How Many Cats in New York City?

It’s problem of the many week around here. First Peter Unger’s argument for dualism using the many as a vital step, now Robbie Williams on many mountains. Since Robbie’s paper is 9 pages while Peter’s is 158, and I already promised to talk about Robbie’s argument, 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 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.

13 Replies to “How Many Cats in New York City?”

  1. I imagined that sentences such as (3)/(3a) being treated in the way that Lewis suggests in `many but almost one’. That is, in ordinary contexts, we discriminate between objects (and count them) by reference to `almost identity’, rather than strict identity.

    First, say that a is almost identical to b if (roughly) they overlap almost completely. Next, read (S) ‘there is exactly one determinate mountain’ as equivalent to ‘there is a determinate mountain, and every determinate mountain is almost identical to that one’. But every determinate mountain in the relevant range is almost-identical to every other. Then (S) comes out true.

    Lewis’ trick is like Quine’s: if you want to say something odd about language, reinterpret the apparatus of individuation. It seemed to me to work pretty well in this case. No?

  2. I don’t think that works for getting there to be (almost) one mountain. Imagine the following case.

    Most of the surrounding plains are at a height of 1000m. There’s a relatively steep rise to 1500m, then a small plateau, then a very steep rise to 2000m. It seems to me there’s determinately one mountain there. But the land from 1000m to 2000m is a borderline mountain (or actually a mountain on Lewis’s view), and the land from 1500m to 2000m is a borderline mountain (or an actual mountain on Lewis’s view), and they aren’t almost identical. To get around this we’ll have to expand ‘almost identical’ in a way that makes the overall picture look less plausible.

    PS: I think I got this example, or something like it, from Delia. So credit to her if it’s a good example.

  3. Suppose that one group of explorers approaches Delia’s mountain from the northside, and baptises it “mt. gloom”. Another group approaches from the south, and baptises it “mt. bright”. I think we’d want it to be assertable (so, determinate) that “mt. gloom is a mountain”. The same goes for “mt. bright is a mountain”.

    Now enrich the example so that “mt gloom” clearly applies just to the 1500-2000 elevation, and “mt bright” clearly applies to the 1000-2000 elevation. (Manifested in the way the explorers name their forward camps, draw their maps etc). On the “one mountain” line neither group of explorers will be able to correctly utter “mt. N is a mountain”. That seems bad.

    So that’s a little reason to stick with the “many” solution for the “problem of the two”. Of course, some story about counting is then owing. (Again the tactic will be to appeal to the messiness of counting Fs in situations where you have Fs as proper parts of other Fs.)

  4. This is a naive question, but is the substitutional reading of (3a) this?:

    (3as): [EA] (A is determinately a mountain in front of Robbie.)

    (If not, I can’t make sense of these two sentences:
    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.)

    If (3as) is the proper reading, why can we switch from “determinate” the adjective to “determinately” the adverb? The other substitutional reading

    (3as’) (EA) (A is a determinate mountain in front of Robbie)

    doesn’t look true to this non-expert, unless you hold that each demonstration or dubbing latches onto one precisification of the mountain—and it’s hard for me to see in virtue of what that would be true.

  5. Here’s another reason why we better not allow any old term into the substitution class. Forget about the problem of the many for a moment, and suppose we have a finite hill-to-mountain sorites. Let ‘bordy’ be a descriptive name for the shortest mountain in the range, and let x’ denote the tallest thing in the range shorter than x. Then we have: Def(Bordy is a mountain) and Def(Bordy’ is not a mountain). So it appears we have a witness for:

    (EA)(Def A is a mountain & Def A’ is not a mountain).

    …which is not what we want.

  6. I’m with Matt, but also a non-expert. “There are many ways to point to the mountain in front of Robbie, and any such pointing will, determinately, pick out a mountain” seems false to me if it means “…and any such pointing will pick out a determinate mountain.” If I say, “Stand over there” and point, I am pointing to a spot, but I am not pointing to a precise spot.

  7. About the Mt. Gloom/Mt. Bright case; perhaps Henry Jackman’s temporal externalism could help here. (I’m going off a memory of a talk a few years ago, so this may not be reliable.)

    Henry argues that it may sometimes happen that, at one time there is no settled fact of the matter about the meaning of some term, but that later the meaning of the term will be settled in one way or another. Then the truth-value of our current claims is to be determined by the stipulations we go on to make about their use.

    (His case, IIRC, was that at one point people identified platinum as gold, and we might have gone on to use “gold” to refer both to platinum and gold. As things played out, the people who dug up some platinum and said “That’s gold” spoke falsely. But if we’d gone on to use “gold” to refer to both, they would have been speaking truly.)

    So we might say: It may be that eventually everyone will get together and decide whether the mountain is Mt. Bright or Mt. Gloom. When that happens, it will settle that one of “mt. gloom is a mountain” and “mt. bright is a mountain” is true, and the other false. So we shouldn’t say that they’re both determinately true—it may turn out that one of them is determinately false.

    [This won’t solve the problem completely if there are some future histories on which people don’t come to a consensus. IIRC Henry had something to say about that in Temporal Externalism and Epistemic Theories of Vagueness, but I shouldn’t try to reproduce it.]

  8. I agree that we better not run these together.

    *Determinately, that is a mountain
    *That is a determinate mountain.

    But I don’t think we have to do so. In fact I don’t really see a reason to do so. The first carefully exploits a penumbral connection which the second claims not to exist.

    Good points about Mt Doom and about Bordy. I need to think more about how to incorporate these.

  9. My worry is that the move from (3a) to (3as) does seem to run the two together. Is the corresponding
    (1as) Determinately, there is a cutoff in any sorites series
    true on your view? In which case we haven’t made any progress in distinguishing (1) from (3).

    The question is pretty basic here—I’m not even positive which of your two asterisked sentences you mean to be an instance of your substitutionally quantified statement.

  10. I do think (1as) is true, but I think there’s still a difference between (1) and (3). Roughly, there’s no demonstrative that determinately picks out the border in a Sorites series, while there is a demonstrative (actually many of them) that determinately pick out the mountain in front of Robbie. (It’s not determinate what the demonstration picks out, not that any particular thing is the mountain, but whichever thing is the mountain is what’s picked out.)

    The thought is that there’s no suitable substitution in (1x) that makes it determinately true, while there is such a substitution into (3x).

    (1x) … is the borderline.
    (3x) … is a mountain.

    Suitable substitutions are ostensive names or demonstratives with attached demonstrations. (So ‘that’ is OK, and ‘Bordy’ isn’t.)

    I’d really like to have a reason why these are the suitable substitutions. That is a small gap in the theory right now…

  11. It sounds like you need the same thing that the people who talk about de re beliefs need when they want to explain the truth-conditions of “There’s someone that Joe believes to be the shortest spy.” That’s just restating the problem, isn’t it?

  12. Got you. It sounds like you want the same thing that the people who talk about de re beliefs need when they want to explain the truth-conditions of “There’s someone that Joe believes to be the shortest spy.” But I’m just restating the problem, I guess.

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