This system has 10 distinct proper affirmative modalities (plus the improper modality p) and is contained in neither S5 nor K2 [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, LMp, MLLp, LMLp MMp implied by Mp, MLp, LMMp, MLMp LMp implied by LLp, LMLp it implies LMMp, MLMp MLp it implies MMp, MLMp implied by MLLp, LMLp LMLp implied by Lp, LLp it implies LMp, MLp, LMMp MLMp it implies Mp, MMp implied by LMp, LMp, MLLp LMMp implied by LMp, LMLp it implies MMp MLLp it implies MLp, MLMp implied by LLp
The system 10p is: the system S3 plus any one of the following (Arranged as axiom, dual of axiom):
LMLLMp => p p => MLMMLp LMLLMp => Lp Mp => MLMMLp LMLLMp => LLp MMp => MLMMLp
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