From Intuitionistic Epistemic Logic to a constructive resolution of the Knowability Paradox (joint work with S. Artemov)
CUNY Graduate Center
We outline an intuitionistic view of knowledge which maintains the Brouwer-Heyting-Kolmogorov semantics and is consistent with Williamson’s suggestion that intuitionistic knowledge be regarded as the result of verification. We argue that on this view A -> KA is valid and KA -> A is not. We show the latter is a distinctly classical principle, too strong as the intuitionistic truth condition for knowledge which can be more adequately expressed by other modal means, e.g. ~(KA & ~A) “false is not known.” We construct a system of intuitionistic epistemic logic, IEL, codifying this view of knowledge and prove the standard meta-theorems for it. From this it follows that previous outlines of intuitionistic knowledge, responding to the knowability paradox, are insufficiently intuitionistic; by endorsing KA -> A they implicitly adopt a classical view of knowledge, by rejecting A -> KA they reject the constructivity of truth. Within the framework of IEL, the knowability paradox is resolved in a constructive manner which, as we hope, reflects its intrinsic meaning.