Improbable knowing

In T. Dougherty (ed.), Evidentialism and its Discontents. Oxford University Press (2011)
Can we turn the screw on counter-examples to the KK principle (that if one knows that P, one knows that one knows that P)? The idea is to construct cases in which one knows that P, but the epistemic status for one of the proposition that one knows that P is much worse than just one’s not knowing it. Of course, since knowledge is factive, there can’t be cases in which one knows that P and knows that one doesn’t know that P (we can’t strengthen ¬KKp to K¬Kp)! If we can construct such cases, we may be able to use them to understand some puzzling epistemic phenomena.
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