## Friday, February 15, 2008

### ... is the set of ...

The locution "...is the set of..." occurs all over philosophy (2365 times in JSTOR's philosophy collection). Sometimes the phrase is used innocently to introduce a stipulation, to draw out the consequences of a stipulation or as part of the philosophy of mathematics. But some uses are far from innocent. The non-innocent uses I want to focus on are ones in metaphysics where something non-mathematical and non-stipulative is identified with a certain kind of set. An example is David Lewis's identification of a property with the set of the possible objects that exemplify it. There is also a whole slew of examples that use the related locution "...is a function from...to..." (a function is, after all, a certain kind of set): thus, some take the meaning of a sentence to be a function from situations or worlds to truth values.

Such non-innocent identifications of things with sets (and, as a special case, with functions) are almost certain to be false. My first argument is very simple. A set is any (necessarily?) existing object that, together with other objects and a member-of relation, satisfies the axioms of set theory. The set of F's is the set to which all and only F's stand in the member-of relation. But if there is one collection of objects and member-of relation that satisfies the axioms of set theory, there are infinitely many such. With a perverse member-of relation, we can identify any necessarily existing entity as the set of F's. Let o be the number 7 or any other necessarily existing entity (for the sake of the argument, I assume there are sets and that they exist necessarily). Then we can take o to be the set of all possible hot objects, just by reinterpreting the initial member-of relation into another relation, say member*-of, that still satisfies the axioms of set theory. (The sun will then be a member* of o.) So we can't take seriously the idea that "the" set of all possible hot objects is the property of hotness if we think there is non-stipulatively such a thing as the property of hotness.

Unless there is a privileged interpretation of set theory, such identifications simply have no hope. But do we really have reason to suppose there is a privileged interpretation of set theory? I doubt it. I don't have an argument here, just a strong intuition.

But even if there were a privileged interpretation of set theory, anything that one might identify with a set can be equally well identified with some other set. For instance, instead of taking a property as a set of possible objects having that property, one might take it as the set of possible objects lacking that property. All the theoretical benefits are still there. Granted, Lewis's approach might be slightly simpler. But a slight difference in complexity does not justify a strong epistemic reason to believe one theory rather than another. So one's reason for believing Lewis's theory over this competitor will be weak. But there will be many such competitors. (Here is one that seems quite natural: a property is a function f from the set of all possible objects to the set {0,1}, where f(x)=1 iff x has the property.) One's reason for believing Lewis's variant to be true will be low. Moreover, it just seems somehow unlikely that of these variants one would actually be right in an objective sense.

This argument is even more obvious when we look at the idea of identifying meanings with functions. A function f from a set S to a set T is a set of ordered pairs <x,y> where x is in S and y is in T, satisfying the condition that each member of S appears in the first place in exactly one pair in the set. But we can equally well define a function* f from a set S to a set T to be a set of ordered pairs <y,x> where x is in S and y is in T, satisfying the condition that each member of S appears in the second place in exactly one pair in the set. Any reason to think a meaning is a function can be turned into an equally good reason to think a meaning is a function*. And, besides, there is more than one way to define an ordered pair <x,y>.

At the same time, what I said is in an important way unfair to Lewis and others. For a charitable reading (supported by things that Lewis actually says) of claims like "a property is a set of possible objects" is that the writer does not actually believe in properties. Rather, he means that all the work that property-talk does in our language can be equally done by set-of-possible-objects-talk. And that claim is immune to my criticisms. However, it is important to note that this claim either has no metaphysical consequences (after all, any work that talk of elephants does in our language can be equally well done by talk of ordered pairs of the form <7, e>, where e is an elephant, with some modifications of the rest of the sentences, but this does not mean elephants are ordered pairs, pace indeterminacy of translation claims) or else is tantamount to a denial of the real existence of properties. Not that there is anything obviously wrong with the latter. But what would be very bad would be to take the claim to give a realist metaphysics of properties.