Do you apply formal logic to other subjects? If so, can you give an example?

Logic, as a key exercise, is certainly applied in other subjects. For example, when I taught Doctrine, I would discuss classical arguments for and against the existence of God. One such argument is the Problem of Evil and goes something like this: “If God exists, then he is both perfectly good and infinitely powerful. If he is perfectly good, then he is willing to prevent evil. If he is infinitely powerful, then he is able to prevent evil. But if evil exists, then God is either unwilling or unable to prevent it. Evil does exist. Therefore God does not exist.” We use a truth table or write a proof to show that this argument is valid. The students who understand logic know that if an argument is valid, but the conclusion is false (as in this case), then at least one of the premises must be false. This leads to a fruitful discussion about which premise is false, and why.