Wolfgang Schwarz

Egan and Lewis on Properties

Andy Egan, in "Second-Order Predication and the Metaphysics of Properties", argues that there is a bug in Lewis' theory of properties which can be fixed by identifying properties not just with sets but with functions from worlds (and times) to sets. I disagree: there is no bug. But there are some interesting questions about Lewisian properties nearby.

Here's the alleged bug. Consider the second-order property being somebody's favourite property. This property belongs to Green. So on Lewis' account, Green is a member of the set being somebody's favourite property. But at another possible world, Green is nobody's favourite property. So it is not a member of that set. Contradiction. In the parallel case of accidental properties of individuals, Lewis resorts to counterpart theory: If Graham Greene is a writer in our world and not in another world, that's not because Greene both is and isn't a member of the set writer, but because Greene is a member while one of his counterparts isn't. However, this solution doesn't work for Green because properties don't have counterparts.

Says Lewis. But we could fix the bug by saying otherwise, by endorsing counterpart theory for properties, as Mark Heller once proposed. We can still let properties themselves be sets. These sets have different properties at different worlds by having different counterpart sets at those worlds. Egan suggests that instead we should let properties be functions from worlds to sets: the value of being somebody's favourite colour for our world is a set (of properties) including Green; its value for other worlds may be a set not including Green. (Heller's and Egan's proposal, modulo Heller's ersatzism, are related roughly like Sider's temporal counterpart theory and Lewis's temporal parts theory.)

So why is this not a bug in Lewis' account? Because Lewis has two answers to the question how things can have different properties at different worlds (or times, or places).

Often it is said that things have some of their properties relative to this or that. Thirst is not a property you have or lack simpliciter; you have it at some times and lack it at others. The road has different properties in different places; here it is surfaced, there it is mud. Nine has the property of numbering the planets at our world, but not at a possible world where a planet takes the place of our asteroid belt. [...] Relative to the number 18, the number 6 has the property of being a divisor; but not relative to 17.

A property that is instantiated in this relative way could not be the set of its instances. For when something has it relative to this but not to that, is the thing to be included in the set or not? [...] [W]hat is had by one thing relative to another might better be called a relation, not a property. (Plurality, 52f.)

Lewis goes on to say, here and elsewhere, that however some apparent properties, shapes for instance, should not be understood as disguised relations: Some things, temporal parts of world-bound individuals for example, have bent simpliciter.

Egan takes Lewis to claim that no monadic predicate should be understood as expressing a disguised relation. This not only contradicts what Lewis says -- Lewis says that the problems of accidental and temporary intrinsics to which counterpart theory and temporal parts theory are solutions apply only to a few special, intrinsic properties like bent -- it also attributes to Lewis a view that is just insane. Is Lewis supposed to believe that 6 is a divisor relative to 18 in virtue of having a part or counterpart at 18 which is a divisor simpliciter? that 9 fails to number the planets at another world in virtue of having a part or counterpart there that doesn't number the planets? that Einstein is famous nowadays in virtue of having a present temporal part that is famous? Certainly Lewis did not believe this, and certainly he did not just miss all such examples (as witness the quote above). Rather, he believed that these are cases where what might appear to be a property grammatically is in fact a relation.

So we have two answers to the question how something x can be F at one world but not at another: Either x itself is a member of F and some counterpart of x is not (as is the case with F = Bent), or F is in fact a relation and x stands in that relation to one world but not to another (as is the case with F = numbering the planets).

A relation, for Lewis, is a class of ordered pairs. Thus a relation between things and worlds is a class of (thing, world) pairs. Such a class is set-theoretically equivalent to a function from worlds to sets of things: to get the function's value for a given world, simply take the set of all things that are paired with the world in the relation; in the other direction, to get the relation from the function, simply take all pairs (x,y) for which the function's value for y contains x. So if F is a relation between things and worlds, we can equivalenty understand it as a function from worlds to sets of things.

In a passage I've omitted from the quote above, Lewis rejects constructions of properties as functions from worlds to sets of things as "misguided". But I think what he objects to is not the construction per se -- which is after all equivalent to one of his own suggestions --, but rather the claim that these constructions deliver properties, as opposed to relations. (Lewis: "I find such constructions misguided: what is had by one thing relative to another might better be called a relation, not a property"; notice the colon.)

Now return to being somebody's favourite property. We have to ask what kind of thing that is. Is it a genuine intrinsic property in Lewis's sense, or is it something that is had by one thing only relative to another, in the way numbering the planets is had by the number nine relative to our world but not relative to other worlds? Perhaps it depends on the interpretation of "somebody". If that is restricted to the population of whichever world is under consideration -- as Egans understands it --, then clearly the apparent property would count as a relation in Lewis's terms: it is only had relative to one world but not to another.

So there's no need to fix anything in Lewis' theory of properties to get Egan's solution. For Lewis, too, being somebody's favourite property is a relation between worlds and properties, that is (equivalently), a function from worlds to sets of properties.

Hm. That got longer than I expected. The "interesting questions about Lewisian properties nearby" will have to wait until tomorrow.