Kripke argues for the existence of rigid designators, and that when x and y are such rigid designators x=y → □(x=y). (In other words if x an y are identical then they are necessarily identical; i.e. identical in all possible worlds). Based on this Kripke then argues that since the mind and the brain are not necessarily identical, because we can conceive of minds existing without brains, the mind is not identical to the brain in this world (~□(x=y) → x≠y). There are many possible responses to this claim. One is to argue that the conceivability of minds without brains doesn’t show that it is actually possible. There is also another, stronger, argument against the conclusion, which is to maintain that the mind is not a rigid designator. And both of these arguments can be seen to be rooted in Kripke’s own analysis of rigid designators, hence the title.
Let me first tackle the argument that the conceivability of x≠y doesn’t imply ~□(x=y). The best way to see this is to consider two different rigid designators that refer to the same person, let us say Samuel Clemmons and Mark Twain. Certainly it is conceivable that they were different people, and I am sure that some people really don’t know that they are the same person. However they are rigid designators, and Samuel Clemmons = Mark Twain, so □(Samuel Clemmons = Mark Twain). Really all that conceivability shows us is that we simply don’t know enough about the rigid designators to say for sure that they are identical, or are simply thinking about them in the wrong way (if we mistakenly say that Samuel Clemmons was possibly someone other than Mark Twain). It is easy to see where the mistaken belief that conceivability of non-identity implies that necessary identity is impossible comes from. Specifically it comes from the counterpart function (which I introduced here). When we say that □(x=y) we really mean that ∀w∈W(C(w,x) = C(w,y)). Therefore if we can think of a world in which the counterparts of two rigid designators aren’t identical then the rigid designators aren’t necessarily identical. The problem with this lies in the fact that when we consider counterparts for some rigid designator, about which we don’t know all the facts, we are really only considering objects in other worlds that meet some limited description that we have associated with the rigid designator. Thus if we don’t know much about Samuel Clemmons and Mark Twain we might think that surely there could be counterparts of Samuel Clemmons that weren’t the same as Mark Twain (one man has one name and a different man has the other). What we are missing is that the counterpart function operates on the objects themselves, not on the rigid designators, and thus even if we are considering two different rigid designators if they refer to the same object then they will have the same counterparts. But this means that in order not to make mistaken claims about the necessity of identity we must first determine if two rigid designators refer to the same object in this world. And this means that to settle the question of whether the mind is necessarily identical to the brain we must settle it in this world first. And appeals to conceivability and necessity can’t hold any weight without the analysis of our world first, as the example of Samuel Clemmons and Mark Twain shows.
But perhaps we place more weight on our ability to conceive of minds without brains. Perhaps we think that anything with the right kind of first person point of view is conscious (and hence a mind), and surely there are things that can meet that requirement which aren’t brains. In fact it is possible that it might meet all the requirements to be our consciousness and not be a brain. And I find this argument compelling. But if we are to accept it then we must accept that consciousness is not a rigid designator, but is instead a definite description. Why? Well because I can certainly also conceive of worlds in which brains are conscious, and in those worlds they would be identical to minds. But if they are identical to the mind in some worlds, but not in all worlds, then they aren’t necessarily identical to minds. And, as Kripke himself points out, the only way this is possible is if “the mind” is some kind of definite description, and not a rigid designator. And if “mind” is really a definite description then Kripke’s argument falls apart, since the necessity of minds being identical with brains has nothing to do with whether they are identical to brains in this world.
So it seems that either way we go Kripke’s argument against mind-brain identity is flawed. If the mind and brain are rigid designators then conceivability can’t tell us whether they are necessarily identical or not. And if we do think that conceivability can tell us whether they are necessarily identical or not then we must accept that the mind isn’t a rigid designator, and thus the argument can’t proceed. This is good news for physicalists, since we think that there is some kind of mind-brain identity in this world, and it would be unfortunate if the entire project could be shown to be flawed from the outset.