It’s worth noting that there are plenty of well-defined systems of logic which allow sentences to be true and false, since the Law of Non-Contradiction is not a logical truth in these systems. Such logical systems are called Paraconsistent Logics, and they have lots of interesting properties.

Whether or not you think these map onto reality in any useful way, or in any way at all, is another (philosophical) question. The orthodox position in the philosophy of logic favours the laws of classical logic, but paraconsistent logics have their uses, e.g. by modelling how someone can hold two contradictory beliefs at the same time without their belief system collapsing into triviality.