David Lewis, in his 1986 book On the Plurality of Worlds, argues that properties are sets. Any property should be taken as “the set of all its instances — all of them, this- and other-worldly alike. Thus the property of being a donkey comes out as the set of all donkeys, the donkeys of other worlds along with the donkeys of ours.” (p. 50.) For me, Lewis’s claim is one of those philosophical statements, both infuriating and intriguing, of which one doesn’t even understand how it could seem to be sensible to anyone. This is more interesting than a mistake or a disagreement; it at least prima facie indicates a deep difference in one’s approach to philosophy. I wrote on Twitter:

Why would anyone think that “a property is a class of things across possible worlds” is a meaningful and enlightening thing to say? This is not a rhetorical question.

And then I got some good replies and queries by Adam F. Patterson and Arturo Javier-Castellanos. But as I started mentally composing a reply to them, I realised it would make more sense to write it out as a blog post. And so here we are.

The claim that properties are sets of individuals, in this world and other worlds, seems to me somewhere on the border between the false and the unintelligible. Take the property of being bald. We can of course write true sentences starting with the words “being bald is…”; for instance “being bald is having no hair on the top of one’s head” or “being bald is a property shared by Peter Adamson and Michel Foucault”. But the claim “being bald is the set of all bald people” seems, on the face of it, as false (or maybe unintelligible) as “the number three is my girlfriend” and “honesty is purple”. The set of all bald people is a bunch of people. Being bald is clearly not a bunch of people.

There are other ways to make the same point. Properties can be perceived. I can see that someone is bald. But can I see that someone is a member of the set of bald people? The only sense in which I can see that is surely by seeing that the person is bald and then (if I happen to be thinking about set theory) concluding that this person is a member of the set of bald people. Quite in general, it seems that there is an explanatory asymmetry between having a property and belong to the set of things that have that property. That X is P explains that X is a member of the set of things that are P. Not the other way around. You belong to the set of bald people because you are bald; you’re not bald because you belong to the set of bald people. Such an explanatory asymmetry suggests that Lewis’s identity claim can’t be true.

What’s more, suppose that properties are these sets stretched out across all the possible worlds. Then denying that there are other possible worlds would amount to denying that there are properties. But surely that can’t be right? There clearly are properties in this world, and they don’t seem to depend in any sense on the existence of other possible worlds. (Or on the existence of sets, for that matter. Surely the existence of properties is far more obvious, far less problematic, than the existence of sets. I myself firmly believe that there are properties, but I do not believe that there are any sets. Maybe they ‘exist in ZFC’ or something like that; but they’re not real.) And that is as it should be. Our grasp of the world is grounded in our very local experiences. It had better not be the case that those experiences themselves require a grasp of the world as a whole, let alone of the totality of all possible worlds. (And if you want to avoid that, maybe you had better avoid adopting Leibniz’s God’s-point-of-view metaphysics while removing God!)

Having said that, we arrive at the difficult point. All of the above seems obvious. So obvious that it can’t be the case that Lewis has failed to recognise it. Instead, it must be the case — or at least we should assume it to be the case — that Lewis would smile through all of the above and then explain that I’ve approached the entire question from the wrong direction; there there is something wrong in a fundamental way with my approach to the question of properties. But what? Here’s something that Arturo Javier-Castellanos wrote:

Classes are technical I guess, but unlike properties, we know how to individuate them, so if you can analyze properties in terms of things+classes, that looks like progress

and here is the Stanford Encyclopedia article on Properties:

Quine (1957 [1969: 23]) famously claimed that there should be no entity without identity. His paradigmatic case concerns sets: two of them are identical iff they have exactly the same members. Since then it has been customary in ontology to search for identity conditions for given categories of entities and to rule out categories for want of identity conditions (against this, see Lowe 1989). Quine started this trend precisely by arguing against properties and this has strictly intertwined the issues of which properties there are and of their identity conditions.

So perhaps here is what Lewis would say to me: those so-called obvious judgements of yours are all nice and dandy around the kitchen table, but when we start doing philosophy we should be ready to revise our understanding even of something as seemingly basic as properties. And the motivation to do so is that we want our entities to have well-behaved identity conditions; and properties, as Quine has shown, don’t seem to have those. But here I, Lewis, come to the rescue with my suggestion that properties are sets; and sets do have clear identity conditions. If you don’t want to accept that, you had better have an alternative story.

But why would we believe that anything that exists must have clear identity conditions? I’m certain that I exist, but the question whether I am the same person as Victor Gijsbers at four years old is, I’d say, rather hard to answer. Maybe there is no answer. So it seems I can be clear about something’s existence without being clear about something’s identity conditions and even without being clear about whether there are any identity conditions.

Let’s look at an example of a genuine question about the identity of properties. Is being hot the same property as having high mean molecular energy? Great question. How do we go about answering that? Presumably by a combination of linguistic analysis (or conceptual choice) and empirical research. We might want to distinguish between a phenomenological feeling of hotness and a power to cause certain thermometer readings. We might want to distinguish between that power (which could perhaps be generated in many very different ways) and the typical physical state (if any) that underlies that power. If we empirically research whether there is a such a typical physical state, we find that, yes, it consists in having high mean molecular energy. And so there is a sense of ‘being hot’ in which it denotes the same property that is denoted by the phrase ‘having high mean molecular energy’.

At no point in the answering of this question did we have to talk about sets or other possible worlds. Indeed, Lewis’s identity criterion is of literally no help at all in answering the question. Is being hot the same property as having high mean molecular energy? Well, Lewis might say, are the hot individuals across all the possible worlds exactly the same as the individuals with high mean molecular energy? That doesn’t help. We can’t go there and check. The only way we could answer Lewis’s reformulation of the question is by first answering the original question; once we know that A and B are the same property, then, I guess, can we say that all individuals across all possible worlds that are A are also B and vice versa. But surely we can never do it the other way around.

But if Lewis’s identity criterion cannot be used to answer real questions about identity, then what is it for? And how could it motivate a rejection of the obvious truths from which I started? So I end up still being in a state of confusion; unable to see why a theory like his might seem to make sense.

(And of course my confusion runs beyond properties. People in this part of philosophy also claim things like “propositions are sets of possible worlds”. That is just as obviously false. A proposition can be true, it can be something you believe. But a set of worlds can’t be true or false, and you can’t believe a set any more than you can believe a table. I am sure there are many more examples.)