This is basically a request for help directed to people who know more set/class theory than I do. The issue is what we should say about Lewis’s theory of properties given his own comments in “Tensing the Copula” and Andy Egan’s comments in Second-Order Predication and the Metaphysics of Properties.
Lewis’s original theory was that properties are sets of possibilia. That won’t work for two reasons.
- Egan’s reason – It can’t allow that things that exist in more than one world have different properties in different worlds.
- Lewis’s reason – there are proper class many possibilia, and there aren’t sets that big.
It is tempting to reply to Lewis’s reason by just saying that properties are classes rather than sets. But this won’t actually help very much, or at least it won’t help very obviously because of Egan’s problem.
Andy argues, plausibly to my mind, that we should identify properties with functions from worlds to, well something. Andy says that they are extensions, and identifies these with sets because he’s explicitly not worrying about complicated set-theoretic problems. (See his footnote 2.) But that can’t be right. Consider the property of being either red or a set. That doesn’t have an extension, at least if extensions are sets.
There are two things we could do here.
First, we could say that extensions are classes. Now we run into difficulties with the standard set/class-theoretic reduction of functions, since that would imply having a class as a member of (a member of?) a class. So we’d like to avoid that.
Second, we could identify extensions with pluralities. A property is a function from a world to some things. Again, it’s not entirely obvious how this is compatible with a set-theoretic treatment of functions.
So here are some of the questions I was hoping for help with. (Not many questions I know!)
- Is there some way of understanding functions other than as sets/classes of ordered pairs, with each ordered pair to be understood in turn as a set a la, say, von Neumann? Is it metaphysically/mathematically plausible to take functions as primitives?
- Can we have functions from worlds to pluralities? (If so, I’d say that version of Egan’s view is the way to go.)
- Are there other options here we could take?
- Is there any way to explain Lewis’s position to someone (like me, and more importantly my students) who don’t know a lot of set theory?
It’s a little undignified to beg like this, but I’m really unsure what to do on this point, and I simply don’t know enough set theory to make progress on my own.