The first explicit argument for the incompatibility of externalism in the philosophy of mind and a priori self-knowledge is Boghossian’s discrimination argument. In this essay, I oppose the third premise of this argument, trying to show by means of a thought experiment that possessing the “twater thought” is not an alternative, a fortiori not a relevant alternative, to having the “water thought.” I then examine a modified version of Boghossian’s argument. The attempt is made to substantiate the claim that the (...) standard incompatibilist support for its second premise is untenable. Furthermore, a third Boghossian-style argument is rejected on the ground that either its second premise cannot be warranted in the way suggested by incompatibilists or its third premise is mistaken because having the “twater thought” instead of the “water thought” is not relevant. Finally, it is argued that the discrimination argument cannot be saved by invoking closure. The upshot of my discussion is that a compatibilist can dismiss Boghossian-style arguments for incompatibilism without having to deal with fundamental issues concerning self-knowledge and the nature of slow switching. (shrink)