We assume that for most sentences that there is some propositional content. Which is not to say that people think in propositions, or that what language is about is propositions, merely that (most) sentences express some state of affairs, which we can formalize by a proposition. For example, the propositional content of “Dennett is a philosopher” might be Pd, meaning that the property P, the property of being a philosopher, holds for the individual d, Dennett. Now consider another possible world. There might also be a philosopher in that world named Dennett who is nearly the same as our Dennett (they will at least differ in one way, namely by belonging to different possible worlds). If someone in that world said that “Dennett is a philosopher” the propositional content of that sentence is Pe, where e being their Dennett. Now we can ask the question: do these two sentences have the same propositional content? Our first reaction might be to say no, because even though our Dennett and their Dennett are much alike they are still not the same individual. But there are some who hold that there is such a thing as inter-world identity (Kripke for example). If this is the case, and our Dennett and their Dennett are counterparts, then this would seem to mean that d = e and thus that Pd and Pe are really the same proposition, which in turn means that these sentences have the same propositional content, even though they are uttered by speakers in different possible worlds.
But this depends on whether inter-world identity is really equality. And this depends on the nature of the counterpart relation. At the very leas the counterpart relation must be symmetric and transitive. So let us consider a set of possible worlds. Let us define one set as containing our world, a world in which Hitler was a humanitarian (good Hitler), but still was the leader of Germany, born to the same parents, ect, along with a number of worlds that are intermediate between these two. The second set of possible worlds contains the world with good Hitler and a world with an individual, Karl, who led basically the same public life as good Hitler, but came from different parents, and was a better artist.
Now consider the counterpart relation. From our point of view it seems reasonable to say that our Hitler, evil Hitler, and good Hitler are counterparts, and that Hitler and Karl are not counterparts. But it also seems reasonable to say that good Hitler and Karl are counterparts. And this would imply that the counterpart relation is not transitive. Now one countermove to make would be to accept that evil Hitler and Karl are counterparts. But if you accept that then we can just keep creating new counterparts for Karl, and counterparts for those counterparts, until we have accepted that Hitler is a counterpart to some arbitrary individual who is nothing like him. And we can pick this arbitrary individual to be a counterpart of someone in our world, thus showing that Hitler has a counterpart in our world, perhaps even someone who was living at the same time as him. And this is ridiculous. On the other hand perhaps some will argue that good Hitler and Karl aren’t counterparts. But I can’t really see how this assertion could be reasonably made. Neither someone’s name, nor their parentage, nor their skill as a painter make them “essentially” who they are, and thus can’t disqualify two individuals from being counterparts.
As an aside: some may be wondering at this point why I don’t argue against inter-world identity simply by showing that the same properties don’t hold for our supposedly identical individuals. (If two things are identical they have all the same properties.) Obviously Hitler has the property of being a bad person, and good Hitler lacks that property, and so clearly they aren’t the same person. I think that this is a reasonable thing to claim, but those who hold that inter-world identity exists may reject the idea that the properties in question are really the same, and that the property of being bad is really relativized to a single world, meaning that they are bad in a certain world. And thus it is possible to claim that good Hitler is bad in some other world then the one he is actually in. Of course this response does mean that sentences spoken in different possible worlds don’t have the same propositional content, by virtue of not using the same predicates.
But anyways, back to the counterpart relation. As things stand we have two options. One is to accept that the counterpart relation isn’t transitive, and thus that there isn’t “really” inter-world identity. The other is to hold that the counterpart relation is transitive, but that whether two individuals are counterparts is not something that we can know, because it isn’t based on any of their properties. Of course this is at least slightly ridiculous, since we are inventing some essence that ties these individuals together, while at the same time has neither a causal effect nor can be known to us.
So, if sentences don’t have the same propositional content in different possible worlds then this changes our approach to certain facts about rigid designators (terms that designate the “same” object or kind of objects in all possible worlds). It is claimed that if two rigid designators are identical in this world (refer to the same thing) then they refer to the same thing in all possible worlds. But if sentences don’t have the same propositional content then this tells us nothing of interest. Let us say that we agree that water is necessarily identical to H2O. Even so this fact tells us nothing about any world other than our own, because there is no such thing as “water” or “H2O” in these other possible worlds. Even though people in those worlds might speak about “water” their sentences don’t have the same propositional content, and so they aren’t speaking about our water, and thus we can conclude nothing about the relation of their “water” to their “H2O”, or to our water or our H2O for that matter. On the other hand if we have embraced some kind of essentialism as a solution to inter-world identity then we are faced with a similar problem, which is that given any possible world we can’t know if their “water” is the same as our water, and thus we can’t draw any conclusions about it. We can’t even find a possible world where there “water” is the same as our water by stipulation for two reasons: one is that we have no evidence that any other possible worlds have “water” that is the same as our water, because we don’t know what it takes for two thing to be essentially the same. And secondly, even if we could stipulate a world with the same “water” as ours, we wouldn’t know what other facts may or may not have hold in those worlds (meaning that you couldn’t consider, for example, a world with the same “water” as us and the same “air” as us, because there may be no such world, possibly because of a hidden rule concerning essences that prevents them from having both the same “water” and “air” as us without being the same world). And thus, under either approach to resolving the problems with inter-world identity, any interesting conclusions that we might desire to draw from the claim that two rigid designators are necessarily identical are still blocked.