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Intuitionistic epistemic logic. (English) Zbl 1408.03004
Summary: We outline an intuitionistic view of knowledge which maintains the original Brouwer-Heyting-Kolmogorov semantics for intuitionism and is consistent with the well-known approach that intuitionistic knowledge be regarded as the result of verification. We argue that on this view coreflection $$A\to\mathbf{K}A$$ is valid and the factivity of knowledge holds in the form $$\mathbf{K}A\to\neg\neg A$$ ‘known propositions cannot be false’.{
}We show that the traditional form of factivity $$\mathbf{K}A\to A$$ is a distinctly classical principle which, like tertium non datur $$A\vee\neg A$$, does not hold intuitionistically, but, along with the whole of classical epistemic logic, is intuitionistically valid in its double negation form $$\neg\neg(\mathbf{K}A\to A)$$.{
}Within the intuitionistic epistemic framework the knowability paradox is resolved in a constructive manner. We argue that this paradox is the result of an unwarranted classical reading of constructive principles and as such does not have the consequences for constructive foundations traditionally attributed it.

##### MSC:
 03B20 Subsystems of classical logic (including intuitionistic logic) 03B42 Logics of knowledge and belief (including belief change)
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