Formal System

Formal Systems are ways of thinking about first order Logic. The have four parts:

1. An alphabet: These are the symbols which are used in the formal system to make formulas or sentences.
2. A grammar: This tells you whether a sentence is something that can be inside the formal system. “I am a cow” is an English sentence, and “Cow am I a” is not (English isn’t a formal system, but grammar does the same job in it).
3. A set of axioms: This tells you some sentences that are in the formal system.
4. Inference rules: If you want to make a sentence that is in the system, start with some other sentence and apply an inference rule, and you wind up with something in the formal system. So if “replace an A with a B” is an inference rule, and “ALPHABET” is in the formal system, then so is “BLPHABET” and “ALPHBBET”.