 Another respect in which Millikan’s second core claim needs to be refined concerns the mapping rule that relates representations to their contents. Millikan describes this relationship as an “isomorphism” or “homomorphism” obtaining between the set of representations on the one hand and the set of represented states of affairs on the other (Millikan 2004, ch. 6; 2013b; 2020). The general idea is this: the mapping rule assigns to every representation (of the relevant set) a represented state of affairs, and does so in such a manner that transformation relations between representations correspond to transformation relations between represented states affairs. This condition is fulfilled in a straightforward way in the beaver splash example, where transformations of a splash signal that consist in the substitution of a particular time t* (for t) or location l* (for l) correspond to transformations that consist in the substitution of the very same time t* (for t) or location l* (for l) at the level of the represented state of affairs. But it can also be fulfilled in more interesting ways, e.g. by a set of sentences where the substitution of one name for another (“Ginger” for “Fred”) corresponds to the substitution of one object for another (Ginger for Fred) at the level of the represented states of affairs.  (For an in-depth discussion of this aspect of Millikan’s theory, cf. Shea 2013.)