Generative-AI systems sometimes cause harms, and sometimes people try to excuse the companies that provide them by arguing that they are "general-purpose technologies." The argument is superficially intuitive, but becomes puzzling on reflection: why does the fact that a technology has other uses override the fact that some of those uses are harmful? We unpack the concept of a general-purpose technology and the argument that its provider should be able, morally and legally, to disavow the responsibility that it would ordinarily face. We identify a set of six factors that explain when the appeal to generality is persuasive, and analyze generative-AI systems through this lens.