The paper deals with Hilary Putnam's model-theoretic argument against metaphysical realism. It considers the objections to the argument raised by David Lewis, Mark Heller, James van Cleve, Anthony Brueckner and others, to the effect that Putnam's reasoning fails to undermine versions of metaphysical realism which construe reference along externalist lines. I argue that the version of Putnam's argument that his critics have attacked is indeed powerless against externalist accounts of reference, but that, on a different construal, the argument puts genuine (...) pressure on externalism. On the version of the argument that I develop, the thrust of the model-theoretic considerations is that an externalist construal of reference would make grasp of reference facts impossible. (shrink)