An axiomatic version of Fitch's paradox

Synthese 190 (12):2015-2020 (2013)
Abstract
A variation of Fitch’s paradox is given, where no special rules of inference are assumed, only axioms. These axioms follow from the familiar assumptions which involve rules of inference. We show (by constructing a model) that by allowing that possibly the knower doesn’t know his own soundness (while still requiring he be sound), Fitch’s paradox is avoided. Provided one is willing to admit that sound knowers may be ignorant of their own soundness, this might offer a way out of the paradox
Keywords Knowledge  Epistemology  Knowability  Paradox  Fitch’s paradox
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DOI 10.1007/s11229-011-9954-0
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References found in this work BETA
A Logical Analysis of Some Value Concepts.Frederic B. Fitch - 1963 - Journal of Symbolic Logic 28 (2):135-142.

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Discussion
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2012-02-14
notation : I use ! for 'not'

Perhaps you can avoid paradox but you have to admit this very strange proposition :
K !K x ->  !P K x
If you know that you ignore (x) it's impossible that you know (x)

I don't see how it could be compatible with the knowability principle :
x ->  P K x
else you can't have
(x) and (K !K x)

(excuse me if this message is out of place, I ignore the policy of tis forum,
excuse also my probable mistakes in english)
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