GARY KEMP THE INTERPRETATION OF CROSSWORLD PREDICATION (Received in revised form 20 July 1998) 1. In what sorts of relations may an object as it actually is stand to either a merely possible object, or to an object, not as it is, but as it might have been? And what explains the difference between those relations which can be exemplified in that way, and those which cannot? Understood in terms of the possible worlds interpretation of modality, this is merely a restriction to the actual-world case of a larger pair of questions: What relations are suitable candidates for crossworld relations? And what makes them so? To begin with examples, it makes sense to say: (1) Even if Mary had not existed, John could have been taller than Mary actually is. On the possible worlds interpretation, this holds just in case there is some possible world in which Mary does not exist, but in which John’s height exceeds the height that Mary has in the actual world. It is irreducibly crossworld: it compares Mary in the actual world with John in some non-actual world – Mary as she is with John as he might have been. (1) entails (2) John could have been taller than Mary actually is. Although, unlike (1), this does not assert the existence of a Mary- less possible world, its truth does not evidently require that Mary exist in some world at which John is taller than her. Its truth requires at most a crossworld comparison of John as he might have been with Mary as she is. Clearly (2) does not entail (3) It could have been the case that John is taller than Mary. Philosophical Studies 98: 305–320, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.