This system has 12 distinct proper affirmative modalities (plus the improper modality p) and is contained in K2 but not in S5 [Pledger, 1972, p270-271]
The distinct affirmative modalities of the syatem are:
p implied by Lp it implies Mp Lp implied by LLp it implies p, LMLp Mp it implies MMp implied by p, MLMp LLp it implies Lp, LLMp MMp implied by Mp, MMLp LMp implied by LLMp, LMLp it implies LMMp, MLMp MLp it implies MLMp, MMLp implied by MLLp, LMLp LLMp implied by LLp it implies LMp, LMLp, MLLp MMLp it implies MMp implied by MLp, LMMp, LMMp LMLp implied by Lp, LLMp it implies LMp, MLp, LMMp MLMp it implies Mp, MMLp implied by MLp, LMp, MLLp LMMp implied by LMp, LMLp it implies MMLp MLLp it implies MLp, MLMp implied by LLMp
The system 12q is: the system S3 plus any one of the following: (Arranged as axiom, dual of axiom)
LMLLMp => LLMLp, MMLMp => MLMMLp
12qb = 12q +
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