If possible worlds can be inconsistent, as in Priest's impossible worlds semantics, then a world where K(p ∧ ¬Kp) holds need not be ruled out by modal logic's standard duality axioms.

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Duality axioms (in modal logic)(as the traditional constraints being potentially violated): Standard rules in logic that establish relationships between possibility and necessity—for instance, 'it's possible that X' is the opposite of 'it's necessary that not-X.'
Impossible worlds semantics(as used in logic): A framework for modal logic that allows for considering not just possible scenarios, but logically impossible ones too, to better understand how meaning works.
K(p ∧ ¬Kp)(as the symbolic notation in the statement (K = knows, p = a proposition, ∧ = and, ¬ = not)): A logical formula meaning 'someone knows that [p is true AND they don't know p]'—it describes a paradoxical situation where knowledge contains its own contradiction.
Modal logic(logic): A system of logic that deals with concepts like possibility, necessity, and what could or must be true.

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