On their alethic reading, the (T)-Schema, the (D)-Schema, and the (K)-Schema codify three of the most basic principles of possibility and its dual (necessity). This talk discusses these schemata on a broadly epistemic reading, and in particular as candidate principles about conceivability and its dual (inconceivability of the opposite). As will be shown, the question whether (T) and its classical dual equivalent, as well as (D) and (K) hold on this reading is not only a logical one but involves a distinctively metaphysical controversy between realist and antirealist views on the relation between truth on the one hand and various cognitive conditions such as knowability, conceivability, and thinkability on the other. It will be argued that the stance we take with regard to the metaphysical dispute has consequences for our assessment of the plausibility not only of (T) and its classical equivalent, but also of (D) and---when that stance is combined with a structural account of propositions---of (K) as well; with all four taken in the above epistemic sense. A second upshot will be that the same sensitivity to metaphysical background commitment also applies to our view as to whether or not inconceivability of the opposite coincides with apriority.

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