It’s problem of the many week around here. First Peter Unger’s “argument for dualism”:http://www.nyu.edu/gsas/dept/philo/faculty/unger/power/Ch7.pdf using the many as a vital step, now Robbie Williams on “many mountains”:http://weka.ucdavis.edu/%7Eahwiki/pub/Main/RobertWilliams/problemofthemany3.pdf. Since Robbie’s paper is 9 pages while Peter’s is 158, and I “already promised to talk about Robbie’s argument”:http://tar.weatherson.org/archives/004148.html, I’ll talk about his today.
I always thought that my various views about vagueness were in some amount of tension. What I say about _truer_ requires that the supervaluationist’s precisifications be theoretically unimportant, at best things we can construct out of what is theoretically important, i.e. the _truer_ relation. But what I say about the Problem of the Many seems to require that precisifications matter quite a lot. I’ve never really figured out how to resolve that tension, and that’s basically why I’ve never written a book on vagueness. Now Robbie argues (among other things) that I don’t have a tension in my views, I have an outright contradiction. This is worrying.
The first part of the contradiction is what I say about the Sorites. Following Kit Fine and Rosanna Keefe and Patrick Greenough, I give a pragmatic explanation for the superficial appeal of Sorites reasoning. I say that the reason the true claim (1) is often rejected is that it is mistaken for the false claim (1a).
(1) There is a cutoff in any Sorites series.
(1a) There is a determinate cutoff in any Sorites series.
(I think Robbie somewhat overstates how much this is my idea – Kit and Rosanna and Patrick were really there first. This is my fault for being sloppy with accreditation in the past.) If that’s right, I should predict that the plausibility of (2) would stand or fall with the truth of (2a).
(2) There is a mountain in front of Robbie.
(2a) There is a determinate mountain in front of Robbie.
But, I say, (2) is plausible even though (2a) is not true. Contradiction.
Robbie suggests I resolve the contradiction by accepting the ‘insane’ claim that there are millions of determinate mountains in front of him. That won’t do for two reasons. First, it’s false. Insanity I can live with, not falsity. Second, it makes the wrong prediction about (3).
(3) There is exactly one mountain in front of Robbie.
(3a) There is exactly one determinate mountain in front of Robbie.
On the position Robbie offers me, there are millions of determinate mountains facing him as he starts his climb, so (3a) is false. So (3) should be unacceptable. But (3) is acceptable.
In conversation in _The Cellar_ I believe Robbie offered the following alternative solution to his puzzle, which I more or less accept. We can treat (2a) as being ambiguous between a false sentence where the quantifiers are interpreted objectually, and a true sentence where the quantifiers are interpreted substitutionally. It might be that the substitutional interpretation is what matters. This has the nice advantage of having vague quantifiers without vague objects, because all the vagueness can come in the subtitutends. This looks like it has all the advantages of theft over honest toil, in other words it is my kind of solution.
(I just noticed that “Wo”:http://www.umsu.de/wo/archive/2005/02/22/Substitutional_Quantifiers says something similar, and said it literally while I was writing this post. Great minds think alike!)
This is correct as far as it goes, but it doesn’t go far enough. What if the mountain in front of Robbie is unnamed? Or worse still, if he correctly utters (4).
(4) There is exactly one unnamed mountain in front of me.
A purely substitutional reading won’t help then. My first pass at an answer is to say that the quantifiers range over possible demonstrations. So _that_, accompanied by the right pointing, is a possible substitution. But this leads to yet another puzzle which I can’t entirely solve.
There are many ways to point to the mountain in front of Robbie, and any such pointing will, determinately, pick out a mountain. So it looks like there are again multiple determinate mountains in front of Robbie. I have two possible replies to this, neither of which is successful.
The first is that in the formalism for _There is exactly one determinate mountain in front of Robbie_, although the substitutends are demonstrations, the identity claim is defined over mountains. This isn’t yet a formal proposal, because I don’t know how to formalise it.
The second is that the substitutends are not demonstrations, but possible demonstration types, where two demonstrations that pick out the same object are of the same type. I’m more sympathetic to this approach, for the substitutends, although officially demonstrations, now correlate with mountains in just the right kind of way. I think this way I can get (2a), and even (3a), to turn out to be true, just as I need.
Much thanks to Robbie, and to Wo, for suggestions pushing this along.
