Predicates are term-to-sentence devices, and operators are sentence-to
-sentence devices. What Kaplan and Montague's Paradox of the Knower de
monstrates is that necessity and other modalities cannot be treated as
predicates, consistent with arithmetic; they must be treated as opera
tors instead. Such is the current wisdom. A number of previous pieces
have challenged such a view by showing that a predicative treatment of
modalities need not raise the Paradox of the Knower. This paper attem
pts to challenge the current wisdom in another way as well: to show th
at mere appeal to modal operators in the sense of sentence-to-sentence
devices is insufficient to escape the Paradox of the Knower. A family
of systems is outlined in which closed formulae can encode other form
ulae and in which the diagonal lemma and Paradox of the Knower are the
reby demonstrable for operators in this sense.