The formalism of quantum theory leads to significant conceptual challenges when applied to scenarios in which multiple agents represent each other as quantum systems. The famous Wigner's friend scenario, which has led to many advances in the project of interpreting quantum theory, is one instance of this. More recently,  Frauchiger and Renner (2018) demonstrated that a naive application of the quantum formalism to more sophisticated scenarios involving four interacting agents leads to explicit contradictions. While the Frauchiger-Renner protocol has often been understood as a challenge to be solved, in this talk it is leveraged constructively to put strict constraints on the sorts of knowledge claims quantum agents are warranted in consistently making. Specifically, by adopting the formalism of epistemic logic -- a familiar tool in contemporary philosophy, but unknown to physics -- we show that certain plausible epistemic axioms must be violated in any theory of knowledge that adequately describes the knowledge obtained by quantum agents. In particular, the axioms required for deriving Fitch's knowability paradox are shown to be inadequate. The philosophical consequences of this will be discussed.