Sunday, September 03, 2006

Ontic vagueness: the shape of the debate.

(cross-posted on theories n things)

One of my projects at the moment is writing a survey article on ontic vagueness. I've been working on this stuff for a while now, but it's time to pull things together. (And writing up comments on Hugh Mellor's paper "Micro-composition" at the RIP Being conference got me puzzling about some of these issues all over again.)

One thing I'd like to achieve is to separate out different types of ontic vagueness. The "big three", for me, are vague identity, vague existence, vague instantition. But there also might be: vagueness in the parthood relation, vague locations, vague composition, vagueness in "supervening" levels (it being ontically vague whether x is bald); vagueness at the fundamental level (it being ontically vague whether that elementary particle is charged). These all seem prima facie different, to me. And (as Elizabeth Barnes told me time and again until I started listening) it's just not obvious that e.g. rejecting vague identity for Evansian reasons puts in peril any other sort of ontic vagueness, since it's not obvious that any other form of ontic vagueness requires vague identity.

[Digression: It's really not very surprising that ontic vagueness comes in many types, when you think about it. For topic T in metaphysics (theory of properties, theory of parts, theory of persistence, theory of identity, theory of location etc etc), we could in principle consider the thesis that the facts discussed by T are vague. End Digression]

Distinguish (a) the positive project of giving a theory of ontic vagueness; and (b) the negative project of defending it against its many detractors. The negative project I guess has the lion's share of the attention in the literature. I think it helps to see the issues here as a matter of (i) developing arguments against particular types of ontic vagueness (ii) arguing that this or that form of ontic vagueness entails some other one.

Regarding (i), Evans' argument is the most famous case, specifically against vague identity. But it won't do what Evans claimed it did (provide an argument against vagueness in the world per se) unless we can argue that other kinds of ontic vagueness give rise to vague identity (and Evans, of course, doesn't say anything about this). Vague existence is another point at which people are apt to stick directly. I think some of Ted Sider's recent arguments against semantically or epistemically vague existence transfer directly to the case of ontically vague existence. And we shouldn't forget the "incredulous stare" maneuver, often deployed at this point.

Given these kind of answers to (i), I think the name of the game in the second part of the negative project is to figure out exactly which forms of ontic vagueness commit one to vague existence or vague identity. So, for example, one of the things Elizabeth does in her recent analysis paper is to argue that vague instantiation entails vague existence (at least for a states-of-affairs theorist). Implicit in an argument due to Katherine Hawley are considerations seemingly showing that vague existence entails vague identity (at least if you have sets, or unrestricted mereological composition, around). (I set both of these out briefly and give references in this paper).
Again, you can think of Ted Sider's argument against vague composition as supporting the following entailment: vague composition entails vague existence. And so on and so forth.

[A side note. Generally, all these arguments will have the form:

Ontic vagueness of type 1
Substantive metaphysical principles
Ontic vagueness of type 2.

What this means is that these debates over ontic vagueness are potentially extemely metaphysically illuminating. For, suppose that you think that ontic vagueness of type 2 occurs, but that ontic vagueness of type 1 is impossible (say because it entails vague identity). Then, you are going to have to reject the substantive metaphysical principles that provide the bridge from one to the other. For example, if you want vague instantiation, but think vague existence is, directly or indirectly, incoherent, then you have an argument against states-of-affairs-theorists. The argument from vague existence to vague identity won't worry someone who doesn't believe in or in unrestricted mereological fusion. Hence, if cogent, it can be turned into an argument against sets and arbitrary fusions (actually, it's in that form --- as an argument against the standard set theoretic axioms --- that Katherine Hawley first presented it). And so forth.]

So that's my view on what the debate on ontic vagueness is, or should be. It has the advantage of unifying what at first glance appear to be a load of disparate discussions in the literature. It does impose a methodology that's not in keeping with much of the literature by defenders of ontic vagueness: in particular, the way I'm thinking of things, classical logic will be the last thing we give up: though non-classical logics are often the first tool reached for by defenders of ontic vagueness (notable exceptions are the modal-ish/supervaluation-ish characterizations of ontic vagueness favoured in various forms by Ken Akiba, Elizabeth Barnes and, erm, me). I'll have to be up front about this.

Still, I'd like to use the above as a way of setting up the paper. It can only be 5000 or so words long, and it has to be comprehensible to advanced undergraduates, so I may not be able to include everything, particularly if the issues allude to complex areas of metaphysics. But I'd like to have an as-exhaustive-as-possible taxonomy, of which I can extract a suitable sample for the paper. I'd be really interested in collecting any discussions of ontic vagueness that can fit into the project as I've sketched it. And I'd also be really grateful to hear about other parts of the literature that I'm in danger of missing out or ignoring if I go this route, and any comments on the strategy I'm adopting.

Some examples to get us started:

If composition is identity, then it looks like vague parthood entails vague identity. For if it's vague whether the a is part of b, then it'll be vague whether the a's are identical to b.

Indeed, if classical mereology holds, then it looks like vague parthood entails vague identity. For if it's vague whether the aa's are all and only the parts of b, then mereology will give us that that object which is the fusion of the aa's is identical to b iff the aa's are all and only the parts of b. Since the RHS here is ex hypothesi vague, the LHS will be too.

If the Wigginsean "individuation criteria" for Fs are vague, it looks like vague existence will follow when it's vague whether the conditions are met.

Friday, September 01, 2006

An argument for conditional excluded middle

(cross-posted from Theories n Things)

Conditional excluded middle is the following schema:

if A, then C; or if A, then not C.

It's disputed whether everyday conditionals do or should support this schema. Extant formal treatments of conditionals differ on this issue: the material conditional supports CEM; the strict conditional doesn't; Stalnaker's logic of conditionals does, Lewis's logic of conditionals doesn't.

Here's one consideration in favour of CEM (inspired by Rosen's "incompleteness puzzle" for modal fictionalism, which I was chatting to Richard Woodward about at the Lewis graduate conference that was held in Leeds yesterday).

Here's the quick version:

Fictionalisms in metaphysics should be cashed out via the indicative conditional. But if fictionalism is true about any domain, then it's true about some domain that suffers from "incompleteness" phenomena. Unless the indicative conditional in general is governed in general by CEM, then there's no way to resist the claim that we get sentences which are neither hold nor fail to hold according to the fiction. But any such "local" instance of a failure of CEM will lead to a contradiction. So the indicative conditional in general is governed by CEM

Here it is in more detail:

(A) Fictionalism is the right analysis about at least some areas of discourse.

Suppose fictionalism is the right account of blurg-talk. So there is the blurg fiction (call it B). And something like the following is true: when I appear to utter , say "blurgs exist" what I've said is correct iff according to B, "blurgs exist". A natural, though disputable, principle is the following.

(B) If fictionalism is the correct theory of blurg-talk, then the following schema holds for any sentence S within blurg-talk:

"S iff According to B, S"

(NB: read "iff" as material equivalence, in this case).

(C) The right way to understand "according to B, S" (at least in this context) is as the indicative conditional "if B, then S".

Now suppose we had a failure of CEM for an indicative conditional featuring "B" in the antecedent and a sentence of blurg-talk, S, in the consequent. Then we'd have the following:

(1) ~(B>S)&~(B>~S) (supposition)

By (C), this means we have:

(2) ~(According to B, S) & ~(According to B, ~S).

By (B), ~(According to B, S) is materially equivalent to ~S. Hence we get:

(3) ~S&~~S

Contradiction. This is a reductio of (1), so we conclude that

(intermediate conclusion):
No matter which fictionalism we're considering, CEM has no counterinstances with the relevant fiction as antecedent and a sentence of the discourse in question as consequent.


(D) the best explanation of (intermediate conclusion) is that CEM holds in general.

Why is this? Well, I can't think of any other reason we'd get this result. The issue is that fictions are often apparently incomplete. Anna Karenina doesn't explicitly tell us the exact population of Russia at the moment of Anna's conception. Plurality of worlds is notoriously silent on what is the upper bound for the number of objects there could possibly be. Zermelo Fraenkel set-theory doesn't prove or disprove the Generalized Continuum Hypothesis. I'm going to assume:

(E) whatever domain fictionalism is true of, it will suffer from incompleteness phenomena of the kind familiar from fictionalisms about possibilia, arithmetic etc.

Whenever we get such incompleteness phenomena, many have assumed, we get results such as the following:

~(According to AK, the population of Russia at Anna's conception is n)
&~(According to AK, the population of Russia at Anna's conception is ~n)

~(According to PW, there at most k many things in a world)
&~(According to PW, there are more than k many things in some world)

~(According to ZF, the GCH holds)
&~(According to ZF, the GCH fails to hold)

The only reason for resisting these very natural claims, especially when "According to" in the relevant cases is understood as an indicative conditional, is to endorse in those instances a general story about putative counterexamples to CEM. That's why (D) seems true to me.

(The general story is due to Stalnaker; and in the instances at hand it will say that it is indeterminate whether or not e.g. "if PW is true, then there at most k many things in the world" is true; and also indeterminate whether its negation is true (explaining why we are compelled to reject both this sentence and its negation). Familiar logics for indeterminacy allow that p and q being indeterminate is compatible with "p or q" being determinately true. So the indeterminacy of "if B, S" and "if B, ~S" is compatible with the relevant instance of CEM "if B, S or if B, ~S" holding.)

Given (A-E), then, I think inference to the best explanation gives us CEM for the indicative conditional.

[Update: I cross-posted this both at Theories and Things and Metaphysical Values. Comment threads have been active so far at both places; so those interested might want to check out both threads. (Haven't yet figured out whether this cross-posting is a good idea or not.)]