Modal logic

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