The Sources of Unknowability and Self-refutation in Epistemic and Dynamic Epistemic Logic

Abstract: In this paper, we define a formula $\varphi$ to be unbelievable if $\Box\varphi$ is unsatisfiable, and unknowable if $\Box\varphi\land\varphi$ is unsatisfiable. We then analyze the sources of unknowability and unbelievability in different classes of frames K, KD, KD45, and S5. We first show that any unbelievable formula is a fixed point of the Moore function defined by $f(\varphi)=\varphi\land\lnot\Box\varphi.$ Our main result shows that in S5, the static notions of unknowability and unbelievability in epistemic logic, and the dynamic notions of always informativeness and eventual self-refutation in dynamic epistemic logic are all equivalent to ``Moorean phenomena.'' We also generalize the result to the multi-agent case, showing that although all unknowable formulas still manifest Moorean phenomena, new mechanisms arise due to their interactive nature. Finally, we briefly analyze whether the Brandenburger-Keisler paradox in epistemic game theory can be considered Moorean.