I was recently asked - where does logic come from? And as an example, he asked about the LNC. Fortunately, I knew what that was.
The LNC is the Law of Non-Contradiction. It says that something cannot be true and false at the same time. And I was immediately reminded, as one is, of Schroedinger's cat. Most people know about this so I won't explain it, follow the link if you need to. Schroedinger's cat is both alive and dead, it's in a superposition of states until the cat is observed. Alive and not alive. You can see why the LNC made me think of the cat.
Now consider geometry and the proposition that parallel lines never meet. That's true for plane (Euclidean) geometry, but it isn't true for spherical geometry. So which geometry is "right"? Both of them; you use the geometry that's most useful in solving the problem that you're facing. So if you want to lay out your garden, you'd use plane geometry, because it's simpler than spherical. But if you want to determine the best route from London to Chicago, you'll use spherical geometry.
I think it's the same with logic. There could be hundreds of different logics; some include the LNC and some don't. We use the generally accepted laws of logic because they lead to the ability to solve the most common problems. For example, in our world, a cat cannot be both alive and not alive. Our ordinary experience means that the LNC is more useful than a logical system without it.
But perhaps when we think about Quantum Mechanics, it would be more useful to adopt a set of logical laws that do not include the LNC. And then the cat can be both alive and not alive, and there's no paradox.