The apparent conceivability of a zombie — a physically indistinguishable duplicate of a conscious person that nevertheless lacks consciousness — seems to show that the relation between brain processes and conscious experience is contingent. But this is probably an illusion of contingency, due to the limitations of our current concepts and tricks of the first-person imagination. The mental is at present conceptually irreducible to the physical, but the strict supervenience of the former on the latter suggests the presence of a concealed necessary connection, which could become transparent to us only through a third type of concept that we would have to create — as part of a theory that yields the necessary connection between the mental and the physical as a logical consequence. Such conceptual creations have been important elsewhere in the development of science.