Here’s a quick thought. Logicians tell us that different systems of Logic exist which are incommensurable, yet consistent within themselves. Thus, Fuzzy logic, which makes propositions not strictly true or false, but true to some degree, (between 1 and 10) is an example of a consistent logic. Similarly the Mathematical intuitionists tell us that their logic, which rejects the law of excluded middle, is consistent. Further, take the true value of a conditional of the form “if P then Q” where P is false, modern logic counts the whole conditional as true, whereas the ancient Stoics would have called the whole conditional false. Both seem consistent.
A Logical empiricist may think it a matter of sheer linguistic arbitration whether we use one logic or another. So long as the language is consistent it is legitimate.
However, what exactly is consistency other than an application of the principle of non-contradiction? If that is all it is, then the principle of non-contradiction is an indispensable principle which is the litmus test for the legitimacy of any and all systems of Logic (not just those intended to supplement Modal Logic, which is the application of Logic to possibility, necessity and impossiblility).
Perhaps this could be weakened to a principle that not all propositions are true and false.