One of the tasks of language learning is the discovery of the intricate division of labour between the lexical-semantic content of an expression and the pragmatic inferences the expression can be used to convey. Here we investigate experimentally the development of the semantics-pragmatics interface, focusing on Greek-speaking five-year-olds' interpretation of aspectual expressions such as "arxizo" ("start") and degree modifiers such as "miso" ("half") and "mexri ti mesi" ("halfway"). Such expressions are known to give rise to scalar inferences crosslinguistically: for instance, "start," even though compatible with expressions denoting completion (e.g. "finish"), is typically taken to implicate non-completion. Overall, our experiments reveal that children have limited success in deriving scalar implicatures from the use of aspectual verbs but they succeed with "discrete" degree modifiers such as "half". Furthermore, children are better at spontaneously computing scalar implicatures than judging the pragmatic appropriateness of scalar statements. Finally, children can suspend scalar implicatures in environments where they are not supported. We discuss implications of these results for the scope and limitations of children's ability to both acquire the lexical semantics of aspectuals and to compute implicatures as part of what the speaker means.