More than true and false

When I find a contradiction in an argument, I feel the rush of being able to deliver a knock out. But, maybe that's overly simplistic. In Buddhist philosophy, statements can be more than just true or false. They can be:

1. Always true
2. Always false
3. True and false
4. Neither true nor false

The programmer in me has a visceral reaction against 3 and 4 because they literally don't compute. But, I'm familiar with paradoxes and the uncertainty principle. So, intuitively I know that my logic system is insufficiently descriptive. For example:

This statement is false.

If the statement is true, then it's actually false. But, if it's false, then it's actually true. So, while I understand that statements can be both true and false at the same time, I'm lost when asked how to think about them.