Interesting logic problems

Question

I've just began a course on logic and learned the following :

• De Morgan's laws

• Normal forms

• How to represent a logical formula (using or, and, not operators) using binary trees

• How to get the conjunctive normal form of a formula

It's obviously not much, but the subject interested me a lot and it was enough to study the 2-SAT and the 3-SAT problems. These two problems were very interesting to study.

I am wondering if there are other classical/interesting algorithms problem in logic (just as 2-SAT, 3-SAT) that can be studied with the little I know about logic?