This note is a part of my Zettelkasten. What is below might not be complete or accurate. It is also likely to change often.
A formal system consists of just two things:
Using these two, we can create theorems. Theorems are states in the formal system that can be reached from axioms using rules.
An example format system could be:
Using these two things, we can make theorems like MII, MIU, etc. But x and y are not part of the system. As something outside the system, we can assign meaning to them, but inside the system they don't have any meaning.