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.

Gabriel Uzquiano proposed something in a conference Q&A that was, I think, either one of the first two proposals or some combination of them, and which sounded to me as if it ought to work. I’ve also had conversations with Agustin Rayo about this kind of thing, and he seemed less optimistic. They’d do a much better job than I would, though, of explaining the grounds for optimism or pessimism…

This comment is for the technical information:

One common treatment of classes is just to paraphrase them away – the only way actual objects are related to classes is by the membership relation, and if we replace “x is a member of C” with “phi(x)”, where phi is the defining formula of C, then we can paraphrase away every reference to C. So classes aren’t really objects. Then we can take extensions to be “classes”, so we get an extension for the property of being either red or a set. And we can also get an extension for being non-self-membered – the extension of this property isn’t an object, so we don’t have to worry about whether or not it falls under any extension.

Of course, this resolution seeks to explain properties in terms of classes, but relies on a further reduction of classes to formulas, and thus likely back to properties.

At any rate, if classes aren’t objects, then we can’t represent a function whose range includes classes by the ordinary set of ordered pairs. The only other common representations of functions that I know of are the representation as a class (ie, a formula “phi(x,y)” such that it is provable that for every x there is a unique y satisfying phi(x,y)), or as an algorithm. In the class case, we can take a property as a formula of this sort where x ranges over worlds and y ranges over formulas encoding classes of elements of that world, and this is how we would represent the attempted solution of my first paragraph.

The only representations of functions as algorithms that seem to be common are the codes of functions from N to N as Turing machines – and this only captures the partial recursive functions and not all functions. I don’t know if there’s any good way to calculate on worlds. And I’m sure you could come up with a property whose function should be uncomputable anyway.

As for taking functions as primitives, this is much more commonly associated with category-theoretic foundations for mathematics as opposed to set-theoretic ones. I don’t know enough about that to say how it works there. (I only understand category theory when I put it on set-theoretic foundations, but I think that might defeat the purpose as far as some algebraists, and Carnegie Mellon philosophers, are concerned.)

However, even taking functions as primitives, we might end up with some sort of Russell problem. How would we represent the property of being a function? Or being a function that isn’t a member of its own range? Or something of the sort.

I don’t know anything about what a plurality might be other than a set or a class, so I can’t say anything about that.

This comment is for a different question:

A lot of these worries are related to the fact that we want to identify a property with some mathematical entity that exists at every world (rather than a physical object, which merely has counterparts at the other worlds). The Russell-type worries suggest that such an object may have to be a member of itself, or may have to encompass too many things to be a set (or a class, or whatever).

But it seems that Andy’s worry is that if a property is one of these world-independent objects, then any of

itsproperties must be had non-contingently. Perhaps I’m missing something important, but why can’t we just solve this problem by taking sets and other mathematical entities to be world-bound themselves, but with counterparts? We can even require the counterpart of a set S to have all the same members as S. Then the property “being Elmer Fudd’s favorite property” can be the set that contains the actual set of (actual and possible) green things (assumign “being green” is actually his favorite property), and the w-set of (actual and possible) yellow things (assuming w is a world where “being yellow” is Fudd’s favorite property), and so on. This set will have counterparts in every world, and I would guess that each of these counterparts would be the representation of “being Elmer Fudd’s favorite property” in the appropriate world (though maybe there’s some reason for this cross-world property to be different in different worlds?)I suppose there might be some issues with an object that exists at one world being an element of a set that exists at another. But it doesn’t immediately seem any worse to me than an object existing at one world being an element of a set that exists separately from any world.

Brian,

Functions from possible worlds to classes are problematic for the reason you mention. The standard reduction of functions to sets or classes of ordered pairs would require classes to be members of ordered pairs, but, unfortunately, proper classes are never members. Nevertheless, I wonder whether you could not sometimes “simulate” functions from possible worlds to classes with the help of a relation, by which I mean, in this context, a class of ordered pairs. To be more specific, maybe you could “simulate” functions from possible worlds to classes or individuals and/or sets. The relation would take possible worlds in its domain and some individuals and/or sets in its range. If R is such a relation, then define R_w to be the class {y: e R}. Think of R_w as a class of individuals and/or sets coded by w in R. The suggestion, then, is that we use the relation to “simulate” a function from worlds to classes. Given R, we might think of each class of the form R_w (for each w in the domain) as the “value of the function” for w. In other words, if we think of the property “being a set” as given by a certain relation that takes possible worlds in its domain and sets in its range, then the extension of the property “being a set” at a possible world w would be S_w, namely, the class {y: e S}. Because S is a class of ordered pairs of worlds and sets, it doesn’t require any proper classes to be members. So it looks like we should be able to simulate talk of functions from worlds into classes of individuals and/or sets without assuming proper classes to be members.

Unfortunately, this will not work for properties had by proper classes like, for example, “being a class”. Or, more precisely, it will not work as long as we take relations to be classes of ordered pairs and ordered pairs are identified, as usual, with sets of a certain sort. For proper classes would never be components of ordered pairs and we wouldn’t be able to represent a relation with proper classes in its range. However, in the appendix of “Parts of Classes” (together with John P. Burgess and A. P. Hazen) and in “Mathematics is Megethology”, David Lewis considers (for a different reason) the possibility of encoding “ordered pairs” by mereological means. I seem to remember that the coding requires some substantial assumptions about the mereological structure of reality, but at least it suggests that there might be some way of extending the method used above to cover the case of properties like “being a class”.

I am a complete beginner in this sort of thinking about classes and properties – please forgive my soon to be exposed ignorance. However, one idea caught my attention right away: the idea of worlds. Please correct me if I am wrong, but it seems that all of the discussion here assumes that it is possible that there is more than one world.

I don’t see how that is possible: if I discover what appears to be another world, haven’t I merely discovered more of the one that I am in? In other words, what possible experience would qualify as evidence of the existence of another world? Try as I might, I cannot imagine it. Help!