 A quite different application of impossible worlds to epistemic states has been proposed by JC Beall 2009, in relation to the  Church-Fitch knowability paradox.  The Church-Fitch reasoning is supposed to show that is contradictory to suppose that any truth can be known (as some anti-realists claim). The reasoning is this. Suppose all truths are knowable but some are not in fact known. Then some truth \(A\), is not known: \(A \wedge \neg KA\). By assumption, it’s possible to know this: \(\Diamond K(A \wedge \neg KA)\). Seemingly good reasoning (in classical epistemic logic) implies that it’s possible both to know and not know that \(A\): \(\Diamond(KA \wedge \neg KA)\), which of course isn’t possible at all. The usual moral to draw is that not all truths are knowable. Beall’s idea, by contrast, is that the knowability principle can be maintained (and the corresponding kind of anti-realism along with it). His suggestion is to deny distribution of knowledge over conjunction: the principle that knowing that \(A \wedge B\) implies knowing that \(A\) and knowing that \(B\). He achieves this using impossible worlds in which conjunctions may be true even if their conjuncts are not.