Mumford's argument for dispositions not grounded in non-dispositional properties is:
- Some subatomic particles are simple.
- Simples have "no lower-level components or properties".
- All properties of subatomic particles are dispositional.
- If x has a grounded dispositional property, the ground of that property is "among the lower-level components or properties" of x.
However, even after this modification, it seems that 3 is dubious. For instance, suppose an up quark is simple. But then, isn't simplicity a property, and a non-dispositional one at that, so that the up quark has a non-dispositional property?
Perhaps, then, we need to have a sparse view of properties. But simplicity seems sufficiently natural to qualify as a property even on sparse views. A different way to save 3 would be to restrict to intrinsic properties, and have a very narrow view of intrinsic properties: having an intrinsic property does not depend on an entity's relations to other entities or the lack of such relations. But being simple depends on not standing in a whole-to-part relation to anything else. In fact, the same restriction lets us not worry about the proton counterexample to 3, since being composed of quarks is not an intrinsic property, then. Note, however, that such a notion of intrinsic properties is narrow enough—for instance, squareness might not be an intrinsic property, since maybe an object is a square only because its parts are arranged a certain way.
But let us continue with such a narrow sense of intrinsicness. Is 3 true? Maybe not. Bill the Up Quark seems to have all sorts of non-dispositional properties: being identical with Bill, being u seconds old, having a worldline that is shaped in such-and-such a way, etc. Maybe some of these don't count as properties on a sufficiently austere sparse account of properties. But all that needs significant argument.
Actually, I am not completely sure about 4, or more precisely the conjunction of 3 and 4. Suppose Occasionalism is true. Then properties like "charge" are grounded in God's dispositions to move particles around. In such a case, we might keep 4 but deny that anything other than God has dispositional properties, or else we might keep 3, but allow that the particles' dispositions are grounded in God's dispositions. Now, Occasionalism is false. But the same issue comes up if one thinks that laws of nature are grounded in global entities—say, fields—that push particles around. Now, in such a case, we can try to ask about the entities that push particles around—don't they have dispositional properties, and aren't they simple? And quite possibly the answer is positive. But a lot more work is needed here.