□
◇
□

modal logic

modal logic extends truth into possibility and necessity. instead of asking “is this true?”, we ask: what must be true? what could be true?
□P — true in all accessible worlds ◇P — true in at least one accessible world
duality (click)
◇P ≡ ¬□¬P
□P ≡ ¬◇¬P

possibility is just “not impossible”

kripke semantics

a model is a triple:

W — set of worlds R — accessibility relation V — valuation
truth definition (click)
w ⊨ □P iff ∀v (wRv → v ⊨ P) w ⊨ ◇P iff ∃v (wRv ∧ v ⊨ P)

frame properties

reflexive → T transitive → S4 euclidean → S5
why S5 is strong (click)
every world sees every other world. possibility becomes globally stable.

proof (animated)

1. □P ∧ □Q
2. □P
3. □Q
4. P ∧ Q holds in all worlds
5. □(P ∧ Q)