The California Critical Thinking Dispositions Inventory (CCTDI) is a commonly used tool for measuring critical thinking dispositions. However, research on the efficacy of the CCTDI in predicting good thinking about students’ own deeply held beliefs is scant. In this paper we report on our study that was designed to gauge the usefulness of the CCTDI in this context, and take some first steps towards designing a better method for measuring strong sense critical thinking.