This is exactly what intuitionistic logic is all about. Instead of doing operations on variables which represent the truth value of a proposition, we can do operations on variables which represent the justification for a proposition. All we have to do is forego the law of excluded middle (A or not A is true). Which makes sense because we can't say we always have justification for A or have justification to refute A. Sometimes there are questions you can ask which you can never find the answer for or constrict justification.