Is Lewis's identity theory a type-type identity theory?

David Lewis’s 1983 identity theory of mind holds that:

1. For each mental state type M there is a causal role R_M such that to be a state of type M is to fulfill R_M.

2. For each actually occurring mental state type M, the causal role R_M is fulfilled by physical states and only physical states.

It is normal to take Lewis’s identity theory to be a type-type identity theory.

But a type-type identity theory identifies being a state of type M with some physical state type. So whether Lewis’s identity theory is a type-type identity theory depends then on whether fulfilling R_M counts as a physical state type.

Here are two accounts of what makes a type T be a physical type:

3. Everything falling under T is physical.

4. Necessarily everything falling under T is physical.

If (3) is the right account of the physicality of a state type, then Lewis’s theory is a type-type identity theory, because everything that fulfills R_M is physical according to (2).

However, (3) is an inadequate account of the physicality of a type. Consider the type ghost. That’s paradigmatically not a physical type. But in fact, trivially, everything that is a ghost is physical, simply because there are no ghosts. If one objects that only instantiated types count, then we can note that by (3) the type ghost-or-pig also counts as a physical type, whereas it surely does not.

It seems to me that (4) is a much better account of a physical type. However, on (4) for Lewis’s theory to count as a type-type identity theory, he would need a version of (2) strengthened by deleting “actually” and inserting “Necessarily” in front. And Lewis’s arguments do not establish such a stronger version of (2). Lewis’s arguments are quite compatible with R_M having non-physical realizers in other possible worlds.

That said, perhaps (4) is not the right account of the physicality of a type either. Consider the type believed by God to be an electron. Necessarily, everything falling under this type is an electron, hence physical. But because the definition of the type makes use of supernaturalist vocabulary, the type does not seem to be physical. This criticism points towards an acocunt of physical type like this:

5. The type T is expressible wholly in terms that natural science uses.

It’s essential for this to fit with Lewis’s theory that causation be one of the terms that natural science uses. But now imagine that we live in a world where one being causes spacetime, and it’s a non-physical being. Clearly, the type cause of spacetime is expressible wholly in natural scientific vocabulary, but given that the one and only instance of this type is non-physical, it sure doesn’t sound like a physical type! Indeed, if this (5) is how we understand physical types, then a type-type identity theory does not even imply a token-token identity theory!

We might try to combine (3) with (5):

6. Everything falling under T is physical and the type T is expressible wholly in terms that natural science uses.

But now imagine that there is no being that causes spacetime and all spatiotemporal entities, but that it is possible for there to be such a being, and that any such being would necessarily be non-physical. In that case causes spacetime and all spatiotemporal entities satisfies (6) trivially, but is surely not a physical type, because the only possible instances of it would be non-physical. (If one objects that types need to be instantiated, just disjoin this type with the type pig, as we did in the ghost case.)

So perhaps our best bet is to combine (4) with (5). But any account on which (4) is a necessary condition for the physicality of a type is an account that goes beyond Lewis’s, because it requires the stronger version of (2) with actuality replaced by necessity.

I conclude that Lewis’s account isn’t really a type-type identity theory, except in the inadequate senses of physicality of type given by (3), (5) or (6).