April 25, 2007

Essences and Impossible Antecedents

The first issue of The Reasoner has just appeared online. Berit and I contributed a paper, "Why Counterpossibles are Non-Trivial", in which we give three reasons for rejecting a vacuous reading of counterpossibles. One reason is that a non-trivial reading facilitates an analysis of essences. While Kripke's wooden table, Tabby, is necessarily a member of the set {Tabby}, it is not essential to Tabby that it be a member of that set. Neither is it essential to Tabby that seven is prime. It is tempting to offer the following explanation. If there hadn't been sets, Tabby might still have existed; and if seven hadn't been prime, Tabby might still have existed. But this sort of explanation requires, for its non-triviality and informativeness, that counterpossibles be non-trivial and informative. At some of the closest (impossible) worlds where there are no sets (or numbers), Tabby exists. By contrast, Tabby fails to exist at all the closest worlds where there is no wood. It is essential to Tabby that he be wood, but not essential to Tabby that seven is prime. Generally, x is essentially F iff x would not have existed if nothing had been F

14 comments:

Aidan said...

I'm interested in how this bears on the following issue in developing a platonist version of the Iterative Conception:

'Another route we could try would be to see the modality involved [in the dependency relation between sets and their members] as some kind of necessity. We might say, perhaps, that one object presupposes another if the one would not have existed without the other. [...] What this amounts to if the thing is a collection is that it would not have existed without if its members had not. But once again we run into difficulties very soon, since in the case of pure collections -- the empty collection and other collections depending on it alone -- the platonist presumably believes that the members exist necessarily and so the antecedent cannot be realized. We might hope to deal with applied collections by the proposed route and then treat pure collections by analogy as some sort of special case, but even this does not work, for although it is no doubt true that the singleton of my goldfish would not have existed if Bubble had not existed, the platonist is equally committed to the converse: Bubble would not have existed if its singleton had not. So the platonist cannot, even in the applied case, appeal to counterfactual reasoning to explain the relation of priority between collections.'
(Potter, 'Set Theory and its Philosophy': 39)

The hope would be that taking your suggestion seriously might allow us to characterize the modality involved here in a suitably non-symmetrical fashion, and perhaps resist Potter's argument that this route is hopeless for pure collections. Or am I barking up the wrong tree?

Aidan said...

Sorry, I've realized I should do some spelling out of what I was driving at above. The suggestions were the following. First of all, that we might capture the asymmetrical dependency relation between an impure set and its members as follows: it is essential to that set that it have just those members, but not essential to the members that they be members of that set. There are some obvious problems (that sadly weren't obvious to me 5 minutes ago); it looks like we're still in trouble explaining the relation between {Bubble} and {{Bubble}}.

The second thought was that when dealing with pure sets, even though we may hold that the members exist necessarily, we might be able to have differential judgments concerning different statements of the form:

'If the members of S had not existed, then neither would S itself',

while on the standard Lewisian account all these would be vacuously true.

I should learn to think as fast as I can post comments :(

Joe Salerno said...

Aiden, I'm not quite seeing Potter's problem for platonism. Maybe you can say more about pure and impure sets, what the platonist wants to say about what is essential to what, and why the standard account contradicts this.

Aidan said...

Ok, sure. The problem is that it's a core thought behind the iterative conception that members are prior to the sets they form. What this comes to a little more formally is a requirement that if a set S was formed at level M in the iterative hierarchy, then all of the members of M must have been formed by level M-1.

The problem for the platonist is initially just to make sense of this notion of priority, since the standard metaphor used to describe this relation sneak in illicit temporal references. For example, it's natural to say that priority is the requirement that all the members of a set S formed at stage M must have been formed at some earlier stage in the hierarchy. But that talk's only harmless so long as we're sensitive to just how metaphorical it must be for the platonist, for whom there is not literally any asymmetric temporal ordering on the stages (Potter constrasts the platonist here with the constructivist who thinks we construct the hierarchy stage by stage).

So Potter thinks the platonist has to make sense of this relation of priority between sets and their members without implicitly appealing to temporal notions. They have to instead hold that there is a asymmetric metaphysical dependency relation holding between sets and their members. But what could that relation be?

In the passage I quoted the suggestion is that the dependency relation is something like the following: If the members hadn't have existed, then the set they comprise wouldn't either. The problems are meant to be 1. that in the case of pure sets (i.e. those built out of just the empty set), this conditional has an impossible antecedent and 2. in the case of impure sets (i.e. those built out of urelements), the relation is a symmetrical one after all, since if the set didn't exist, neither would its members.

In your move away from the standard account, I saw potential for holding that some statements of the form:

If the members of (pure) set S hadn't have existed, neither would S itself,

false, despite the fact that the antecedent is unrealizable. And if statements of this form can be false, then perhaps the *true* instances could mark out some sort of dependency relation holding after all.

Secondly, it seemed to make room for the relation between an impure set and its members to be asymmetric. So my singleton metaphysically depends on me since it couldn't exist without me, but I could exist without it in the following sense: If there were no sets, Aidan might still have existed. But this needs 'If there were no sets, Aidan might still have existed' to be non-trivial, otherwise once again the truth of such statements can't allow us to discriminate cases where there is a genuine dependency relation holding.

As I suggested in my last post, it's unlikely that this latter suggestion cuts any ice. At best it works for the special case where the impure set in question actually has urelements as members. But take the singleton of my singleton. For there to be a dependency there, we'd want to have the following be (non-vacuously) true:

If there were no sets, {Aidan} might still have existed.

But this seems manifestly false (once we're holding that it's not vacuously true). So it looks like my original comment was way off-track. I hope I've at least managed to make clear why I thought there was something that might be worth exploring here (and I hope I haven't overstayed my welcome exploring it on Joe's patch!).

Joe Salerno said...

Right. Since we are distinguishing levels anyway, how's this natural suggeston? Formulate the condition this way: if there were no M sets, then {Aidan} (i.e., the M-1 set) would not have existed.

Aidan said...

Sorry for taking so long to respond - we had a conference here this weekend that I was co-organizing.

That suggestion does look like it'll generalize to the cases I was worried about. I guess I was assuming that the closest (impossible) worlds in which there are no sets of level M would also be worlds in which there were no sets at all - that these impossibilities are pretty coarse-grained, so to speak. Am I right in thinking your suggestion would force us to give that assumption up? And if so, are there reasons to think that's a real cost, or is there something to be said in its favor? I not seeing things clearly enough to be able to answer these myself.

Brit Brogaard said...

Hi Aidan
Interesting discussion. Impossible worlds are as fine-grained as possible worlds. We take them to be sets of sentences that satisfy interesting and non-trivial constraints depending on the theoretical role they are supposed to play. So, there are worlds where there are sets yet no sets of level M. Whether the closest worlds are worlds where there are no sets is a different story. As I see it, worlds with sets but no sets of level M and worlds without sets are tied for closeness.

Mike said...

x is essentially F iff x would not have existed if nothing had been F

I might not have this analysis right, but here's a counterexample. Let F be the property of being on the largest island. Take a possible world in which it is true that everything not on the largest island instantly dies. It is true there that if nothing had been on the island (including me), then I would not have existed. But surely it is not an essential property of me that I am on the largest island.

Joe Salerno said...

Thanks Mike. You have a described a world w where the existence of anything living depends on its being on the largest island. So it seems right, as you say, that, at w, had nothing been on the largest island then you would not exist. But it doesn't follow that it is an essential property of you, in the actual world, that you are on L. That would require the relevant counterfactual be true in the actual world.

However, in many of the worlds closest to the actual world where nothing is on the largest island, you do exist. So, in the actual world, it is false that if nothing had been on the closest island then you would fail to exist. Therefore, being on the largest island is not an essential property of you.

Mike said...

I'm not sure I follow this, Joe,

However, in many of the worlds closest to the actual world where nothing is on the largest island, you do exist. So, in the actual world, it is false that if nothing had been on the closest island then you would fail to exist. Therefore, being on the largest island is not an essential property of you.

It does turn out in the world w that I discuss that 'being on the largest island' is an essential property of me. That far we seem to agree, but you tell me. Surely, if that is an essential property of me at w, then it is an essential property of me in every world in which I exist. But then it is actually an essential property of me.

Now of course I agree that the counterfactual is false in the actual world. But so what? All I need is that be true in some world. If it is true in some world or other, then 'being on the island' is an essential property of me. And it is a short argument from that premise to the conclusion that it is essential in every world in which I exist, including of course @.

Joe Salerno said...

Mike, you just agreed that the relevant counterfactual is false in the actual world. So your example is not a counterexample the theory presented in the post.

Along the way you revealed your implicit assumption---viz., that having an essential property in one world entails having the essential property in every world. This is an interesting thesis. I definitely do not think that it is obviously true.

Think about essences in the simplest way as those properties that an object has in every metaphysically possible world in which it exists. If my logic of metaphysical necessity is governed by S5, then it seems that your thesis holds. For if there is an accessible world w1, where i have essential property P, then, by S5 principles, i have that property essentially in every possible world in which I exist, including the actual world.

However, S4 is also a competitor for the logic of metaphysical necessity. In that logic, the worlds are not symmetrically related. So even if there is an accessible world w in which i have property p is every accessible world w' in which i exist, it does not follow that I have that property in the actual world (let alone essentially).

Brit Brogaard said...

I don't see why the island example is a counterexample even if S5 is the correct logic for metaphysicial modality.

As I see it, the force of the counterexample stands and falls with the truth of 'if nothing had been on the largest island, then I would not exist' in some possible world w1.

As 'something is on the largest island' is bound to be a priori, 'nothing is on the largest island' is impossible. But then there is no *possible* world w1 of the described sort.

While our bi-conditional is metaphysically necessary, it need not hold in impossible worlds. So there is no guarantee that I have the essential property of being on the largest island in all the described impossible worlds w1.

And while I do have the essential property of being on the largest island in some impossible world w1, it doesn't follow that it is an essential property of me in the actual world that I am on the largest island.

Mike said...

However, S4 is also a competitor for the logic of metaphysical necessity. In that logic, the worlds are not symmetrically related. So even if there is an accessible world w in which i have property p is every accessible world w' in which i exist, it does not follow that I have that property in the actual world (let alone essentially).

Well, I guess it does seem a problem that your analysis is not compatible with the most widely accepted logic of necessity. But set that aside. The problem I have in mind does not depend at all on the choice of S4 or S5.
To get the counterexample, I needn't assume anything like @ is accessible from w (the island world). So let's suppose it isn't.
The problem is that, by your analysis, it is true in w that the beings on the island in w have the essential property of being on that island. But clearly being on the island is not an essential property of the beings in w. No doubt, there is a world w' accessible from w (perhaps @, perhaps not, it doesn't matter) in which those very beings exist and are not on that island. To make good the claim that being on the island is an essential property of the beings on the island in w you'd have to defend the claim that in every world w' accessible from w in which those islanders exist, they are on the largest island. That's wildly implausible, seems to me.
But even that wouldn't be enough. Here's one of lots of other counterexamples. Certainly there are worlds in which, on your analysis, Tabby has the essential property of being made by carpenter C at t. Consider a world w in which it is true that, if carpenter C does not make Tabby from W at t, some official will immediately destroy W. In such a world it is easy to make the counterfactual true, 'if nothing had been made by C at t, then Tabby would not exist'. But then, on your analysis, Tabby has the essential property in w of being made by C at t. But that's bizarre. Certainly there is some possible world w' accessible from w, in which Tabby was made by some non-carpenter.

Mike said...

As 'something is on the largest island' is bound to be a priori, 'nothing is on the largest island' is impossible. But then there is no *possible* world w1 of the described sort.

Brit,

You seem to want to say that w1 is an impossible world, but I'm not following the argument. All that's true in w1 is that there are some beings on the largest island and, no being not on the island exists. How does it turn out a priori true that there are islanders? But, second, how do we arrive at the conclusion that w1 is impossible?