2. Plural Semantics

It might seem that if nothing has parts, then much of our everyday talk is untrue. It's tempting to hold that when I say, ‘I made a bookcase at the weekend’, or ‘My bicycle's front wheel is missing a spoke’, I'm implying that there's something that has parts. If so, then if nothing has parts, what I say implies something false. And that means that what I say is false.Footnote ⁶ But I believed what I said. It seems, then, that anyone who denies that anything has parts must deny many of our everyday thoughts and assertions.

However, van Inwagen disagrees: he claims that, although nothing has parts, these thoughts and assertions are often true. In van Inwagen's view, sentences which seem to imply the existence of composite entities – that is, entities that have parts – do not genuinely do so.

Van Inwagen offers us paraphrases of these sentences. For instance, he paraphrases

Note that C* involves plural quantification (‘There are xs’) rather than the more commonly-encountered singular quantification (‘There is an x’).

It is not at all clear what van Inwagen thinks the relation is between the original sentence and the paraphrase. He claims that C* ‘describes the same fact’ as C but denies that the two sentences ‘express the same proposition’ (1990: 113, 112). On one hand he says:

On the other, he says that the sentence ‘There are two valuable chairs in the next room’ is ‘“essentially existential”, meaning that … a correct paraphrase – correct by the most pedantic and literalistic standards – of this sentence into the language of formal logic must start with an existential quantifier’ (1990: 102).

It is, then, hard to fathom exactly what van Inwagen is offering us. What ought he to offer us? Van Inwagen claims that sentences like C only appear to imply the existence of composite entities: in his view, they don't really do so. To back this up, we would like van Inwagen to tell us what the truth-conditions of these sentences are, in a way that makes clear that these conditions can obtain even in a world where there are only simples (things without parts).

It would be unsatisfying just to be told what the truth-conditions of these sentences are, sentence by sentence. We would prefer to be given a systematic way of working out which truth-conditions each sentence possesses. To do this, van Inwagen needs to explain what contributions the words in these sentences make to their truth-conditions. In one sense of the term, then, we would like him to supply us with a semantic theory for this part of English.

I will now sketch a theory of this sort, and – for simplicity's sake – impute it to van Inwagen. I am not confident that van Inwagen would endorse this theory, even though his paraphrases are consequences of it; but since it provides what we would like van Inwagen to provide, I think it is worth discussing whether or not van Inwagen believes it.Footnote ⁸

Consider the true sentence

It might seem as if this contains a singular term, ‘My bookcase’, referring to a composite entity; if so, it can't be true unless there are such things as composite entities. Van Inwagen is thus faced with a choice. He can either affirm or deny that ‘My bookcase’ refers. If he affirms this, he must deny that its referent is composite. On this view, there are bookcases, but they are simples. Van Inwagen chooses the other option: he denies that there is any entity that the phrase ‘My bookcase’ refers to. Van Inwagen doesn't think that the phrase is a singular term which fails to refer, like ‘Zeus’; if he did, he could hardly agree that B is true. Rather, van Inwagen denies that this phrase is a singular term at all.

What account, then, does van Inwagen give of the meaning of this phrase? Van Inwagen thinks that ‘My bookcase’ is a plural term: a phrase, like ‘The natural numbers’ or ‘The British Empiricists’, whose function is to pick out several things at once, not just one.Footnote ⁹ Exactly which things, though?

Metaphysicians commonly assume that every material object is made of simple micro-physical particles, leaving it to science to determine the nature of these things. Perhaps they are quarks; or perhaps quarks will turn out to be built of particles of some more fundamental type. Those who believe in composition think that ‘My bookcase’ refers to the entity that is composed by a certain group of simple particles. Van Inwagen denies that these particles compose anything; if they did, it would have them as parts, and he thinks there's nothing that has parts. Van Inwagen's view is that ‘My bookcase’ refers, plurally, to these simple particles themselves.

We apply various predicates to pieces of furniture and other material objects. For instance, we sometimes describe them as heavy, dilapidated, or wooden. Those who believe in composite entities will probably regard these as singular monadic predicates. They are monadic because they have just one argument place (unlike, say, ‘… is the brother of …’), and they are singular because this argument place is to be occupied by a singular term. Van Inwagen agrees that these predicates are monadic, but he holds them to be plural ones: he thinks that each of them takes a plural argument. They're like ‘… are sisters’, or ‘… form a square’. That's just what we should expect from someone who thinks that apparent singular reference to a composite entity is really plural reference to some simple ones.

What goes for ‘… is heavy’ also goes for predicates like ‘… is a bookcase’. Where the believer in composites claims that there is an entity constituted by very many simple particles, van Inwagen believes that there are just very many particles arranged in a particular way. Believers in composite objects do not think that every collection of simples constitutes a bookcase: for that to happen, the particles have to be within a bookcase-shaped region of space, be of the right sorts to constitute the right chemical substances, and so on. Say that some simples are ‘arranged bookcasewise’ if and only if they satisfy this condition.Footnote ¹⁰ Then van Inwagen's view is that ‘is a bookcase’ just means ‘are arranged bookcasewise’.

Van Inwagen's strategy for interpreting talk about tables and chairs is to interpret as plural what might look like singular reference and predication. This explains how sentences which seem to refer to composite objects can be true, even though there are no composite objects.Footnote ¹¹ Van Inwagen's view thus has considerable affinities with other projects of reductive paraphrase: for instance, the phenomenalist's project of interpreting sentences which seem to refer to material objects as long conjunctions of conditionals about experiences; the modal fictionalist's project of interpreting modal talk as talk about what is true according to David Lewis's modal realism; and the modal structuralist's project of interpreting mathematics as talk about what the axioms of our mathematical theories entail.Footnote ¹² There is at least one respect in which van Inwagen's project has greater initial plausibility than these. For van Inwagen attributes to each sentence about material objects a logical structure that closely resembles its syntactic structure: he interprets ‘There is’ as a quantifier and ‘table’ as a predicate, for example. In contrast, statements about material objects do not look like long conjunctions of conditionals; and modal fictionalism and modal structuralism are similarly unfaithful.

3. The Threat

Van Inwagen (Reference Van Inwagen1990: 100) writes: ‘There are no tables.’ Elsewhere, he denies that there are chairs, statues, houses, sticks, stones, severed limbs, ships, or ‘any other visible objects except organisms’ (1990: 1, 128).

On van Inwagen's view, as I have explained it so far, he is contradicting himself. On van Inwagen's view, ‘is a table’ is a disguised plural predicate, meaning ‘are arranged tablewise’. Thus ‘There are tables’ means ‘There are some things that are arranged tablewise’. (Note that this uses the plural existential quantifier rather than the more usual singular one. Compare: ‘There are some things arranged in a square’.) Presumably, ‘There are no tables’ expresses the negation of this claim: it says that there are no things arranged tablewise. Yet van Inwagen does not deny that there are things arranged tablewise. So he should think that ‘There are no tables’ is false. He has no business asserting it (that is, asserting the proposition it expresses).

Van Inwagen is aware of this threat. He responds to it by claiming that ‘There are no tables’, and sentences like it, ‘express different propositions in different contexts of utterance’ (1993: 711; see also 1990: 101, 103). Ordinary utterances of them express claims about whether there are things arranged tablewise. But, says van Inwagen, when they are uttered in ‘the philosophy room’ – that is, in the context of a metaphysical debate – they express different propositions instead. I'll call this doctrine of van Inwagen's contextualism.

Once we take his contextualism into account, it is no longer clear that van Inwagen is contradicting himself. He only ever says ‘There are no tables’ when he's discussing metaphysics; he never asserts it ‘in the ordinary business of life’ (to borrow a phrase from van Inwagen Reference Van Inwagen1990: 102). And it's not clear that there's a clash between van Inwagen's belief that there are things arranged tablewise and the proposition that, in the philosophical context, is expressed by ‘There are no tables’.

But that's only because we have not been told what proposition this is. Perhaps if we knew what it was, we could argue that it's incompatible with some other part of van Inwagen's view. We must therefore ask: what exactly is the content of ‘There are no tables’ when it is asserted in the philosophy room?

This question is particularly pressing, because some sceptics claim not to understand what is being discussed when metaphysicians engage in such ontological debates. Eli Hirsch (Reference Hirsch1993: 690) writes:

Explaining what propositions philosophers are debating would lay Hirsch's worry to rest and remove the suspicion that the metaphysicians aren't making any sense.

We will be clear what it is that van Inwagen takes himself to be denying once we have examined the reasons he offers for denying it. The key claim is this (1990: 100): ‘If there were tables, they would be composite material objects’. It suggests the following argument:

1. (1) If there are tables, they are composite;

2. (2) Nothing is composite; therefore

3. (3) There are no tables.Footnote ¹⁴

As (3) expresses something van Inwagen wants to assert, it cannot be read as ‘There are no xs arranged tablewise’. Presumably, then, the argument has the obvious form:

If so, then we must find a reading of ‘is a table’ on which it takes a singular argument. Charity demands that it be a reading on which (1) is not a crazy thing to think, or – to be more accurate – not too crazy for van Inwagen to think.

A hypothesis that meets these requirements is that van Inwagen is using ‘table’ here with the meaning his opponents think it has. Many of these opponents think that phrases like ‘My table’ are singular terms; that these often succeed in referring; and that their referents are composed of entities arranged tablewise. If we use the word ‘table’ to apply to these referents, then ‘is a table’ takes a singular argument. And, read this way, (1) is definitely not too crazy for anyone to think. It is a common and appealing thought: after all, it seems that we sometimes remove parts of tables (when we chip them, for instance). And it follows from the meaning of ‘tablewise’ that tables, if they are composite, are composed of things arranged tablewise (compare the explanation of ‘bookcasewise’ given in §2).

Hirsch is a vociferous defender of ‘common sense ontology’ (see his Reference Hirsch2002). He believes that there are things composed of things arranged tablewise. When, in the philosophy room, van Inwagen says that there are no tables, he is simply asserting the negation of this belief. Since Hirsch has the concept of negation, he is in a position to understand what van Inwagen means.

4. Contextualism contra Merricks

We have seen that van Inwagen responds to the threat of self-contradiction by claiming that ‘There are no tables’, and sentences like it, are context-dependent. Van Inwagen restricts his plural semantics to everyday uses of these sentences; although he does not explain what he thinks these sentences mean when philosophers are discussing the ontology of material objects, we have just seen that a way of doing so is available to him. I will now argue that his contextualism provides him with a response to an objection posed by Trenton Merricks.

Roughly speaking, Merricks agrees with van Inwagen that there are no composite objects, but disagrees with him about the semantics of talk about material objects. According to Merricks, many of our everyday utterances and beliefs do imply that there are composite objects. When I say, ‘My bookcase is heavy’, I say something that cannot be true unless there is something that has parts. Since there is nothing like that, the sentence I utter and the belief it expresses are both false.

Actually, I have just given a simplified version of Merricks's position. Merricks does think that there are some composite objects, namely, human beings, and he is agnostic as to whether there are other composite organisms (2001: 114–6). But I will ignore this in what follows.

For Merricks (Reference Merricks2001: 1), eliminativism is the doctrine that there are no books, statues, rocks, tables, chairs, or any other macro-physical objects.Footnote ¹⁵ Commenting on van Inwagen's interpretation of talk about tables, chairs, and so on, Merricks writes:

Merricks's line of thought is this. Van Inwagen's ontological view is striking and surprising; we can best explain this by saying that his ontological view is in massive conflict with our pre-theoretical beliefs about what there is; but van Inwagen's semantic views prevent him from endorsing this explanation. So van Inwagen has reason to drop his semantic views.

It is certainly true that that people often find van Inwagen's view striking and surprising. But van Inwagen's contextualism provides him with a way of explaining this. He can deal with Merricks's objection by claiming that we confuse different readings of ‘There are no tables’. I imagine him saying something of the following sort:

Now this alternative has been put forward, it is no longer clear that Merricks's proposed explanation of the strikingness of van Inwagen's view is the best available. Unless Merricks can provide reasons to think that his explanation is superior, van Inwagen's contextualism allows him to argue that he is at no theoretical disadvantage when it comes to explaining people's amazed reactions to his ontology.

5. Explaining the Context-dependence

As we have seen, claiming that sentences like ‘There are no tables’ express different propositions in different contexts of utterance is useful to van Inwagen: it has allowed him to deal both with the threat of contradiction and with Merricks's objection.

However, these advantages impose a duty: van Inwagen ought to provide us with a credible explanation of how this context-dependence takes place. For instance, does the word ‘table’ make different contributions to the meanings of the sentences it occurs in, depending on the context? Or is some other mechanism in play? If van Inwagen cannot provide us with a credible explanation, then it will be hard for us to accept that his contextualism is true. Without that claim to rely on, van Inwagen would have to start looking for new responses to the threat of contradiction and Merricks's argument. So there is considerable theoretical reason to ask: can van Inwagen explain how ‘There are no tables’, and sentences like it, manage to express different propositions in different contexts?

We should begin by making it clear just what it is that van Inwagen is obliged to explain. Van Inwagen's view is that utterances of ‘There are tables’ ordinarily express the claim that there are some things arranged tablewise; but in the philosophy room, they express another proposition. In §3 we identified this proposition as the view that there are tables, where ‘table’ has the meaning van Inwagen's opponents think that the word ordinarily has. In this sense, tables are the referents of singular terms like ‘My table’. Van Inwagen needs to give an account of how the same sentence expresses these two different propositions in the two contexts. No such account is presented in his writings;Footnote ¹⁷ in the next section, I will assess the prospects for providing one.

6. Failed Explanations

The most obvious explanation for van Inwagen to use is that words like ‘table’ are ambiguous. We have already mentioned different senses of ‘table’. The idea would be that, whilst these words ordinarily function as plural predicates, in the philosophy room they transform themselves into singular ones. To state the proposal crudely: ‘is a table’ means ‘are arranged tablewise’ – except when philosophers use it, in which case it applies to all and only the things composed by things arranged tablewise.Footnote ¹⁸

The most obvious explanation is awful. Neither believers in composition nor opponents like Merricks have any need to commit themselves to these ambiguity claims: they can say that ‘There are no tables’ always expresses the claim that there are none of a certain kind of composite entity. But Grice's Razor, ‘Senses are not to be multiplied beyond necessity’ (Grice Reference Grice1989: 47), tells us that these theories are to be preferred to this one, other things being equal. Since we should minimise the amount of ambiguity we postulate, for van Inwagen to postulate ambiguity would be playing into his opponents' hands.

Were he to adopt the explanation we are currently discussing, van Inwagen could not confine himself to claiming that ‘chair’ is ambiguous. He denies that there are chairs, statues, houses, sticks, stones, or severed limbs, ships, or ‘any other visible objects except organisms’ (1990: 1, 128); he would therefore have to say that every predicate which seems to pick out a sort of visible object is ambiguous. On one reading, the one used in ontological enquiries, these predicates take a singular argument; on the other, everyday reading, they take a plural argument. There is no obvious independent motivation for any of these claims.

It is therefore natural to look for other ways for van Inwagen to explain the context-dependence. For instance, can van Inwagen appeal to quantifier domain restriction?

Quantifier domain restriction is highly familiar. Often, when we say ‘All the Fs are G’, what we say would be ludicrous if we meant all the Fs that there are. We are better interpreted as meaning that all the contextually salient Fs are G. And the same goes for existential quantification. Often, ‘There is an F’ means something closer to ‘There is an F in the contextually salient domain of things’. We are usually more interested in whether there is a café round here than whether there is a café in the universe.

Quantifiers do not have to be singular to be restricted. In the right context, for instance, ‘Some people write things together’ means ‘Some people who are philosophers write things together’. Domain restriction is one way to get singular quantification from plural quantification, as I will now explain.

Consider the plural predicate ‘are one’, defined as follows: the xs are one iff there is a z such that for every y, y is one of the xs iff y is z. The xs are one iff there is exactly one of them. For instance, there is only one current UK prime minister; so the current UK prime ministers are one.

‘Cook things’ is a plural predicate. ‘Some people cook things’ means that there are some people such that they cook things. Now let us restrict the domain of the quantifier ‘Some’ to things that are one. The result: some people are such that they cook things and are one. Or, more naturally, there is somebody who cooks things. Plural has turned to singular.

Might the shift from the ordinary to the metaphysical context restrict the domain of plural quantifiers to things that are one? On this picture, in the ordinary context, ‘There are tables’ expresses the claim that there are things arranged tablewise, whereas in the metaphysical context it expresses the claim that there are things arranged tablewise that are one.

The problem with this proposal is that it generates the wrong meaning for ‘There are tables’ in the metaphysical context. When van Inwagen's opponents assert that there are tables, they mean that there are things which are composed of entities arranged tablewise and which we refer to with singular terms like ‘My table’. Van Inwagen's opponents do not mean that there is a single thing that is arranged tablewise; very likely, they will think that this is false. So the proposal fails. It is possible to combine quantifier restriction with an ambiguity claim: the position would be that shifting to the metaphysical context both restricts the quantifier to things that are one and also shifts the sense of ‘is a table’ from ‘are arranged tablewise’ to ‘is a table’. But this position is little better than the ambiguity-without-domain-restriction view I have already dismissed.

Other mechanisms are sometimes invoked to explain context-dependence. For instance, van Inwagen might appeal to ellipsis: when in the ‘ordinary business of life’ we say ‘There are tables’, our utterance is short for ‘There are things arranged tablewise’. But as ‘arranged tablewise’ is not a part of ordinary English but an artificial predicate introduced by philosophers, this is a non-starter.

Judging by this survey of the options, it looks as if van Inwagen will find it very difficult to explain the context-dependence he invokes. At the very least, the burden is on him to nominate a semantic mechanism that makes the shift happen.

Contextualism offered to rescue van Inwagen from apparent contradiction; it also promised a reply to Merricks's objection. I now show how these benefits can be obtained without postulating context-dependence.

7. The Benefits of Invariantism

I recommend a partial retreat. Van Inwagen should claim that ‘There are tables’ picks out the same proposition in all contexts. He should say that ‘There are tables’ always combines a plural existential quantifier and a plural predicate; both inside the philosophy room and out, the sentence means that there are things arranged tablewise. By his lights, this proposition is true. So van Inwagen should stop asserting ‘There are no tables’ – indeed, he should assert: ‘There are tables’. And the same goes for ‘There are chairs’, ‘There are bookcases’, and so on. Call this stance invariantism.

Invariantism immediately deals with the threat of contradiction. The worry was that van Inwagen could not consistently assert ‘There are no tables’, given his ontological and semantic commitments. Once he stops making these assertions, no-one can sensibly worry that he is making them incoherently.

Merricks argued that van Inwagen cannot adopt the best explanation of why people find his view striking and surprising. If van Inwagen adopts invariantism, then there will be nothing to explain. For he will cease to assert ‘There are no tables’, ‘There are no apples’, and so on: in other words, he will cease to assert the claims that make people gasp. Shorn of these claims, it is not clear that his view will be so startling.

However, van Inwagen's claim that nothing has parts also causes incredulity. He should withdraw this claim too. Given the notion of some things being among other things, he can interpret everyday talk of parthood: ‘… is part of …’ should be interpreted as ‘… are among …’ (compare Morton Reference Morton1975: 314, Hossack Reference Hossack2000: 424). This predicate takes two plural arguments (as it does in ‘Joe's parents are among his relatives’): the xs are among the ys if every one of the xs is one of the ys but not vice versa.Footnote ¹⁹ Thus ‘The arm is part of the chair’ becomes ‘There are things, the xs, arranged armwise and there are things, the ys, arranged chairwise, and the xs are among the ys.’ Van Inwagen should think that claims like this are often true. So he should say that some things are parts of other things. The problem is not with endorsing the sentence ‘Some things have parts’; it is with interpreting this sentence by means of singular resources rather than plural ones.

If he were to adopt my recommendations, what startling claims would van Inwagen now be committed to? None, it seems. Merricks's objection has been undermined by removing the phenomenon for which it demanded an explanation.

Invariantism offers the same theoretical benefits as contextualism. But is it defensible, or are these benefits outweighed by invariantism's theoretical costs? I will now show that invariantism is defensible, by defending it. In each of the next two sections I will present an objection to invariantism and defend invariantism from it.

8. Against the Misinterpretation Objection

Van Inwagen's distinctive semantic view is that what might look like singular reference and predication is in fact plural. Once he swaps contextualism for invariantism, he must extend this interpretation to speakers in the philosophy room. Then he will say that ‘There are tables’ always expresses the proposition that there are some things that are arranged tablewise, and that ‘There are no tables’ always expresses the negation of that proposition.

It is natural to worry that adopting this view would commit van Inwagen to misinterpreting his philosophical opponents. For instance, when Merricks says ‘There are no tables’, we are wrong if we interpret him as asserting that there are no entities arranged tablewise. Merricks (Reference Merricks2001: vii) explicitly says that there are some entities arranged in that way. What Merricks denies is that there is anything which has these entities as parts. When he says, ‘There are no tables’, his statement seems to succeed in communicating this. Similarly, when a believer in composite objects replies to van Inwagen by saying, ‘No, you're wrong: there are tables,’ we will interpret them wrongly if we take them to be saying just that there are things arranged tablewise.

It therefore appears that van Inwagen will run into trouble if he endorses invariantism. However, this appearance is an illusion: the worry can be dissolved by appeal to pragmatics, as I will now explain.

Merricks sets out what he thinks he means: he presents a philosophical account of the truth-conditions of some of his own utterances. Merricks asserts: ‘There are no tables’. And he explains what truth-conditions he thinks this sentence has. On his view, ‘There are no tables’ expresses the claim that nothing is a table – in the singular, ‘philosophical’ sense of table explained above (§3).

Now it is common ground that we should take people at their word. We should accept the following principle of interpretation:

In other words, we should interpret philosophers as if their own theory of the meaning of their utterances were true, whether or not we agree with that theory.

Using this principle, van Inwagen can explain how, by uttering (or writing) ‘There are no tables’ assertorically, Merricks succeeds in telling us that he believes that nothing is a table (rather than that there are no things arranged tablewise) – even though the sentence does not express the claim that nothing is a table. For Merricks asserts, contra van Inwagen, that it does pick out that claim; so we should interpret him as trying to assert this when he utters the sentence assertorically. If no defeating conditions obtain (and in this case they seem not to), we should think he believes what he's trying to assert: we should think he believes that nothing is a table.

A similar story applies when a believer in composite objects asserts: ‘There are tables’. What they say is the proposition that sentence expresses – on the invariantist view, the proposition that there are some things arranged tablewise. But what they communicate is that they believe the proposition they are trying to assert. They believe that the sentence expresses the claim that there is something that is a table (where ‘there is’ is a singular quantifier), and so we should understand them as trying to assert this proposition.

None of this reasoning is out of the ordinary. When we meet someone who thinks, wrongly, that ‘refute’ means ‘deny’, we soon realise that they do not mean what they say; and we interpret them charitably. When they tell us, ‘He refuted Joe's claim, even though Joe was right’, we do not ascribe contradictory beliefs to them.

So van Inwagen can explain how his opponents convey their philosophical views, without having to claim that the sentences they use change their truth-conditions from context to context. Moreover, van Inwagen can explain why Merricks and company are tempted to misinterpret everyday talk of tables using singular quantification. Ever since logicians started to deal with quantified sentences in the nineteenth century, they have dwelt on singular quantification at the expense of plural. This bias is reflected in philosophers' education: the logic of singular predication and singular existential and universal quantifiers is a standard part of the syllabus, whereas the logic of plural predicates and quantifiers most decidedly is not. (For proof, simply browse through some undergraduate logic textbooks.) It is quite standard to interpret sentences of the form ‘There are Fs’ using a singular quantifier and a singular predicate. If van Inwagen is right, this tradition is mistaken. For lots of our ordinary talk concerns chairs, statues, houses, sticks, stones, and other visible objects; it is the sort of talk van Inwagen thinks we should interpret using plural quantification. English contains a large number of sentences which very plausibly should be interpreted using plural resources (‘The blocks form a square’, for instance); but plurals are not part of the standard philosophical tool-kit. This explains why few philosophers have considered whether sentences like ‘There are chairs’ should be understood as van Inwagen understands them.

9. Against the Unstatability Objection

As we saw above (§3), Hirsch claims not to understand what van Inwagen means when he denies that there are chairs, tables, and so on. If van Inwagen adopts invariantism, he will cease to issue these denials; so Hirsch will have no legitimate complaint to make about them. But invariantism raises a related worry: can invariantists state their own view? Van Inwagen initially presented his position by saying that there are no tables, and that nothing has parts. But invariantists are happy to assert sentences such as ‘There are tables’ and ‘Tables have parts’; according to invariantism, ‘There are no tables’ and ‘Tables do not have parts’ are false in both ordinary and ontological contexts. The worry is that adopting invariantism would block van Inwagen from stating his view. An invariantist version of van Inwagen's position threatens to be not self-refuting, but self-suppressing.

Invariantism has nothing to say about the interpretation of the technical terms ‘singular’, ‘plural’, ‘quantifier’, or ‘predicate’. So it does not threaten van Inwagen's entitlement to make his semantic claims. But van Inwagen's view is not exhausted by his semantic claims alone; for these miss out his ontological doctrine, that nothing has parts. And ‘Nothing has parts’ is just what an invariantist cannot say.

But let us remind ourselves why van Inwagen wanted to say ‘Nothing has parts’. His aim was to resolve puzzles about constitution such as the bookcase puzzle I set out in §1. These puzzles assume that there is (singular quantifier!) an entity that has some things as parts; for example, in the bookcase puzzle, the bookcase is supposed to be constituted by the wood and the screws. Van Inwagen's preferred way of dealing with these puzzles is to reject this assumption. In his view, if we use a singular quantifier then we can truly say that there is no such thing as the bookcase. For that reason, van Inwagen denies that there is any relation of parthood relating the bookcase to anything else.

Invariantists interpret ‘… is a part of …’ as ‘… are among …’: the xs are a part of the ys just when the xs are among the ys. Suppose the ys are one, in the sense of ‘are one’ introduced in §6. Then it follows that for no xs are the xs part of the ys.Footnote ²⁰ So it is a consequence of the invariantist's interpretation of ‘part’ that if the bookcase were one, it would not have parts.

While van Inwagen cannot say that nothing has parts, he can (and must) say that for all xs, if the xs have parts then they are not one; only things that are not one have parts. Equivalently, he can say that if ‘Something’ expressed the singular existential quantifier, then ‘Something has parts’ would be false. These formulations capture van Inwagen's distinctive ontological doctrine: so invariantism does not bring unstatability.

10. Conclusion

Van Inwagen's contextualism enabled him to counter Merricks's objection and to deal with the threat of contradiction. I have shown that these benefits can be obtained without having to postulate any context-dependence (§7); I have defended the invariantist position from two objections (§§8, 9); and I have criticised contextualism (§6). My main conclusion is that thus van Inwagen should prefer invariantism to contextualism.

Let me offer three further conclusions. First, it is sometimes suggested that sentences of the form ‘There are Fs’, ‘Fs exist’, and so on, take on a special sense in the context of ontological discussions; these sentences have an unsophisticated everyday sense and a lofty philosophical one (Carnap Reference Carnap1950; Fine Reference Fine2001; Hofweber Reference Hofweber2005; Dorr Reference Dorr and Kalderon2005: §7 [though see footnote 27 for a qualification]). If invariantism is correct, then this suggestion is untrue, at least in the case of discussions of visible objects. Sentences about the existence of chairs and tables do not change their truth-conditions when we enter the philosophy room.

Second, as I noted in §2, van Inwagen has affinities with advocates of other paraphrase positions: philosophers who claim that, for some sortal ‘F’, sentences that might seem to entail the existence of Fs do not genuinely do so, and go on to offer paraphrases to make clear what they do entail. We may generalize the point I made in §2 to reach the following conclusion: these philosophers ought not merely to offer us claims about the truth-conditions of the sentences they are seeking to account for, but should also offer a semantic theory which explains how the parts of the sentences give rise to these truth-conditions.

Finally, these projects of reductive paraphrase are subject to problems that mirror the threat of contradiction I have been trying to help van Inwagen to escape. Suppose a paraphraser denies the existence of Fs and provides an account of the truth-conditions of F-discourse which reveals that its sentences can be true even if there are no Fs. The general form of the objection is this: the paraphrasers are committed to regarding the sentence ‘There are Fs’ as true, and are thus committed to the existence of Fs in spite of themselves.Footnote ²¹ A natural way for a paraphraser to respond to such an objection is to deny that they must regard ‘There are Fs’ as true and to attempt to disrupt the reasoning that is supposed to show they are saddled with that commitment.Footnote ²² My discussion of van Inwagen suggests another way for paraphrasers to respond to these objections: insist that ‘There are Fs’ is true, and find a way to express their ontological claims which does not involve asserting ‘There are no Fs’.Footnote ^23, Footnote ²⁴