Previous research has revealed considerable variability in children’s performance when computing scalar inferences (Noveck 2001; Papafragou and Musolino 2003; Guasti et al. 2005, among others). One promising explanation for this variability attributes it to differences in children’s ability to generate the alternatives from which these inferences are derived (Reinhart 2006; Barner et al. 2011; Singh et al. 2016; Tieu et al. 2016). According to this ‘Alternatives-based’ approach, children’s difficulties are due to processing limitations or limitations in lexical knowledge. These limitations are only expected to affect the generation of alternative sentences that require lexical replacement. Therefore, children are predicted to readily generate alternative sentences (and associated scalar inferences) based on material from other sources, for example, those based on subconstituents of the asserted sentence. This paper investigates children’s behaviour with one such sentence, where an existential expression is in the scope of a universal quantifier (e.g., Every pig carried some of his rocks). Such sentences have been associated with two scalar inferences: the inference that Not every pig carried all of his rocks, and the stronger inference that None of the pigs carried all of his rocks (Chemla and Spector 2011, among others). The Alternatives-based approach predicts that children will be successful in deriving at least one of these inferences. The results of our experiment are consistent with this prediction, with children deriving inference-based interpretations at adult-like rates. We also observe that adults and children differ in which of the two possible inferences is preferred. We discuss how this pattern is consistent with the hypothesis that children favour strong interpretations (Crain et al., 1994).