# Pledger's S3 extension 12s

[Pledger, 1972]

## Notes

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 => 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

[Pledger, 1972, p275]

## Basis for

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

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This page was last modified on November 21st, 2004