Wolfgang Schwarz

Non-existing properties

For many things, there is no set that contains just those things. There is no set of all sets, no set of all non-self-members, no set of all non-cats, no set of all things, no set of pairs (x,y) such that x is identical to y, no set of (x,y) with x part of y, no set of (x,y) with x member of y.

If Lewis is right and there are proper-class many possibilia, there is also no set of possible philosophers, no set of possible dragons and no set of possible red things. However, if Lewis is right and there are proper classes, there will be proper classes of all these things. But there will still not be a class of all classes, a class of all non-self-members, a class of all non-cats, etc.

So whether we identify properties with the sets or with the classes of their possible instances (or with functions from worlds and times to sets or classes of instances), many properties will not exist. There will be no such thing as being a class, not being a cat, no identity, no parthood, no membership. At least for membership, Lewis explicitly agreed: "I do not believe in the membership relation" ("Tensing the copula", p.8).

If most properties are proper classes, there will also be hardly any interesting higher-order properties, for proper classes are never members of other classes. There is no such thing as being perfectly natural, being intrinsic or being somebody's favourite property. That last property is suspicious anyway: Couldn't my favourite property be being a set (which exists if there are proper classes)? And couldn't it be being somebody's favourite property -- in which case that property would have to be a member of a member of itself?

So if properties are construed set-theoretically out of possible instances, 1) many predicates will fail to express a property; 2) many (apparent) property names will fail to name a property; 3) for many things there will be no property which just those things have in common.

It would be wrong to blame the set-theoretic construction. There is a good reason why these classes are missing in set theory: the alternative assumption is inconsistent. We have to choose between consistency, avoiding (1), (2) and (3) and the principle

*) If some Fs are not Gs, the property F is not identical to the property G.

For instance, if we identify properties with the fusions of their instances, we get a consistent theory avoiding (1), (2) and (3), but also without (*): my left arm is a body part but not a body, yet the fusion of all body parts is the same as the fusion of all bodies.

It seems to me that consistency and (*) are non-negotiable. Thus we must live with (1), (2) and (3): there is no property expressed by "does not belong to itself", nor is there a property named by "the property of not belonging to itself", and for the vast majority of things, there is no property they all have in common (otherwise there would have to be 2^n properties for n things, which is impossible, because "thing" is unrestricted here and also applies to properties).

So how bad are (1), (2) and (3)? First, they create problems in formal semantics. What semantic value are we to assign names and predicates that denote or express non-existing properties? These problems are closely related to Tarskian problems about languages not being able to be their own meta-language. I'm not sure if there is a satisfactory solution. Frege suggested to de-reify semantic values, to not assign any thing at all to our predicates, but 'functions', where functions are not things, so that it is actually wrong to use the first-order predicate "function"; we should use a second-order, plural predicate instead. Maybe that could work. But even if not, I don't think the problem is very serious from a philosophical point of view. Suppose it cannot be solved. What follows? That we can always only give a formal semantics of the kind we know it for languages with restricted domains. That's unfortunate, but not really a philosophical earthquake, I believe.

Perhaps more serious are the consequences for some philosophical theories of properties. Lewis -- as I understand him -- argues that we need (abundant) properties because there are so many truths about them that cannot plausibly be explained away: "humility is a virtue", "redness is a sign of ripeness", "there are undiscovered fundamental properties", and so on. But then what about "parthood is transitive", "membership is intransitive" or "there is a relation in which everything stands to itself"?

This time, we have no option but to explain them away: "parthood is transitive" really only means that whenever x is part of y and y is part of z, then x is part of y. (It's not a problem that we use the predicate "is part of" here: we're not committed to any entities by using predicates; we're not committed to the Russell properties by saying, truly, that Red does not instantiate itself.) "There is a relation in which everything stands to itself" probably has to be interpreted substitutionally. That's ugly, and it certainly weakens Lewis's point against nominalists who argue that all apparent talk about properties can be paraphrased away: "[E]ven if such paraphrases exist -- even if they always exist, which seems unlikely -- they work piecemeal and frustrate any systematic approach to semantics."

Funnily, it turns out that Lewis himself later commits himself to paraphrase away all talk about properties. He continues to identify properties with classes, but on his structuralist account of classes, there is no such thing as, say, the class of all possible philosophers. Something is that class only relative to a singleton relation; and any singleton relation is as good as any other. When we talk about classes, we quantify over all of them. Or rather, we don't really quantify over them, because the singleton relations belong to the non-existing relations. But we can plurally quantify over mereologically construed ersatz-pairs to get the same effect. This carries over to properties. Talk about properties takes place inside plural quantifiers over mereological ersatz-pairs. Lewis agrees (again in "Tensing the Copula", p.9): "properties understood as classes are not the same from one [set-theoretical] hierarchy to another. Within any one hierarchy we can quantify over classes. So we can simulate quantification over properties by embedding a genuine quantifier over classes within a simulated quantifier over hierarchies".

At this stage, one might wonder if we can't just scrap the embedded quantifiers over classes and interpret talk about properties as plural throughout: that this apple instantiates Green means that it is one of the green things, "humility is a virtue" is a third-order plural predication. At some places in "Tensing the Copula", it sounds like this is indeed what Lewis has in mind. The proposal has the advantage of doing away with properties-as-things completely, thus circumventing all problems and paradoxes about non-existing collections. (It is more or less the same proposal as Frege's.) Robert Black ("Against Quidditism", p.98) reports, on the problem of quantifying over collections of worlds that are too large to exist: "Lewis now plans to avoid this problem by utilising George Boolos's idea that second-order logic amounts to plural quantification. Thus rather than quantifying over unlimited collections of possible worlds, we can quantify plurally over the possible worlds themselves, however many of them there may be".

Unfortunately, it is far from clear how this is supposed to work in the end. We would need polyadic third-order quantification, which plurals don't deliver. So it seems that Lewis left us with a rather unfinished account of abundant properties. It would all be simple if naive set theory were consistent and sets sui generis entities in Lewis's metaphysics. But since neither is the case, the story isn't simple at all. I'm not sure how exactly it goes.