Editing Fitch's paradox of knowability (section)

== The knowability thesis ==
Rule (C) is generally held to be at fault rather than any of the other logical principles employed. It may be contended that this rule does not faithfully translate the idea that all truths are knowable, and that rule (C) should not apply unrestrictedly. Kvanvig contends that this represents an illicit substitution into a modal context.

[[Gödel's incompleteness theorems|Gödel's Theorem]] proves that in any recursively axiomatized system sufficient to derive mathematics (e.g. Peano Arithmetic), there are statements which are undecidable. In that context, it is difficult to state that "all truths are knowable" since some potential truths are uncertain.

However, jettisoning the knowability thesis does not necessarily solve the paradox, since one can substitute a weaker version of the knowability thesis called (C').
{|
| (C')|| ∃''x''(((''x'' & ¬'''K'''''x'') & '''LK'''''x'') & '''LK'''((''x'' & ¬'''K'''''x'') & '''LK'''''x'')) || – There is an unknown, but knowable truth, and it is knowable that it is an unknown, but knowable truth.
|}
The same argument shows that (C') results in contradiction, indicating that any knowable truth is either known, or it is unknowable that it is an unknown yet knowable truth; conversely, it states that if a truth is unknown, then it is unknowable, or it is unknowable that it is knowable yet unknown.