The division of labor between semantic and pragmatic contributions of <i>almost</i> and other proximatives has long been controversial. A watershed in this dispute is Sadock’s (1981) proposal that <i>I almost won</i> only conversationally implicates, rather than entailing, <i>I didn’t win</i>. Neither this “radical pragmatic” line nor a pure entailment account covers the full range of data, including the non-cancelability of the polar component and the distribution of polarity items. This gap prompts the construct of assertoric inertia (Horn 2002a), exploiting the distinction between what is entailed and what is asserted. I buttress that approach here with additional arguments, address the role of other semantic and pragmatic factors, and revisit the viability of assertoric inertia in the light of other recent work.