There is debate over how the integration of non-verbal quantifying and verbal counting relates to the representation of number principles. A stringent representational test would be one in which a child obeyed a number principle where it ran counter to a characteristic procedure. We devised a test relying on the uniqueness principle for using evidence from a miscount in inferring a counterfactual cardinal number. All the 5-year-olds passed, as did half the preschoolers. Subtests probed associated number-skills. We suggest that a crucial preschool step is to start conceptualising error by categorising relations between counting and miscounting. That step is taken at a similar age to passing a representational theory of mind test but the two were uncorrelated.