Maybe this is a dumb question but, wouldn't it be possible to just
"program" yourself some kind of paradox into a contract, for example, some
variant of the Paradox of the Court
https://en.wikipedia.org/wiki/Paradox_of_the_Court where I have to pay
someone or not, then request in a CFJ to know if I have to pay them or not?
Then, have that CFJ gain a verdict of "Paradox" (and not because of the
case itself, but because of the contract you've engineered to make the CFJ
read from it that value of "Paradox", to avoid "PARADOXICAL is not
appropriate if (...) the undecidability arises from the case itself or in
reference to it.")
Then claim a win via the Paradox rule.