User:TMDrew/fitch

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 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, the knowability thesis does not solve the paradox, since the paradox can be reformulated by substituting Rule (C) which states that all truths are knowable with rule (C') which states that there is some particular truth that is unknown.

The proof can also be modified to show that one need not believe that all truths are knowable in order to derive the paradox. Replace (C) with (C'): The proof proceeds almost exactly the same way: This proof establishes the weaker claim that if a truth is knowable, then it is known