Truth-functions are not material functions. If e.g. an affirmation can be produced by repeated denial, is the denial—in any sense—contained in the affirmation? Does “\~\~p” deny \~p, or does it affirm p; or both? The proposition “\~\~p” does not treat of denial as an object, but the possibility of denial is already prejudged in affirmation. And if there was an object called “\~”, then “\~\~p” would have to say something other than “p”. For the one proposition would then treat of \~, the other would not.