Despite the recent upsurge in research on abstract concepts, there remain puzzles at the foundation of their empirical study. These are most evident when we consider what is required to assess a person's abstract conceptual abilities without using language as a prompt or requiring it as a response-as in classic non-verbal categorization tasks, which are standardly considered tests of conceptual understanding. After distinguishing two divergent strands in the most common conception of what it is for a concept to be abstract, we argue that neither reliably captures the kind of abstraction required to successfully categorize in non-verbal tasks. We then present a new conception of concept abstractness-termed 'trial concreteness'-that is keyed to individual categorization trials. It has advantages in capturing the context-relativity of the degree of abstraction required for the application of a concept and fittingly correlates with participant success in recent experiments. This article is part of the theme issue 'Concepts in interaction: social engagement and inner experiences'.