When faced with a sentence like, "Some of the toys are on the table", adults, but not preschoolers, compute a scalar implicature, taking the sentence to imply that not all the toys are on the table. This paper explores the hypothesis that children fail to compute scalar implicatures because they lack knowledge of relevant scalar alternatives to words like "some". Four-year-olds were shown pictures in which three out of three objects fit a description (e.g., three animals reading), and were asked to evaluate statements that relied on context-independent alternatives (e.g., knowing that "all" is an alternative to "some" for the utterance "Some of the animals are reading") or contextual alternatives (e.g., knowing that the set of all three visible animals is an alternative to a set of two for the utterance "Only the cat and the dog are reading"). Children failed to reject the false statements containing context-independent scales even when the word "only" was used (e.g., "only some"), but correctly rejected equivalent statements containing contextual alternatives (e.g., "only the cat and dog"). These results support the hypothesis that children's difficulties with scalar implicature are due to a failure to generate relevant alternatives for specific scales. Consequences for number word learning are also discussed. (Contains 3 figures.)