Pledger's S3 extension 12s

[Pledger, 1972]


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

[Pledger, 1972, p270-271]

Based on

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  => MLLMp,  LMMLp  => MLMp
LMMp  => MLLMp,  LMMLp  => MLLp
MLMp  => MLLMp,  LMMLp  => LMLp
LMMLp => LMLLp,  MLMMp  => MLLMp
LMMLp => MLLMp,  self dual

[Pledger, 1972, p275]

Basis for

12sa = 12s + Axiom S (MMp) [Pledger, 1972, p279]

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