Recent studies use semantic structural priming to show that various cases of linguistic strengthening happen through a common mechanism: generation of implicatures through alternative-based (scalar) reasoning. In this paper, we used priming to investigate another group of cases, where strengthening is postulated to follow from the tendency to systematically neglect structures that verify a sentence by virtue of an empty configuration (neglect-zero): empty-set quantifiers ('at most/fewer than') and disjunction under a universal quantifier. We report data indicating semantic priming between these two structures, but not between them and scalar 'some'. We propose that 1. there is a common mechanism in use for strengthening constructions postulated to follow from the neglect-zero tendency, and that 2. this mechanism is different from the one involved in alternative-based reasoning.