Stanford’s unconceived alternative argument is inductively based on the history of science and tells us that when a scientist is choosing a theory T₁ at time t₁ over a set of less promising alternatives, she is concurrently failing to conceive valid theoretical alternatives to T₁, i.e., theories that will be accepted by a scientific community at later times, thus displacing T₁. The aim of the present paper is to argue that the actual strength and reach of Stanford’s argument sensibly vary according to the status of the unconceived alternatives at time t₁, i.e, whether they are conceivable (theories that could be conceived by scientists at t₁, but in fact are not) or inconceivable (theories which can not be conceived at t₁ as they are incompatible with scientists’ background knowledge at t₁). As Stanford does not explicitly address this issue, we give reasons to conclude that alternatives considered in the unconceived alternative argument are supposedly conceivable at time t₁, and we investigate the consequences of this conclusion for the alleged novel induction the argument draws upon. We then investigate what are the implications for Stanford’s analysis if inconceivability is considered as a possible status of an unconceived alternative at t₁, and we argue that, in this case, Stanford’s antirealism has to be severily restricted to specific phases of theory-change, thus making room for tamed forms of realism.