Modal logic extends classical logic by adding operators for necessity (□) and possibility (◇). This powerful framework allows us to reason about not just what is true, but what must be true and what could be true.

The basic axioms of modal logic (in the system K) are:

  1. □(p → q) → (□p → □q) [Distribution Axiom]
  2. If p is a theorem, then □p is a theorem [Necessitation Rule]

From these simple beginnings, we can develop rich systems for reasoning about:

  • Knowledge and belief (epistemic logic)
  • Time (temporal logic)
  • Obligation and permission (deontic logic)
  • Programs and computation (dynamic logic)

Modal logic has applications ranging from philosophy to artificial intelligence, showing how mathematical abstractions can help us understand fundamental concepts of reasoning.