Operators in the paradox of the knower
The Paradox of the Knower can be developed in a formal arithmetic equipped with the usual introspective machinery for the Gödel diagonalization theorem to obtain. This theorem says that with every well-formed formula \(Wx\) in the system, there is a sentence \(S\) such that \(S\equiv W\ulcorner S\urcorner\) (\(\ulcorner S\urcorner\) is the Gödel number of \(S\)). Now, if this system contains a predicate symbol, \(K\), with the premises: \(K\ulcorner A\urcorner\to A\), \(K\ulcorner(K\ulcorner A\urcorner\to A)\urcorner\), \(I(\ulcorner A\urcorner, \ulcorner B\urcorner)\to (K\ulcorner A\urcorner\to K\ulcorner B\urcorner)\), where \(I\) is the deducibility relation for the system, then it is inconsistent. This syntactic property becomes a paradox, ``the Paradox of the Knower'', if we believe that \(K\ulcorner A\urcorner\) can be interpreted as ``\(A\) is known''. In trying to resolve the paradox, some authors have suggested that the knowledge predicate \(K\ulcorner A\urcorner\) rather ought to be dealt with as an operator, or sentential connective. That is, not taking a term, like a Gödel number, as argument but rather a sentence. The author argues against such views. In particular, he deals with systems where all formulae, \(W\), are recoverably encoded, not by Gödel numbers but by closed formulae, ``counter-formulas'', \([W]\). He finds that the diagonalization theorem still holds, and that the corresponding premises are inconsistent whereby also the interpretation of \(K[A]\) as ``\(A\) is known'' becomes paradoxical. However, if \(K\) operates directly on non-encoded formulas, a corresponding diagonalization theorem need not hold, and \(KA\to A\), etc., need not give an inconsistency. The author concludes that the diagonalization theorem and the Paradox of the Knower are not merely a question of whether a sentence-generating function takes terms or sentences as arguments. Rather, it is the ``effective expressive power'', of the system, that matters.
scientific article; zbMATH DE number 490766