We investigate the manner in which the power of the weak axiom of revealed preference is affected by the completeness of the choice environment. We fully characterize those domains on which the weak axiom coincides with strong rationalizability for arbitrary choice correspondences. We also provide a related result that characterizes those domains on which the strong rationalizability of a choice correspondence is equivalent to (i) the satisfaction of the weak axiom, and (ii) the strong rationalizability of its restrictions to suitable collections of small sets. Our proof technique involves a generalization of many of the differential concepts of classical demand theory to the abstract choice model. We conclude with an application to the problem of aggregating incomplete preferences.