# Inference Rules of Modal Logic

I'm currently reading the book "An Introduction to Non-Classical Logic." Currently, I'm being introduced to modal logic for the first time.

This book seems to prefer to present the reader with the idea that a tableau is the primary way to prove a statement. For example, take this example of a tableau rule from the chapter on propositional logic.

_A → _B

_↓ ____↓

~A __ ~B

As somebody who has been studying classical logic for a while, I looked at this and thought "Conditional Exchange." However, the concept of inference rules or the idea of (in my mind) a "more formal" proof using rules such as Modus Ponens, Modus Tollens, Hypothetical Syllogism, etc. never came up. I figured this was because the book assumed that the reader had prior knowledge of these rules, but it doesn't appear that any formal inference rules will be introduced for modal logic.

This leads me to the question, what are the inference rules (or whatever may be analogous) of modal logic?

• Modal logic can be axiomatised. Then only modus ponens is used as a rule. There are different systems with different axioms, but the common ones are ◻(A→B)→(◻A→◻B) and ◻A→A. Feb 1 '20 at 4:52
• Classical inference rules are inherited. The additional modal axioms are listed in SEP, Modal Logics. Feb 1 '20 at 6:35