## Sunday, May 07, 2006

### The argument from vagueness

In his book Four-Dimensionalism: An Ontology of Persistence and Time, Ted Sider presents an improved version of David Lewis's argument from vagueness (from On the Plurality of Worlds). The argument has generated a lot of attention (see links in the end) because of its powerful metaphysical conclusion, i.e., four-dimensionalism (or, in Lewis's version, universalism for mereological composition), and its rather innocent-looking premises about vagueness.

For brevity, let us here ignore the more complicated argument for four-dimensionalism, and just focus on Lewis's version with an important strengthening made by Sider (if you want to see Sider's full argument, here is a preprint of an article called 'Four-dimensionalism'). The argument is roughly as follows: There are three broad positions on mereological composition: (i) nihilism, no non-empty set of objects form a mereological sum, i.e., a fusion; (ii) universalism, all non-empty set of objects form a mereological sum; and (iii) retrictivism, some non-empty set of objects form a mereological sum.

Now, our intuitions tell in favour of (iii). Some objects, like the particles of my hand, form another object, namely my hand. However, some sets of objects, like the set of my left hand, a deck of card, and the planet Saturn, does not form a fusion. As pointed out by Lewis, distance, cohesion and other criteria decide whether or not we believe that some objects have a fusion. But, he continues, these criteria cannot be stated non-vaguely. To see this, imagine that we take all the particles of my hand and move them all 0,00001 nanometer in different directions. Due to the insignificance of the distance, the object will still form a hand. Then move them all again, and a third time, and a fourth, etc. The question becomes: When are the particles too scattered to form a hand?

We find ourselves in a sorites paradox. There is no sharp cut-off where the particles n nanometers apart form a hand, but at n+0000,1 nanomets they seize to form a hand. So, according to Lewis and Sider, the criteria for restrictive composition would have to be vague. Furthermore, they both agree that ontic vagueness is simply untenable (this is an assumption, if you like ontic vagueness, you shouldn't be worried by the argument), so if there is any vagueness it has to be linguistic vagueness, i.e., vagueness owing to the terms involved in formulating the criteria. (Sider also brings up and rejects epistemicism as an answer (see Sider's book, pp. 130-132).)

However, to which terms should we attribute the vagueness? Lewis argues that all that is needed to formulate the criteria is classical first-order logic with identity and a special predicate for identity of overlap. But, Lewis claims, none of these terms are candidates for vagueness, thus there cannot be linguistic vaguness. Consequently, restrictivism is impossible. We are left with nihilism and universalism. Both are counter-intuitive positions, but Lewis and Sider opt for the latter (ibid., ch. 5.6 for the case against nihilism).

There are many weaknesses to the argument, and I will not list them here (although see the papers listed at the end). What I am interested in is the crux of the argument, namely the alleged non-vagueness of the terms listed by Lewis. Of course, some commentators prone to believe that logical terms (including identity) are non-vague, made the case for the vagueness of the notion of identity of overlap. This term is surely the weakest link in the argument, and Sider admits that this needs fixing. Fortunately, however, he has a brilliant way of embedding Lewis's claim into strictly logical language.

For, how we set the restrictions on composition will affect the overall number of objects in the domain. (The domain Sider has in mind is the quantication domain of the 'absolutely' unrestricted quantification. Whether such a quantifier can be stabilised is a widely debated topic. See here for a paper by Williamson and here for a paper by Rayo. Sider has responded to critics in the paper 'Against vague existence' here. See also a recent post by Carrie Jenkins.) More precisely, the number of simple objects (atomic, if you like) and the criteria for composition will determine the number of existing objects. Sider's idea is to argue that since the number of existing things (i.e., the size of the domain D) can be expressed in first-order logic with identity (but without the special predicate for identity of overlap), it cannot be vague. Because if it were, that would entail that some of the logical terms would be vague.

Assuming that Sider is right that logical terms are non-vague, it seems that he has entrenched Lewis's result. Restrictivsm fails, and the only options are nihilism and universalism. Personally, I think the issue hinges on whether Sider can make a convincing case for the non-vagueness of the quantifiers, something which rests heavily on the intelligibility of unrestricted quantification. This discussion, however, I will put off for some other time. The issue I want to raise, rather, is the relation between the formulation of the criteria for composition and the number of existing things. Why is that the indeterminacy transmits from the former to the latter?

Note first of all that this is not due to logical consequence. Although the number of existing things is a logical consequence of the number of simpels and criteria for composition, it is not in general the case that indeterminacy transmits through logically valid inferences. The most obvious examples are those in which the conclusion is a logical truth (and thus follows from anything, including vague premises). Of course, it is tempting to reply that in these examples the premises don't really do any work, while in Sider's argument they are crucial. However, by the model-theoretic definition of logical consequence, the number of existing things (in the unrestricted domain) is precisely an instance of a logical truth. (See Etchemendy 1990)

Nevertheless, our intuition still tells us that Sider is correct that vagueness will transmit from the number of simples and criteria of composition to the number of existing things. What is it then that ensures us of this. My best guess is that this is connected to mathematics rather than logic. Whereas the number of simples, being determinate, is non-vague, the criteria for composition is in a sense a (set-theoretically) fuzzy function. It takes a determinate number of things as functional argument but yields an indeterminate number of things as functional value.

This is just a proposal. In general I am interested in how indeterminacy transmits through arguments, so any comments would be appreciated. Here, as promised, is a list of relevant literature:

Achille C. Varzi 'Change, Temporal Parts, and the Argument from Vagueness', Dialectica, 59 (2005), 485-498 (preprint).

Daniel Nolan 'Vagueness, Multiplicity, and Parts', forthcoming in Noûs.

Daniel Z. Korman 'Unrestricted composition and the argument from vagueness' (unpublished).

Ned Markosian 'Two arguments from Sider's Four-Dimensionalism', Phil. and Phen. Research, 68 (2004), 665-673.

A more complete bibliography can be found in Korman's paper.

Categories: Philosophy, Logic, Metaphysics, Vagueness

andri said...

alright, now we're talking. more of this and less of chess, and i might even bookmark you. well done.

Bernhard said...

less of this and more chess, and I might bookmark you:)

Andreas said...

You're raising some very interesting issues here. Perhaps it would be helpful to attempt a clarification of some of them:

We need a precise formulation of the question of indeterminacy-transmission. If we can take it that your example involving logical truth shows that it is at least sometimes the case that indeterminacy does not transmit across entailment what we want to know is whether there are any cases where it does and, crucially, what are the characteristics of those cases. That is, can we reach a set of conditions that has to be fulfilled for indeterminacy-transmission? Indeed, reaching such conditions would seem to be the only way of going about answering the first question.

First, there is an issue of whether this question should be approached from a generic point of view, i.e. a view that just construes 'indeterminately' as a syntactic operator I defined in the standard way in terms of a determinately-operator D, or whether we shold approach the issue from the perspective of a certain philosophical interpretation of those operators.

Secondly, there is a distinction to be made between an argument being indeterminacy-transmitting and a conditional being indeterminacy-transmitting. Now, one might think that the latter is a void case, since the definition of at least material conditionals presupposes determinacy. However, the case of material conditionals is only void if we are construing indeterminacy as having to do with the truth-status of the sentences involved, i.e. if one were to adopt an epistemicist interpretation of I it would not be obvious that the question of indeterminacy-transmission for conditionals is a non-starter.

Now, if we confine ourselves to arguments, there are really two things one might mean by an argument being indeterminacy-transmitting. First, one might say that an argument is indeterminacy-transmitting iff if I appears somewhere in the premises, then I appears somewhere in the conclusion. Secondly, one might say that an argument is indeterminacy-transmitting iff if I appears somewhere in the premises, then the conclusion is prefixed by I.

(The same distinction between two senses of indeterminacy-transmission can be made for the case of conditionals, I suppose.)

I believe that it is actually very hard to see how one could have an opinion about either case unless one begins by assuming a certain interpretation of I. The obvious reason is, as indicated above, that it seems that on a Williamsonian epistemicist interpretation which preserves bivalence even for indeterminate p these questions become very different than on any other interpretation.

A further, but significantly distinct, issue is whether there is any motivation for epistemicism to be gained from these investigations. That is, supposing one came to the result that one wanted at least some arguments to be indeterminacy-transmitting and that they could only be so given an epistemicist interpretation of I, would that be an argument for epistemicism?