Friday, November 25, 2005

Counting Vaguely Identical Objects

Dominic Hyde gave three talks on vagueness during his recent visit here, the most recent of which focussed on an argument from Pinillos against vague identity (NB you need a subscription to Mind to get to the full text from this link).

Suppose a ship (b) leaves port and two ships (a and c) dock, but (for the usual reasons having to do with the replacement of bits) we want to say it is indeterminate whether a=c, and indeterminate whether b=c, yet determinately not the case that a=c. Pinillos argues this cannot be right, and a key premise of his argument is that the set of ships which left port - {b} - is distinct from the set of ships which docked - {a,c} - because these two sets have different cardinalities. (This leads him to conclude that there must be something in one set which is not in the other - so that there is some pair consisting of one member from each set such that it is (determinately) the case that the two things in that pair are not identical - contradicting the assumption that all the relevant identity claims are indeterminate.)

I won't go into the details of Dominic's response to the Pinillos argument, since (as far as I know) it's not yet publicly available. But my own reaction (different from Dominic's) was to wonder whether the defender of this sort of vague identity should accept that the two sets are distinct. Maybe she should say that it's indeterminate whether the set of ships which left port is identical to the set of ships which docked.

Sure, there is a strong intuition that something is wrong with '{b} = {a,c}', but perhaps this could be explained as an intuition to the effect that, because of the difference in cardinality between the sets, this identity claim is necessarily not true, which does not (for people - like Dominic - who are into three-valued logics) entail that it is false. (By analogy, a claim Dominic argued for in one of his other talks here is that although we have a strong intution that claims like 'Patch p is red and non-red' are not true, we should not therefore regard them as false when p is a borderline case.)

In fact, is the defender of vague identity even compelled to accept the claim that the sets have distinct cardinalities? If it really is indeterminate whether a=b and indeterminate whether b=c, then the set of ships which left port contains b, and it is indeterminate whether it contains a and indeterminate whether it contains c. And the set of ships which docked contains a and c, and it is indeterminate whether it contains b. Maybe, therefore, we should say that the cardinality of each set is indeterminate, in such a way that it is not determinately the case that the two sets have distinct cardinalities.


Daniel Elstein said...

Hi Carrie, sorry if this is annoyingly off-topic, but something bugs me about the way Pinillos presents the issue (and to some extent this goes for you too). It looks as though you can think that identity is vague without thinking that it's indeterminate in the technical sense, where the technical sense is that if it is indeterminate whether x=y, then it's not true that x=y and it's not true that ¬x=y. This isn't a very intuitive sense of vagueness. More plausibly, when it's vague whether x=y my credence is divided between x=y and ¬x=y, whereas on the strict indeterminacy view I seem to have to have zero credence in both alternatives, since to have non-zero credence in a sentence is to hold it to be epistemically possible that that sentence is true. Since Pinillos' argument relies on it being not true that ¬x=y when it is indeterminate whether x=y, his argument seems not to touch the idea of vague identity that I find plausible (and all the business about three-valued logic doesn't matter). But I suppose this just follows from the fact that proponents of vague identity don't agree with me about what vague identity is!

More relevantly, I'm dubious about your final suggestion that the defender of indeterminacy can deny that the sets have different cardinalities. You say:

If it really is indeterminate whether a=b and indeterminate whether b=c, then the set of ships which left port contains b, and it is indeterminate whether it contains a and indeterminate whether it contains c.

But when the set contains a, that's because a=b, and so the cardinality of the set is still 1; and similarly for c. More precisely, it's supertrue (and so determinate) that that set has cardinality 1, because it is 1 on any assignment of identities. And it's supertrue that the other set has cardinality 2. But I'm sure you're going to tell me that I'm not allowed to use that terminology when we're talking about indeterminacy...

Carrie Jenkins said...

Hi Daniel,

It's not obvious to me that vague identity cases are cases where I ought to assign credence roughly 1/2 to x=y and roughly 1/2 to ¬x=y; that looks more like a characterization of a state where I lack the relevant information about x and y. Still, this isn't something I have particularly strong intuitions about.

On your second point, it's not that I have anything against supervaluationism in general (though I don't think it's obvious that it's the best way to go either) but Dominic's project is to develop a non-supervaluationist truth-functional approach, so relying on supervaluations to support the Pinillos premise that the sets have different cardinalities will look question-begging, insofar as the Pinillos argument is supposed to have force against all vague identity views.

Daniel Elstein said...

I'd say that there should be a presumption (based on parsimony, naturalism etc.) that vague identity involves some division of credences between identity and non-identity, rather than believing in a third state, vague identity. Of course, intuitions might end up defeating the presumption, but what I see is the opposite presumption being made. But it's usually silly to argue about burden of proof stuff...

I think my worry about your argument for indeterminate cardinality is just that I think you need to give a reason why the fact that it's indeterminate whether a set contains b (for instance) should make it indeterminate what the cardinality of the set is. Even if bringing in supervaluationism does beg the question, I have a pretty strong intuition that if the only way that b can be in a set is by being identical to an object that I already know is in the set, whether b is in the set makes no difference to the set's cardinality. Compare: I know that the set of authors of Alice in Wonderland contains Carroll. I believe that [Dodgson is a member of the set iff Dodgson = Carroll]. So if I am ignorant about whether Dodgson = Carroll, this ignorance does not lead me to be uncertain of the cardinality of the set.

Carrie Jenkins said...

On your intuition that "if the only way that b can be in a set is by being identical to an object that I already know is in the set, whether b is in the set makes no difference to the set's cardinality". Suppose it is indeterminate whether b is in the set because b is indeterminately identical to a, which we already know is in the set. Your intuition, if I've got you right, is that the two sorts of indeterminacy 'cancel out', as it were, by the time we get to thinking about the cardinality of the set.

I can feel the force of the intuition, but I still wonder whether it could be argued most of that force comes from having supervaluationist intuitions (or other intuitions that we should be suspicious of if we want a truth-functional logic) lurking in the background. The intuition in question seems to be similar to the intuition that, for Bill and Ben two borderline tall guys with Ben a bit taller, it's determinately true that:
If Bill is tall then Ben is tall.
But (this is something that I raised in discussion after Hyde's talk) we might be able to explain away this sort of intuition as tracking the *non-falsity* of that conditional, rather than its truth. If so, perhaps we could do something similar with the intuition that, where a=b is indeterminate, it's determinately true that:
{a} has cardinality 1.
That is, perhaps we can explain away the intuition as an intuition tracking the non-falsity, rather than the truth, of this cardinality claim.

Daniel Elstein said...

I guess you could do that. But consider the following argument: if it's indeterminate whether a=b, and indeterminate whether b=a, then isn't it also indeterminate whether a=a? After all, we can't guarantee that the two indeterminacies cancel out: i.e we can't guarantee that
a=b iff b=a
I can feel the force of the intuition that they do cancel out, but perhaps we can explain this intuition away as tracking the non-falsity, rather than the truth, of the claim that a=a.

Now, you might say that it's analytic (or something) that a=a, but it seems analytic to me that 'tall' is monotonic, and so that if Ben is taller than Bill then [if Bill is tall then Ben is tall]. The worry is that unless you allow indeterminacies to cancel out, everything will be indeterminate if anything is.

Carrie Jenkins said...

Agreed, you can't simply apply that strategy willy-nilly. But I feel that the original argt has a bit more going for it intuitively than does the analogous move in your a=a case. Still, as I said, I don't have very strong intuitions or views in this area, and even less of an idea what to say to people who seem to have intuitions different to mine ...!

Angel Pinillos said...

Carrie, I think that once you accept that the sets {b} and {a,c} have distinct cardinalities, you cannot also say that the sets are indeterminately identical. This is because one set will have a genuine property (not an "indeterminate property") that the other one lacks (i.e. one will have cardinality 1 and the other will not).

However, you could avoid this by taking up the suggestion of your last paragraph in which it is denied that the cardinalities of the sets are distinct. But now the problem is to come up with a principled method for counting objects that covers both the normal cases and the cases of vague identity. This isn't a trivial matter at all. In the paper, I tried to show how Parsons' methods didnt quite work. When I gave this paper at an APA conference, Parsons reply (as I remember it) was to come up with distinct methods for counting; one for regular objects and one for vague ones. I found this problematic.