such a predicate would be `are arranged cubically', which would be applied to some non-overlapping spheres arranged in a 2x2x2 cube, or in a 3x3x3 cube, etc. In most cases where we have a predicate that we would be interpreted in a introductory logic class as a one-place predicate applying to composite physical objects, e. g. `is a chair', `is a table', `is a stone', van Inwagen paraphrases as a variably multigrade plural predicate that applies to some simples, e. g. `are arranged chairwise', `are arranged tablewise', or `are arranged stonewise'. Those predicates that are interpreted in introductory logic as two-place predicates applying to composite physical objects, van Inwagen paraphrases as two-place variably multigrade plural predicates: `This is heavier than that' becomes `These are collectively heavier than those'.

Here are several examples of how the nihilist and van Inwagen would paraphrase sentences which apparently quantify over physical objects. `There are exactly two chairs' is paraphrased as `There are xs and ys, the xs are distinct from the ys 9, the xs are arranged chairwise, the ys are arranged chairwise, and there are no zs distinct from the xs and from the ys such that the zs are arranged chairwise'. `Some tables are heavier than some chairs' is paraphrased as `There are some xs and some ys such that the xs are arranged tablewise, the ys are arranged chairwise, and the xs are collectively heavier than the ys'.

One sort of sentence that is troublesome for the mereological nihilist is the sort of sentence that compares things in number without saying how many of those things there are, such as `There are exactly as many chairs as there are tables', or `There are more chairs than there are tables'. The most straightforward way to render these sentences into "the language of logic" is to use plural quantification, and a primitive two-place plural relation `are equinumerous with':10 `There are exactly as many chairs as there are tables' becomes `There are some xs and some ys such that each of the xs is a chair, every chair is one of the xs, each of the ys is a table, every table is one of the ys, and the xs are equinumerous with the ys'. But this rendition of this sentence is not available to the nihilist, for appears to quantify over tables and chairs. The nihilist will have to find some other way of paraphrasing this sentence. But before looking at how the nihilist may try to paraphrase such sentences, it will be instructive to see how one might try to paraphrase these sentences into first-order form, without use of plural quantifiers.

One way of rendering these sentences into first-order logic without using plural quantifiers is to make use of an ontology of sets: `There are exactly as many chairs as there are tables' becomes `There are two sets, S1 and S2, such that every member of S1 is a chair and every chair is a member of S1, every member of S2 is a table and every table is a member of S2, and S1 is equivalent to S2'. 11 But as I have mentioned before, many philosophers find sets suspicious, and these philosophers, prior to the advent of plural quantification, desired another way to render these sentences into the language of logic. It turns out there is a way, but it is subject to some serious limitations.

Suppose we had some blocks, some tetrahedral and some cubical, and all of the blocks were the same with respect to some mereologically additive predicate, such as mass. 12 Then we may say that there are exactly as many cubes as there are tetrahedrons by saying that the sum of the cubes is equal in mass to the sum of the tetrahedrons. But suppose that the cubes and the tetrahedrons are not equal in mass. We may still compare the cubes and the tetrahedrons in number if there is some "representative part" of each cube and each tetrahedron such that these parts are equal in mass. We may then render `There are exactly as many cubes as there are tetrahedrons' as follows: 13 `There exists an x and a y such that x is part of the sum of the cubes and overlaps each cube, y is part of the sum of the tetrahedrons and overlaps each tetrahedron, 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 (or there are no cubes and no tetrahedrons)'. 14 These "maximally connected parts of the sum of x and y" (parts that are connected, and that are not part of any connected part of the sum of x and y) are our representative parts of the cubes and the tetrahedrons.

This paraphrase will not work, however, if any of the cubes (or the tetrahedrons) overlap mereologically (have some part in common). Suppose that there are two cubes that partially overlap, and one tetrahedron. Then the proposed paraphrase will be true, even though there are more cubes than tetrahedrons. For let x be some connected part of the common part of the two cubes. Then there will be some y, a connected part of the tetrahedron, equal in mass to x. The paraphrase does not enable us to pick a representative part of each cube, given that the cubes partially overlap. We need to add a clause to the paraphrase, ruling out the possibility that any of our representative parts (the maximally connected parts of the sum of x and y) are parts of more than one cube, or more than one tetrahedron. We may do this by tacking on the following clause to the paraphrase: `and there is no maximally connected part of the sum of x and y that overlaps more than one cube, or overlaps more than one tetrahedron'.

Adding this clause will not suffice to render the paraphrase materially equivalent to the original sentence, however, if there is some cube (tetrahedron) that is a part of another cube (tetrahedron). For if this were the case, there would be no way to pick an x such that it overlaps each cube, satisfying the first clause of the paraphrase, and such that it does not overlap more than one cube, satisfying the clause that we have tacked on to deal with cases of partial overlap. Now this will probably never be the case with our example, given that we do not consider those cubical (tetrahedral) proper parts of the cubes (tetrahedrons) to be themselves cubes (tetrahedrons). But there are other perfectly respectable predicates for which this will sometimes be the case, i. e. one entity satisfying that predicate will be a proper part of another entity satisfying that predicate. Consider the predicate `is a statue-arm or is a statue' (which is, I submit, just as respectable as `is a statue' and `is a statue-arm'), and suppose that we have one statue with two arms, and three chairs. It would be true, then, to say that there are exactly as many things that are statues or statue-arms as there are chairs. But we will not be able to pick out a representative part of either of the statue-arms, that is, a maximally connected part of a statue arm that does not overlap any statue. The proposed paraphrase will come out false.

Suppose then that we decided to drop the added clause, thinking perhaps that we were wrong to think that there could be two everyday objects (things satisfying normal, respectable sortals) that merely partially overlap, i. e. two things x and y such that x and y have a part in common, but x is not a part of y and y is not a part of x. For the first attempt at paraphrase will be adequate for the case in which there is one two-armed statue and three chairs. There is an x that is a part of the sum of the statues and its arms 15, and overlaps each statue or statue-arm viz. a sum of some connected one-gram part of the right arm, a connected one-gram part of the left arm, and a connected one-gram part of the body of the statue; and there is a y that is a part of the sum of the chairs and overlaps each chair, viz. a sum of a connected one-gram part from chair 1, a connected one-gram part from chair 2, and a connected one-gram part from chair 3; and each maximal connected part of the sum of x and y is the same mass as every other such maximal connected part, for each is one gram in mass; and x is the same mass as y, for they are both three grams in mass. The original paraphrase is true, as is the sentence to be paraphrased.

Alas, this will not generalize, even assuming that no things partially overlap. For the original paraphrase cannot deal with a case in which we have one two-armed statue and only two chairs. In this case, it would be false to say that there are exactly as many statues or statue-arms as there are chairs: there are three things that are statues or statue-arms, and only two chairs. But the original paraphrase (without the added clauses) will come out true: there is an x that is a part of the sum of the statues and the statue-arms and overlaps each statue or statue-arm, viz. the sum of a connected one-gram part of the left arm and a connected one-gram part of the right arm; and there is a y that is a part of the sum of the chairs and overlaps each chair, viz. the sum of a one-gram part of chair 1 and a one-gram part of chair 2; and each of the maximal connected parts of the sum of x and y has the same mass as every other maximal connected part of the sum of x and y, for each is one gram in mass; and x has the same mass as y, for each is two grams in mass.

Now we may see what problems will arise for the nihilist who wishes to give paraphrases of sentences comparing objects in number. Suppose that the nihilist wishes to give a paraphrase of `There are just as many cubes as there are tetrahedrons', while quantifying over (and committing herself to) only simples. The nihilist, using plural logic, may make use of the plural relation `are equinumerous with', and so she need not make use of a mereologically additive predicate, or worry about whether the simples are commensurable with respect to that predicate. Instead of picking a representative part of each cube (as we did when trying to paraphrase the sentences into first-order logic without plural quantification), the nihilist must try to pick a representative simple from the cubically arranged simples, and compare these simples in number.

(Recall that the nihilist paraphrases `There is a chair' as `There are simples arranged chairwise'.) As we shall see, the nihilist will face the same problems that we did when trying to render the sentence into first-order form without quantifying over sets.

Here's the nihilist's first shot at paraphrasing `There are exactly as many cubes as there are tetrahedrons': `There are xs and ys such that (1) the xs and ys are equinumerous, (2) for all zs, if the zs are arranged cubically, then exactly one of the zs is one of the xs, and (3) for all zs, if the zs are arranged tetrahedrally, then exactly one of the zs is one of the ys'. Here, the xs and the ys serve as the representative simples for the purposes of numeric comparison. But again, this paraphrase will not be successful if some of the cubes or tetrahedrons overlap. (Or, as the nihilist would have it, there are xs and ys such that the xs are arranged cubically, the ys are arranged cubically, and the xs and ys are distinct, but not completely distinct. See footnote 8.) Suppose that there are three cubes, two of which partly overlap, and two tetrahedrons. Let the xs consist of one simple in the overlap of the two cubes and one simple from the other, and let the ys consist of one simple from each tetrahedron. Then the paraphrase will be true, for all three clauses of the paraphrase are true; but the sentence we are interested in paraphrasing will not be (even "loosely speaking") true: there are more cubes than there are tetrahedrons.

We may try to remedy this defect (as we did when trying to give a paraphrase into non-plural logic) by tacking on two more clauses to the paraphrase: `(4) there is no v such that (a) v is one of the xs, and (b) there are distinct ws and zs such that the ws are arranged cubically, the zs are arranged cubically, and v is one of the ws and one of the zs' and `(5) there is no v such that (a) v is one of the ys, and (b) there are distinct ws and zs such that the ws are arranged cubically, the zs are arranged cubically, and v is one of the ws and one of the zs'. These two clauses are there to make sure that our representative simple is not "part of" more than one cube.

Once again, the paraphrase will work only if there is no cube (tetrahedron) that is a part of another cube (tetrahedron). (Or, as the nihilist would say, if there are no distinct xs and ys such that the xs are arranged cubically, the ys arranged cubically, and the xs are among the ys.) For if there is a cube (some xs that are cubically arranged) that is a part of (are among) another cube (some distinct ys that are cubically arranged), then it will be impossible to pick a representative simple for each cube. That is to say, we will be unable to pick some xs such that for all zs cubically arranged, exactly one of the zs is one of the xs, satisfying clause (2), and such that there are no distinct, cubically arranged, ws and zs such that none of the xs is both one of the ws and one of the zs, satisfying clause (4). Again, it is unlikely that this will ever be the case for our predicates `are cubically arranged' and `are tetrahedrally arranged': if the xs are cubically arranged, then there are no distinct ys among the xs that are also cubically arranged. (That is, no proper part of a cube is itself a cube.) But again, there are other perfectly respectable predicates for which this may well occur, such as `are arranged statue-arm-wise or arranged statuewise'. 16 Again, suppose that we have one two-armed statue and three chairs: then it will be "loosely speaking" true that there are exactly as many things that are statues or statue-arms as there are chairs. But the nihilist cannot paraphrase this sentence by the method we have been looking at, for she cannot pick a representative simple for each statue arm; that is to say, she cannot pick a simple that is one of some xs arranged statue-arm-wise that is not one of some simples arranged statuewise. The proposed paraphrase will be false.

The nihilist must, then, find some representative entities other than simples in order to compare the "virtual cubes" and "virtual tetrahedrons" (Material Beings, 112) in number. The nihilist might try regions of space as the representative entities. 17 Then we could paraphrase `There are exactly as many cubes as there are tetrahedrons' as follows: `There are some regions of space, the Rs, and some regions of space, the Ss, such that (1) the Rs are equinumerous with the Ss, (2) the occupants of each of the Rs are arranged cubically, and (3) the occupants of each of the Ss are arranged tetrahedrally'. This paraphrase is much less problematic than the previous one, in which simples were used as the representative entities, for there is no problem with partial overlap, or with one virtual object being a part of another. As long as each of our virtual objects (simples arranged cubically or tetrahedrally) occupy a unique region of space, the paraphrase will work. But this a strong metaphysical assumption, one which van Inwagen, at least, does not wish to make (Material Beings 50). Indeed, if current physical science is correct, it is not only metaphysically possible for two distinct objects to exactly occupy one region of space, it is physically possible:

At this point [near absolute zero], . . . somehow all the atoms will "go schlump," to use [physicist Carl] Wieman's words, all occupying the same place at the same time. . . . And, as Wieman is quick to point out, the atoms don't even have to be atoms. "In principle, you could do this with locomotives: If you took two locomotives at normal temperatures and put them on the same place on the same track, you'd get a giant crash. But if you got precisely identical locomotives cold enough, and the combined spins of all the particles that made them up added up to an integer, you could put a whole pile of them together. (David H. Freedman, "The Biggest Chill," 64-5)

Suppose then that there are three (virtual) cubes, two of which occupy exactly the same region of space, and two (virtual) tetrahedrons, occupying different regions of space. Then the proposed paraphrase will be true, for there will be regions of space, the Rs and the Ss, such that the Rs are equinumerous with the Ss, the occupants of each of the Rs are arranged cubically, and the occupants of each of the Ss are arranged tetrahedrally; but the sentence we are trying to paraphrase will not be even loosely speaking true: there are more cubes than there are tetrahedrons.

The final option for the nihilist is to use sets as the representative entities. The nihilist may paraphrase `There are exactly as many cubes as there are tetrahedrons' as follows: `There are some sets, the Rs, and some sets, the Ss, such that (1) the Rs are equinumerous with the Ss, (2) the members of each of the Rs are arranged cubically, and (3) the members of each of the Ss are arranged tetrahedrally'. This paraphrase does not have any problems with overlap, or with co-occupation of the same region of space, but it does commit the nihilist to the existence of sets. And for many philosophers, this will be reason enough to reject this paraphrase.

There are other sentences that will give the nihilist trouble as well, perfectly grammatical sentences of English. Consider this sentence: `There are some statues that are admired only by those critics who admire only those statues'. Or, if we wish to make the meaning of the sentence more perspicuous: `There are these statues; and anyone who admires one of them is a critic who doesn't admire anything that is not one of them'. Or, in somewhat formal plural logic: `There are some xs such that each of the xs is a statue, and for every y, if y admires one of the xs, then y is a critic and y doesn't admire any z that is not one of the xs'. How can the nihilist or van Inwagen, who does not admit the existence of statues, paraphrase this sentence? (Let's assume that we're admitting the existence of critics; for van Inwagen certainly does.) Here's the first attempt: `There are some xs such that (1) each of the xs is one of some ws arranged statuewise, (2) for all y, if y admires some ws arranged statuewise and at least one of the ws is one of the xs, then (a) y is a critic, and (b) there are no zs such that y admires the zs, and there is nothing that is one of the zs and one of the xs'. This paraphrase will work, provided that statues cannot overlap, a reasonable assumption about statues.

But again, it is not a reasonable assumption about things that are either statues or statue-arms, for a thing that is a either a statue or a statue-arm (in virtue of its being a statue-arm) may be a part of a thing that is also a statue or a statue-arm (in virtue of its being a statue). Suppose then, that we have a statue, Goliath, and that two particularly eccentric critics, Mary and Paula, admire the arms of Goliath and admire nothing else, and that no one else admires the arms of Goliath; and suppose further that John, who is not a critic, admires Goliath, indeed John admires every statue there is, but John does not admire the arms of Goliath. 18 In this case, the sentence to be paraphrased, `There are some statues or statue-arms that are admired only by those critics who admire only those statues or statue-arms' is (at least "loosely speaking") true: the statues or statue-arms are the arms of Goliath, and the critics are Mary and Paula. But the proposed paraphrase, `There are some xs such that (1) each of the xs is one of some ws arranged statuewise or statue-arm-wise, (2) for all y, if y admires some ws arranged statuewise or statue-arm-wise and one of the ws is one of the xs, then (a) y is a critic, and (b) there are no zs such that y admires the zs, and there is nothing that is one of the zs and one of the xs', is false. For no matter what xs we pick, if each of the xs is among some things arranged statue-arm-wise, then clause (2) will be false: John admires some ws arranged statuewise or statue-arm-wise, and at least one of the ws is one of the xs; but John is not a critic.

Sets, of course, would do the job nicely. Here's the paraphrase: `There are some sets, the Ss, such that (1) the members of each of the Ss are arranged statuewise or statue-arm-wise, and (2) for all y, if y admires some ws that are the members of one of the Ss, then (a) y is a critic and (b) there are no zs such that y admires the zs and the zs are the members of one of the Ss. But this paraphrase commits the nihilist to the existence of sets; and this, for many philosophers, is too high a price to pay for eliminating puzzles about composition.

I have herein shown that there are many perfectly acceptable English sentences that cannot be adequately paraphrased by the mereological nihilist, or by one who comes close to mereological nihilism, as does van Inwagen, without use of set theory. The choice, then, seems clear: believe that there are chairs, tables, statues, and statue-arms, and that these ordinary objects are composites; or believe in sets. I know which ones I believe in.

Notes:

1. The proponent of the ontological denial, may, of course, seize the first horn with respect to some of the sentences in question, denying that they are true, and the second horn with respect to others, offering a paraphrase. She must, in fact, seize the first horn with respect to some of the sentences, at the risk of refuting the ontological denial itself. It would be absurd to deny that there are numbers, and then offer a true paraphrase of the sentence `There are numbers', at least insofar as this sentence is uttered "in all ontological seriousness," or "in the philosophy room."

2. Van Inwagen does assume an ontology of events and, although he is rather coy about it, a substantial mereology of events, i. e. an ontology of composite events. Strictly speaking, van Inwagen's thesis should be stated thus: "There are no composite physical objects other than living organisms."

3. David Lewis, in Parts of Classes, has argued that all sets other than the singletons are themselves composite entities. I shall ignore this matter in this paper, holding the nihilist and van Inwagen only to denying the existence of composite physical objects.

4. The most famous of these sentences is the Geach-Kaplan sentence, `Some critics admire only one another'. One may prove that a sentence cannot be rendered into first-order logic without use of set theory by showing that there is an interpretation of the non-logical vocabulary of the sentence such that the reinterpreted sentence is true in all non-standard models of arithmetic but false in the standard model: for any first-order sentence of arithmetic is true in the standard model iff it is true in all the non-standard models (otherwise, the non-standard models wouldn't be models of arithmetic). See George Boolos' "To Be Is to Be the Value of a Variable (Or to Be Some Values of Some Variables)," Journal of Philosophy 81 (1984).

5. I am interpreting van Inwagen here as implying that we also do not need to use quantifiers that purport to quantify over pluralities, aggregates, or sets.

6. Note that there is no reason that a plural referring expression must refer to more than one thing. Van Inwagen's "proposed answer" to the special composition question, for example, is formulated as follows: "For all y, the xs compose y if and only if the activities of the xs constitutes a life (or there is only one of the xs)" (Material Beings, 82, emphasis added).

7. The ontological non-committal nature of plural quantifiers has been argued by Boolos, "To Be Is to Be the Value of a Variable (Or to Be Some Values of Some Variables)," op. cit., and Lewis, op. cit., 61-71.

8. I shall call a plural predicate F "distinctly plural" just in case the following inference is invalid: the xs are F; the ys are among the xs; therefore, the ys are F.

9. The xs are distinct from the ys if and only if there is at least one z that is not one of the xs but is one of the ys, or is not one of the ys but is one of the xs. The xs are completely distinct from the ys if and only if there is no z that is one of the xs and is one of the ys. The xs are among the ys if and only if every z that is one of the xs is one of the ys.

10. This two-place variably multigrade predicate may not be defined in first-order logic. To prove this, suppose that some first-order expression F "means" there are exactly as many As as there are Bs, i. e. F is true in exactly those models that assign the same number of elements to the extension of 'A' as they do to the extension of `B'. Consider then the set of sentences S that is the union of {'~F'}, A*={'There is at least one A', 'There are at least two As', . . .}, and B*={'There is at least one B', `There are at least two Bs', . . .}. S is clearly a consistent set of sentences. Since A* and B* are only satisfiable by models with infinite domains, S is satisfiable only by models with infinite domains. But since '~F' is a member of S, and F is true in all models that assign the same number of elements to 'A' and to `B', no model of S assigns a denumerable extension to both 'A' and to 'B', and so any model of S has a non-denumerable domain. But by the Lowenheim-Skolem theorem, we know that every consistent set of first-order sentences is satisfiable by a model with a finite or denumerable domain. Proof adapted from George Boolos, "For Every A There Is a B," Linguistic Inquiry
12 (1981).

11. Equivalence of sets may be defined rigorously in terms of set membership.

12. A predicate P is mereologically additive if and only if it is necessarily the case that if P(x, n) and P(y, m) (where n and m are real numbers) and x is disjoint from (has no common part with) y, then P(the sum of x and y, n+m).

13. This is an adaptation of the method of paraphrase given by Goodman and Quine in "Steps Toward a Constructive Nominalism," Journal of Symbolic Logic 12, 109-110.

14. If the cubes and tetrahedrons 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 cube composed of simples having 1g mass, and a cube or tetrahedron composed of simples having v2g mass, then no maximally connected part of the sum of x and y will have the same mass as every maximally connected part of the sum of x and y. That is, we would not be able to pick a representative part of each cube and each tetrahedron.

15. The sum of the statue and its arms is, of course, identical with the statue.

16. Van Inwagen denies the existence of undetached statue-arms, just as he denies the existence of statues. But he is still in need of some way of paraphrasing sentences that apparently quantify over statue-arms.

17. Regions of space are surely composite entities; but (see note 3) I will not begrudge the nihilist regions of space. Van Inwagen explicitly denies being committed to regions of space in "Reply to Reviewers," Philosophy and Phenomenological Research 53, 718, but I will not hold him to this either.

18. I am assuming here that admiration may be "holistic," in that one might admire a thing without admiring certain (or perhaps even all) of its parts.



References Boolos, George. "To Be is to Be the Value of a Variable (Or to Be Some Values of Some Variables)," Journal of Philosophy 81 (1984), 430-449.

"For Every A There Is a B," Linguistic Inquiry 12 (1981), 465-67.

Freedman, David H. "The Biggest Chill," Discover 14 (1993), 62-70.

Goodman, Nelson, and W. V. Quine. "Steps Toward a Constructive Nominalism," Journal of Symbolic Logic 12 (1947), 105-112.

Lewis, David. Parts of Classes. Cambridge, MA: Basil Blackwell, 1991.

Van Inwagen, Peter. Material Beings. Ithaca, NY: Cornell University Press, 1990.

Reply to Reviewers," Philosophy and Phenomenological Research 53 (1993), 709-719.

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