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This paper examines Hume’s formulations and uses of the conceivability principle (abbreviated as CP: Whatever is conceivable is possible) and the inconceivability principle (abbreviated as ICP: Whatever is inconceivable is impossible). In Hume’s works, we identify different versions of CP and ICP, including proper CP, proper ICP, the weak versions of CP and ICP, the epistemic versions of CP and ICP, and show that Hume not only expresses ICP, but also really maintains it. Assuming an axiomatic characterization of modalities, we argue that if there is a sharp distinction between levels of modalities, then Hume’s conceivability arguments do not hold. But, in a rather different way, we also argue that if Hume’s conceivability arguments hold, then there should be no distinction between levels of modalities. Finally, we argue that after Hume, there are lots of endeavors in logic and philosophy to distinguish different levels of modalities, and to accept new concepts of necessity other than logical necessity.

DOI: 10.1007/s10516-019-09422-5

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