A great deal of discussion in the recent literature has been devoted to the so-called `McKinsey' paradox which purports to show that semantic externalism is incompatible with the sort of authoritative knowledge that we take ourselves to have of our own thought contents. In this paper I examine one influential epistemological response to this paradox which is due to Crispin Wright and Martin Davies. I argue that it fails to meet the challenge posed by McKinsey but that, if it is set within an externalist epistemology, it may have application to a related paradox that concerns the problem of radical scepticism.