This system has 12 distinct proper affirmative modalities (plus the improper modality p) and is contained in both K2 and 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 LMLp, LMLLp it implies LMMp, MLMp MLp it implies MLMp, MLMMp implied by LMLp, MLLp LLMp implied by LLp it implies LMLLp MMLp it implies MMp implied by MLMMp LMLp implied by Lp, LMLLp it implies LMp, MLp, LMMp MLMp it implies Mp, MLMMp implied by LMp, MLp, MLLp LMMp implied by LMp, LMLp it implies MLMMp MLLp it implies MLp, MLMp implied by LMLLp LMLLp implied by LLMp it implies LMp, LMLp, MLLp MLMMp it implies MMLp implied by MLp, LMMp, MLMp
The system 12s is: the system S3 plus any one of the following: (The table is arranged as an axiom, followed by the dual of axiom)
Lp => MLLp, LMMp => Mp Lp => MLLMp, LMMLp => Mp Lp => LMLLp, MLMMp => Mp Lp => LMLLMp, MLMMLp => Mp LMp => LMLLMp, MLMMLp => MLp LMp => MLLMp, LMMLp => MLp MLp => MLLp, LMMp => LMp MLp => MLLMp, LMMLp => LMp LMLp => MLLp, LMMp => MLMp LMLp => LMLLp, MLMMp => MLMp LMLp => LMLLMp, MLMMLp => MLMp LMLp => MLLMp, LMMLp => MLMp LMMp => LMLLMp, MLMMLp => MLLp LMMp => MLLMp, LMMLp => MLLp MLMp => MLLMp, LMMLp => LMLp LMMLp => LMLLp, MLMMp => MLLMp LMMLp => LMLLMp, MLMMLp => MLLMp LMMLp => MLLMp, self dual
12sa = 12s + Axiom S (MMp) [Pledger, 1972, p279]
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This page was last modified on November 21st, 2004