## Friday, September 28, 2007

### Properties as sets of (actual) individuals

One benefit of admitting the existence of possibilia, says Lewis, is the identification of properties with sets of possibilia. I've always been confused as to why this is meant to be better than the actualist saying that proeprties are sets of their instances.

Lewis says, "The usual objection to taking properties as sets is that different properties may happen to be coextensive. . . the property of having a heart is different from the property of having a kidney, since there could have been a creature with a heart but no kidneys." And this is usually the reason I'm given when I ask this question.

But the 'since' is no good! The fact that there could have been a creature with a heart but no kidneys shows only that the property of being a renate might not have been the property of being a cordate. That only tells us that the properties are actually distinct, and hence that the properties aren't actually identical to their actual instances, if we accept the (necessity of) the necessity of non-distinctness. But Lewis *doesn't* accept that, due to his acceptance of counterpart theory.

Assuming contingent identity in general is not incoherent, what would be wrong with someone holding that being a cordate *is* identical to being a renate, but only contingently so? On this view, something might have had being a cordate and lacked being a renate because the actually identical properties might have been distinct. Whenever Lewis holds that two distinct properties are accidentally coextensive this theorist holds that they are contingently identical; properties that Lewis identifies, this theorist claims to be necessarily identical. Would anything go wrong with this?

You might object to the proposal on the grounds that, even if contingent identity in general is okay, it's not okay for sets to be contingently identical. Why? Because sets have their members essentially and because the axiom of extensionality is necessary. Those two claims entail that identical sets are necessarily identical. (Proof: if S and S* are identical they share their members. Given the essentiality of membership they share their members in all worlds. So given the necessity of extensionality they are identical in all worlds.) But I don't find this that convincing from the perspective of the counterpart theorist. I think the counterpart theorist should hold that whether sets have their members essentially is a context-sensitive matter, just as it is a context-sensitive matter whether or not I am essentially human. When we specify a set extensionally - 'the set of a, b, and c' - it's natural to suppose we invoke a context whereby nothing gets to be a counterpart of that set unless its members are the counterparts of a, b and c (what we say if one of a, b or c has multiple counterparts at a world is going to get tricky). But when we specify a set intensionally - 'the set of the Fs' - it's natural to suppose that the counterpart of this set at a world is the set of the things at that world that are F. It doesn't seem objectionable to me, then, to say that 'the set of the cordates' and 'the set of the renates' are contingently identical which, on the current proposal, is just what it is for the proeprties being a cordate and being a renate to be contingently identical. Being such that 2+2=4 and being such that everything is self-identical, on the other hand, will be necessarily identical, because at every world the set of things that are such that 2+2=4 is identical to the set of things that are such thateverything is self-identical - namely, it is the set containing everything at that world.

I'd be interested to hear reasons for not going this way.

Mike said...

The argument in OTPW (pp. 50-2) does note that "the usual objection to taking properties as sets is that different properties may happen to be coextensive". The intuition, I'm guessing, is that some properties are such that they not even contingently identical. Part of the worry, I think, is that otherwise any two unintantiated properties at a world will be identical. But it is difficult to believe that the property of being a talking donkey might be identical to the property of being green.

Ross Cameron said...

I guess I'm scepical about claims concerning intuitions to the effect that properties couldn't have been identical. Let it be settled by best theory, say I. If identifying properties with sets of possible individuals is a benefit, identifying them with sets of actual individuals is even more so (assuming for the moment that there's no independent reason to believe in possibilia). Now I concede it would be a reductio of the view if it had the consequence that there couldn't be a creature with a heart that lacked a kidney. But it doesn't. It *does* have the consequences that being a talking donkey is contingently identical to being the missing shade of blue, and that being green could have been identical to being a donkey. But those are highly theoretical claims, so I'm happy to have them settled either way by best theory.

Mike said...

Ross, you write,

Now I concede it would be a reductio of the view if it had the consequence that there couldn't be a creature with a heart that lacked a kidney. But it doesn't. It *does* have the consequences that being a talking donkey is contingently identical to being the missing shade of blue, and that being green could have been identical to being a donkey.

I suppose I don't see how the latter are any more theoretical than the former. Why not say, let the best theory also determine whether there couldn't be a creature with a heart that lacked a kidney? As it is, it seems invidious. Modal intuitions on hearted-non-kidneyed creatures count decisively against any theory, but the possible identity of the property being green and the property being a talking donkey might be an interesting consequence of the best theory. We seem to be going on modal intuitions in both cases, no?, and the former seem no more evidential than the latter. But maybe I'm missing something.

Ross Cameron said...

I don't have much to say in support of this, and maybe it's just a prejudice, but I kind of think the man on the street will have a pre-theoretic belief that, say, he could have lacked kidneys whilst still having his heart, whereas I expect you'd be hard pressed to find anyone other than a philosophers immersed in theory to give an opinion on the possible contingent identity of properties. That's an empirical question, of course. But it just strikes me as far more costly - far more an affront to pre-theoretical belief - to say that, necessarily, every thing with a heart has a kidney than it does to say that two distinct properties might have been identical. (I'd be prepared to deny either claim given good reason, of course; but I'd need a stronger reason to deny the former than the latter.)

Mike said...

In fairness, we could also ask the guy on the street whether he thinks there might be no difference between everything that's red and everything that's green (or, more simply, between red and green). For what it's worth, that's an affront to the commonsense picture, too. I concede that this just mires the question more. It doesn't take us anywhere toward solving it. That might have something to do with not knowing the solution.

Rich Woodward said...

Hi Ross,

If I remember rightly, Quine says that the only thing deserving the name 'the properties' are intensional entities of some sort. But that rules out, from the off, identifying the properties with sets of actual individuals since that would equate properties with their actual worldly extensions. It's noticable in this regard that Quine is happy to claim that the work that we need properties for might just as well be done by sets of actual individuals. But that wouldn't, for Quine, be to identify the properties with sets of actual individuals.

Louis said...

Hi Ross,

I can think of one motivation that does not rest on claims regarding the coherence of contingent identity for sets or anything else. It is plausible to think that the properties and relations are closed under certain operations. If those operations include, e.g., necessitation, then the result of the application of some of these operations to *cardiate* and *renate* will be the property *being such that necessarily every cardiate is a renate*. If *cardiate* and *renate* are identical, then the result of these operations on the very same raw materials will also be *being s.t. every cardiate is a cardiate*. But it is implausible to think that everything with the one property also has the other.

- Louis