Borderline Composition without Metaphysical Vagueness

Ted Sider argues that composition is unrestricted on the grounds that, were it restricted, there’d have to either be sharp cut-offs with respect to composition or borderline cases of composition, and there can’t be either. The conventional wisdom is that embracing borderline composition is going to lead either to metaphysical vagueness or else some sort of relativism. That not quite right: I know of at least four ways (one due to my blogmate Chad Carmichael) to be a non-relativist borderliner without abandoning a linguistic theory of vagueness. I want to sketch one way here. It’s kind of crazy, but not obviously crazier than unrestricted composition.

Some background: numerical sentences are sentences of the following form that say that there are exactly n concrete things (for some number n, in this case 2): ‘∃x∃y(Cx & Cy & x≠y & ∀z(Cz → (x=z v y=z)))’. One premise of Sider’s argument is that whenever there’s borderline composition, some numerical sentence will lack a determinate truth value. After all, it’ll be indeterminate whether there are just some things, or those things *plus* an additional thing that’s composed of them. The strategy I have in mind is to concede that some such sentence lacks a determinate truth value, but insist that the only indeterminacy at issue is semantic indeterminacy. I’ll develop this in three phases. Phase 1: identifying the representational indeterminacy. Phase 2: eliminating everything from the ontology that might lead to nonrepresentational vagueness. Phase 3: ignoring possible objections. 

Phase 1: There are countless perfectly precise ways for simples to be arranged: they may be in arrangement1, arrangement2, arrangement3, and so forth. These are monadic properties, collectively instantiated by pluralities of simples, and each of which has a perfectly determinate extension. The envisaged borderliner will countenance states of affairs involving mereological simples and their precise arrangements. Such states of affairs are all that will be needed for securing the semantic indeterminacy of the numerical sentence. To see why, let us say that some things compose something iff they are arranged objectwise, where the vague expression ‘arranged objectwise’ may be replaced by any vague restriction on composition (e.g., arranged lifewise.) Assuming that some simples cannot compose more than one thing at a time, the numerical sentence for some number n will be true iff the arithmetic sum of the number of simples and the number of sets of simples whose members are arranged objectwise is n. Because it is a vague restriction there will be borderline cases of being arranged objectwise. Suppose that simples in arrangement1 through arrangement10 are the clear cases of being arranged objectwise, and that simples in arrangement11 are a borderline case of being arranged objectwise. The sparse borderline can then say that the numerical sentence for n is semantically indeterminate insofar as it is indeterminate which of the following two (perfectly determinate) states of affairs it represents as obtaining:

(i) There being m simples and n minus m sets of simples who members are either in arrangement1 or arrangement2 or … or arrangement10.

(ii) There being m simples and n minus m sets of simples who members are either in arrangement1 or arrangement2 or … or arrangement10 or arrangement11.

One might naturally think of these states of affairs as precisifications of the numerical sentence for n.

Phase 2:  Linguistic theorists agree that it’s indeterminate whether Paul is bald, but avoid de re vagueness by denying that there is such a property as the property of being bald; there are only the perfectly precise properties: bald1, bald2, bald3, etc. There also is no such proposition as the proposition that Paul is bald (which would be a borderline instance of the property of being true) or such a state of affairs as the state of affairs of Paul’s being bald (which would be a borderline case of the property of obtaining).

Our sparse borderliner piggybacks this strategy: there is no such property as being arranged objectwise; there are just the precise properties of being in arrangement1, etc. There also is no such proposition as the proposition that ∃x∃y(Cx & Cy & x≠y & ∀z(Cz → (x=z v y=z))), and no such state of affairs as the state of affairs of there being an x and a y such that (Cx & Cy & x≠y & ∀z(Cz → (x=z v y=z))). So there is nothing in the ontology that exhibits de re vagueness. Mission accomplished!

Phase 3: What, me worry?

Comments

[jason]

Hi Dan,

Interesting post. I'm not sure I'm quite following the entire line of thought, so maybe you can clarify something for me a bit. Here are two premises of Sider's argument:

(1) If a numerical sentence is indeterminate for semantic reasons, then some semantically simple expression of that sentence is precisifiable, i.e., not entirely precise.

(2) No expressions of numerical sentences are precisifiable, i.e., every semantically simple expression is entirely precise.

(1) and (2) together entail the negation of what you're arguing for, so I take it that you're denying one of them. But I can't quite tell which one it is -- could you elaborate a bit?

[Dan Korman]

Hi Jason,

The sparse borderliner will say that it's (1) that's false. This is actually a pretty natural way to go for someone who embraces a sparse ontology on which sentences aren't necessarily isomorphic with the states of affairs they represent as obtaining.

By denying (1), I'd have to give up the theoretical machinery available to most linguistic theorists for deriving the precisifications of a sentence from the precisifications of its constituents. But I take it that doesn't show it's not a linguistic theory; only that it's not a standard linguistic theory.

Is it therefore worse than standard linguistic theories? Well, they both have their costs. The nonstandard one is less powerful. The standard ones can't accommodate borderline composition.

[Irem Kurtsal Steen]

I think we knew what the candidate meanings would be all along--this view clarifies that. What we don't know is how come a numerical sentence is indeterminate between these two representations when it has no non-logical parts that can be blamed for it.

[Mama Africa]

great! thanks very much for sharing!

[Kenny Easwaran]

It almost sounds to me like you're denying composition at all! You're replacing it with talk of simples being "arranged objectwise", which you then say is a vague expression. If the only facts are about the simples and their arrangements, then it seems that you're saying there are no composite objects.

[Dan Korman]

Irem --

The sparse borderliner has to deny a certain inheritance principle about vagueness, i.e., that a sentence is vague only if some expression in that sentence is vague. I take it that this is what’s worrying you. I’m not sure how best to motivate denying this principle. Here are a couple (highly controversial!) examples that might be used to motivate denying the principle. First example: suppose that one says ‘there is no beer left’, and the context makes it clear that there is a tacit restriction to *cold* beers. And suppose that there is one beer left which is borderline cold. This sentence would then (arguably) lack a determinate truth value as a result of vagueness, but the vagueness would (arguably) not be the result of any expression *in* that sentence. (NB. The word ‘cold’ does not appear in the sentence.) Second example: ‘The sentence “Paul is bald” is true’, where Paul is borderline bald. The sentence (arguably) lacks a determinate truth value as a result of vagueness, but (arguably) no expression in the sentence (including ‘true’) is vague. I don’t expect everyone to be moved by these examples; there’s a lot of wiggle room.

Mama Africa -- thanks!

Kenny --

I was hoping that the sparse borderliner can play the same sorts of games that other sparse theorists sometimes play (though I admit that I don’t know much about how those games are usually played; I’ve spent too much time in the Platonic heavens, not worrying about nominalist hell!). I imagine that there are plenty of people who want to be relatively sparse about which properties, facts, and propositions exist, and yet in the same breath want to accept ordinary claims that, taken at face value, commit them to these things. For instance, they’ll accept that wisdom is a virtue even though they deny that ‘wisdom’ names some entity. Similarly, my sparse borderliner wants to pound the table and insist that of course there are tables, and that some things neither definitely do nor definitely do not compose tables, and yet weasel his way out of ontological commitment to such entities as *facts* or *propositions* about tables. You probably have a better sense than I do about the range of strategies that one might employ here, and how well they carry over to this case.