I recently started learning about this subject, which is pretty interesting. I had heard about it a while ago, but didn't know much about it besides the basics, and certainly didn't know how it could be applied to real problems since it always seemed pretty abstract. While I did learn about classical logic in college many years ago, I also recently gave myself a refresher on FOL and its axioms and inference rules so I could better understand how temporal logic departs from FOL and builds on it.

Anyway, in an attempt to help me learn the material better, I decided to write up the basic ideas along with several examples of how it could be applied. Perhaps it will be of interest to you as well, although there is certainly nothing new or groundbreaking in my exposition. I did try very hard to make things as simple and concrete as I could while not getting too "hand wavy" as they say.

Caveat: I'm certainly no expert in this field, so if you spot any mistakes, please let me know and I'll revise it (or just submit a PR on GitHub!).