# Common Modal Logics

### System K

- A simplest modal logic
- Propositional logic - PC, in our case Hilbert's
- Necessitation rule -
*X |= □ X*
- Axiom K -
*□ (X ⇒ Y) ⇒ (□ X ⇒ □ Y)*

### System S5

- Often the ending point
- Several ways to S5 from propositional logic
- Several ways to S5 from strict systems

### Strict Systems

- With a superscript
^{0} in the picture
- An alternative to PC as the starting point