My understanding was that you just had to get rid of the law of the excluded middle. Then things like Russell's Paradox can be stated, but evaluate to neither true nor false, and be fine. Is that not correct?
It's more to do with unrestricted circular self-reference than excluded middle but what you're saying makes sense. Without excluded middle there is no contradiction. There is something called Curry's paradox which is similar. Excluded middle is not involved in Curry's paradox but circular self-refernece is still the culprit for the paradoxical conclusion.
