Useful Axioms and Theorems in S5 and Lower Modal Logic.

K L(p ⊃ q) .⊃. (Lp ⊃ Lq)

K1 L(p & q) .⊃. (Lp & Lq)

K2 (Lp & Lq) .⊃. L(p & q)

K3 L(p & q) .≡. (Lp & Lq)

K4 (Lp v Lq) .⊃. L(p v q)

K5 Lp .≡. ~M~p

K6 M(p v q) .≡. (Mp v Mq)

K7 M(p ⊃ q) .≡. (Lp ⊃ Mq)

K8 M(p & q) .⊃. (Mp & Mq)

K9 L(p v q) .⊃. (Lp v Mq)

T Lp ⊃ p

T1 p ⊃ Mp

T2 M(p ⊃Lp)

S4 Lp ⊃ LLp

S41 MMp ⊃ Mp

S42 Lp ≡ LLp

S43 Mp ≡ MMp

S44 MLMp ⊃ Mp

S45 LMp ⊃ LMLMp

S46 LMp ≡ LMLMp

S47 MLp ≡ MLMLp

S5 Mp ⊃ LMp

S51 MLp ⊃ Lp

S52 Mp ≡ LMp

S53 Lp ≡ MLp

S54 L(p v Lq) .≡. (Lp v Lq)

S55 L(p v MMq) .≡. (Lp v Mq)

S56 M(p & q) .≡. (Mp & Mq)

S57 M(p & Lq) .≡. (Mp & Lq)

S58 p ⊃ LMp

S59 MLp ⊃ p

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