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 superscript⁰ in the picture
• An alternative to PC as the starting point