Definition
Modal logic is the logic of necessity and possibility. Ordinary logic deals with what is true; modal logic adds operators for what must be true and what could be true. The two are conventionally written □P for 'it is necessary that P' and ◇P for 'it is possible that P'.
These two operators are interdefinable. To say something is necessary is to say it could not have been otherwise — its denial is impossible. So □P amounts to 'not possible that not-P', and ◇P amounts to 'not necessary that not-P'. A modest extension of ordinary logic, modal logic turns out to have surprising depth.
Why it matters
How it works
The breakthrough that made modal logic tractable was possible-worlds semantics. Picture not just the actual world but a whole space of possible worlds — complete alternative ways things could be. Then □P is true when P holds in every accessible world, and ◇P is true when P holds in at least one accessible world.
Crucially, the accessibility relation between worlds can have different properties — it may be reflexive, transitive, symmetric, or not. Different choices yield different modal systems, validating different principles. This single framework also models other modalities: obligation (deontic logic), knowledge (epistemic logic), and time (tense logic) all reuse the same machinery.