When involved in projects concerning language and its limits, a number of philosophers (Wittgenstein, Heidegger, Frege) have been driven to the conclusion that there are certain things which appear to be both objects and not objects. They have tried to avoid the contradiction by various strategies, such as the apparently desperate one of declaring some of their own assertions to be meaningless. However, a quite different strategy is to accept the contradictions involved. This requires the use of a paraconsistent logic, though this is only a first step. How, given the resources of such a logic, can one understand the thought that something both is and is not an object? And does such an understanding in any way destroy meaning? In this talk I will address all of these issues.