The standard way to run the liar paradox is with truth. But the family of liar paradoxes is wide, and I am trying to find a way of characterizing it. To that end, I am thinking about as many variants as I can. Here is one with desire satisfaction (it's not original to me): Annabelle believes that to have all of one's desires satisfied makes one soft. So, she has a desire to have an unsatisfied desire. One day, as it happens, all of Annabelle's other desires are satisfied. Is this one, too?
We can do this with action fulfillment, too. On Thursday, Patrick tells Annabelle that he will do whatever she asks of him on Friday. On Friday, she asks him to refrain from doing everything that he on Thursday said he would do. Patrick made no other statements on Thursday about what he will do.
These cases suggest that one isn't going to get a solution to liar paradoxes simply by restricting language to remove "truth" from it. For there are just too many concepts that can play the same role as truth does.
By the way, there is a joke I once heard about the sadist and the masochist together in hell. The masochist says: "Hurt me." The sadist says: "No."