Knowing that some state of affairs—expressed by a proposition, p—is possible, and the possibility that one knows that p have, quite obviously, different meanings. This paper focuses only on their logical relationship—whether they entail one another. I will argue for the following three claims: the basic verificationist principles of anti-realism, at least in their simplest forms, and in conjunction with some other, intuitively reasonable principles, do entail that these two concepts are substitutionally equivalent. Our pre-theoretical expectations question this outcome, as (...) counterexamples can be manufactured. I will also argue that this substitutional equivalence has further, highly counter-intuitive implications. Finally, I will argue that some of the standard strategies to avoid the well-known paradoxes of anti-realism fail to solve this new paradox, while others may be able to do that, but only at a considerable price. However, the introduction of a moderate anti-realist truth operator, the actual objective of this paper, does dissolve the paradox, and arguably at an affordable price. (shrink)