> I don't have any problem imagining systems with true, false, both and neither. You just treat them as anonymous values with particular operations, instead of getting stuck on "but if we were in this other system that we're not in, this would violate an axiom!".
It's not quite that simple. We've built up entire codes of ethics and law and branches of mathematics with practical applications like computing that rely utterly on these axioms. These are things used in the real world, they don't just live in the realm of ideas. These systems don't tolerate more than two values of truth. Even "Western" religion, from Judaism to Christianity to Islam is intimately concerned with the external world where things are concrete and ambiguity is not to be tolerated. Big-C Catholicism itself considers the Church to be the only institution capable of determining Truth, what it calls dogma. We're very much handcuffed to using two values of truth here in this side of the world.
Buddhism is heavily oriented on the experience of the individual, individuals are free to use whatever sort of logic they want to make decisions. So it can experiment with ideas that are neither true nor false. But you can't build a computer with it, at some level it all is resolved on individual components which are either on or off. We don't even know yet if quantum computers which break the binary hegemony offer any real advantage.
But being able to integrate other kinds of logic might make entirely new approaches to, say, law, possible. It's not enough to be able to imagine an alternative system, it's in making it consistent. I personally am very excited about this.